EP2164713B1 - Security element having a magnified, three-dimensional moiré image - Google Patents

Description

The invention relates to a security element for security papers or documents of value with a micro-optical moiré magnification arrangement for displaying a three-dimensional moiré image.

Data carriers, such as valuables or identity documents, but also other valuables, such as branded goods, are often provided with security elements for the purpose of security, which permit verification of the authenticity of the data carrier and at the same time serve as protection against unauthorized reproduction. The security elements can be embodied, for example, in the form of a security thread embedded in a banknote, a covering film for a banknote with a hole, an applied security strip or a self-supporting transfer element which is applied to a value document after its manufacture.

Security elements with optically variable elements, which give the viewer a different image impression under different viewing angles, play a special role, since they can not be reproduced even with high-quality color copying machines. For this purpose, the security elements can be equipped with security features in the form of diffraction-optically effective microstructures or nanostructures, such as with conventional embossed holograms or other hologram-like diffraction structures, as described, for example, in the publications EP 0 330 733 A1 or EP 0 064 067 A1 are described.

It is also known to use lens systems as security features. For example, in the document EP 0 238 043 A2 a security thread of a transparent material described on the surface of a grid of several parallel cylindrical lenses is imprinted. The Thickness of the security thread is chosen so that it corresponds approximately to the focal length of the cylindrical lenses. On the opposite surface of a printed image is applied register accurate, the print image is designed taking into account the optical properties of the cylindrical lenses. Due to the focusing effect of the cylindrical lenses and the position of the printed image in the focal plane different subregions of the printed image are visible depending on the viewing angle. By appropriate design of the printed image so that information can be introduced, which are visible only at certain angles. By appropriate design of the printed image while "moving" images can be generated. However, as the document rotates about an axis parallel to the cylindrical lenses, the subject moves only approximately continuously from one location on the security thread to another location.

From the publication US 5 712 731 A For example, the use of a moiré magnification arrangement is known as a security feature. The security device described therein has a regular array of substantially identical printed microimages of up to 250 μm in size and a regular two-dimensional array of substantially identical spherical microlenses. The microlens array has substantially the same pitch as the microimage array. When the micro-image array is viewed through the microlens array, one or more enlarged versions of the microimages are created to the viewer in the areas where the two arrays are substantially in register.

The basic mode of operation of such moiré magnification arrangements is described in the article " The moire magnifier ", MC Hutley, R. Hunt, RF Stevens and P. Savander, Pure Appl. Opt. 3 (1994), pp. 133-142 , described.

In short, moiré magnification thereafter refers to a phenomenon that occurs when viewing a raster of identical image objects through a lenticular of approximately the same pitch. As with any pair of similar rasters, this results in a moire pattern, which in this case appears as an enlarged and possibly rotated image of the repeated elements of the image raster.

From the pamphlets WO 2007/007793 A respectively. EP 1 905 613 A is a stereoscopic surface structure known whose motif image consists of planar motif elements. The three-dimensional effect of such a surface structure is that different motif elements seem to float above or below the image plane at different heights or depths, so that the totality of the motif elements appears arranged on a three-dimensionally curved surface.

The publication WO 2006/125224 A relates to a micro-optical imaging system, which is formed in an exemplary embodiment as a polymer film. To produce three-dimensional synthetic images, a plurality of image projectors are combined, each containing a lens, an optical spacer and a symbol image. Each of the symbol images is calculated taking into account the three-dimensional digital model of the synthetic image, the desired depth position and the depth range to be displayed in the synthetic image, the lens repeat period, the lens field of view, and the graphic resolution of the symbol images.

Based on this, the object of the invention is to avoid the disadvantages of the prior art and to provide a security element with a micro-optical moiré magnification arrangement specify three-dimensional moiré images with impressive visual effects. The three-dimensional moiré images should be able to be viewed as far as possible without limiting the field of view and should be able to be modeled in all design variants using a computer.

This object is achieved by the security element having the features of the main claim. A method for producing such a security element, a security paper and a data carrier with such a security element are specified in the independent claims. Further developments of the invention are the subject of the dependent claims.

• einem Motivbild, das zwei oder mehr periodische oder zumindest lokal periodische Gitterzellen-Anordnungen mit unterschiedlichen Gitterperioden und/ oder unterschiedlichen Gitterorientierungen enthält, die jeweils einer Moiré-Bildebene zugeordnet sind und die Mikromotiv-Bildbestandteile zur Darstellung des Bildbestandteils der zugeordneten Moire-Bildebene enthalten,
• einem zum Motivbild beabstandet angeordneten Fokussierelementraster zur Moiré-vergrößerten Betrachtung des Motivbilds, das eine periodische oder zumindest lokal periodische Anordnung einer Mehrzahl von Gitterzellen mit jeweils einem Mikrofokussierelement enthält,

wobei sich das vergrößerte, dreidimensionale Moiré-Bild beim Kippen des Sicherheitselements für fast alle Kipprichtungen in eine sich von der Kipprichtung unterscheidende Moire-Bewegungsrichtung bewegt.According to the invention, a generic security element includes a micro-optic moiré magnification arrangement for displaying a three-dimensional moiré image having a deep-space structure containing image components to be displayed in at least two moire image planes spaced in a direction perpendicular to the moiré magnification device , With

• a motif image, the two or more periodic or at least locally periodic grid cell arrangements with different Includes grating periods and / or different grating orientations each associated with a moire image plane and containing the micromotif image constituents for representing the image constituent of the associated moire image plane;
• a focusing element grid arranged at a distance from the motif image for moiré-enlarged viewing of the motif image, which contains a periodic or at least locally periodic arrangement of a plurality of grid cells, each with a microfocusing element,

wherein the magnified, three-dimensional moiré image moves when tilting the security element for almost all directions of tilting in a different from the direction of tilting Moire movement direction.

As will be explained in more detail below, in such designs, the visual spatial impression and the spatial experience due to the tilting movement are not in harmony with each other or even contradict each other, resulting in striking, sometimes dizzying effects with high attention and recognition value for the viewer.

The image components of the three-dimensional moiré image to be displayed can be formed by individual pixels, a group of pixels, lines or patches. As explained in more detail below, it is generally advantageous, in particular for complex moiré images, to start from individual pixels of the three-dimensional moiré image as image components to be displayed and for each of these moiré pixels an associated micromotiv pixel and a grid cell arrangement for repeated arrangement of the micromotiv -Picture point in the motive level To determine. However, for simpler moiré images in which easy-to-describe lines or even patches lie in a moire image plane, such as embodiments 1-4 described below, those lines or patches may also be selected as image components to be displayed and the associated associated Micromotif image components and their repeated arrangement in the motif plane for the line or the surface piece to be performed as a whole.

miteinander verknüpft, v =

k, so gelten für die beiden in diesem Fall existierenden Eigenvektoren der Transformationsmatrix k ₁ , k ₂ die Beziehungen v ₁ = m ₁ · k ₁ bzw. v ₂ = m₂ · k ₂, mit den Eigenwerten der Transformationsmatrix m ₁ und m ₂. Bei einer Verkippung in Richtung eines der beiden Eigenvektoren sind Bewegungsrichtung und Kipprichtung daher parallel, während sie sich für alle anderen Kipprichtungen unterscheiden.The fact that the moiré image moves in tilting direction of the security element for almost all tilt directions in a moire movement direction different from the tilting direction takes account of the fact that there may be some excellent directions in which tilt direction and moire direction of movement coincide. For symmetry reasons, there are usually just two such directions: namely, are the moire direction of motion v and the tilt direction k in the plane of the moire magnification arrangement over a symmetric transformation matrix

linked together, v =

k , then for the two eigenvectors existing in this case, the transformation matrix applies k ₁ , k ₂ the relationships v ₁ = m ₁ · k ₁ or v ₂ = m ₂ k ₂ , with the eigenvalues of the transformation matrix m ₁ and m ₂ . With a tilt in the direction of one of the two eigenvectors, the direction of movement and tilting direction are therefore parallel, while they differ for all other tilting directions.

With particular advantage, the three-dimensional moiré image by the parallax when tilting the security element for the viewer in a first height or depth above or below the plane of the security element floating, and appears due to the eye distance in binocular vision in a second height or Depth above or below the level of the security element floating, wherein the first and second height or depth for almost all viewing directions differ.

The specification of a viewing direction includes not only the viewing direction but also the direction of the eye distance of the viewer. Again, the phrase that the first and second heights or depths differ for almost all viewing directions expresses that there may be certain excellent viewing directions in which the first and second heights and depths coincide. In particular, these excellent viewing directions may be just the directions in which the tilt direction and Moire direction of motion coincide.

In an advantageous variant of the invention, both the grid cell arrangements of the motif image and the grid cells of the focusing element grid are arranged periodically. The periodicity length is preferably between 3 μm and 50 μm, preferably between 5 μm and 30 μm, particularly preferably between about 10 μm and about 20 μm.

According to another variant of the invention, both the grid cell arrangements of the motif image and the grid cells of the focusing element grid are arranged locally periodically, the local period parameters changing only slowly in relation to the periodicity length. For example, the local period parameters may be periodically modulated over the extent of the security element, wherein the modulation period is preferably at least 20 times, preferably at least 50 times, more preferably at least 100 times greater than the local periodicity length. Also in this variant, the local periodicity length is preferably between 3 μm and 50 μm, preferably between 5 μm and 30 μm, particularly preferably between about 10 μm and about 20 μm.

The grid cell arrangements of the motif image and the grid cells of the focusing element grid advantageously form, at least locally, respectively a two-dimensional Bravais grid, preferably a Bravais grid with low symmetry, such as a parallelogram grid. The use of low symmetry Bravais gratings offers the advantage that moiré magnification arrangements are difficult to imitate with such Bravais gratings, since, in order to form a correct image, the low symmetry of the arrangement, which is difficult to analyze, must be precisely adjusted , In addition, the low symmetry provides a large amount of freedom for differently selected lattice parameters, which can thus be used as a hidden marking for products secured according to the invention, without this being readily apparent to a viewer on the moire-magnified image. On the other hand, any attractive effects achievable with higher-symmetry moiré magnification arrangements can also be realized with the preferred low-symmetry moiré magnification arrangements.

The microfocusing elements are preferably formed by non-cylindrical microlenses, in particular by microlenses with a circular or polygonal limited base surface. In other configurations, the microfocusing elements can also be formed by elongated cylindrical lenses, whose longitudinal extension is more than 250 μm, preferably more than 300 μm, particularly preferably more than 500 μm and in particular more than 1 mm. In further preferred embodiments, the microfocusing elements are pinhole apertures, slotted apertures, apertured apertured or slotted apertures, aspheric lenses, Fresnel lenses, GRIN (Gradient Refraction Index) lenses, zone plates, holographic lenses, concave mirrors, Fresnel mirrors, zone mirrors, or other focusing or focusing elements also formed with a masking effect.

The total thickness of the security element is advantageously below 50 μm, preferably below 30 μm. The moire image to be displayed preferably contains a three-dimensional representation of an alphanumeric string or a logo. According to the invention, the micromotif image constituents can be present in particular in a printing layer.

• einem Motivbild, das zwei oder mehr, in unterschiedlicher Höhe angeordnete, periodische oder zumindest lokal periodische Gitterzellen-Anordnungen enthält, die jeweils einer Moiré-Bildebene zugeordnet sind und die Mikromotiv-Bildbestandteile zur Darstellung des Bildbestandteils der zugeordneten Moire-Bildebene enthalten,
• einem zum Motivbild beabstandet angeordneten Fokussierelementraster zur Moire-vergrößerten Betrachtung des Motivbilds, das eine periodische oder zumindest lokal periodische Anordnung einer Mehrzahl von Gitterzellen mit jeweils einem Mikrofokussierelement enthält,

wobei sich das vergrößerte, dreidimensionale Moire-Bild beim Kippen des Sicherheitselements für fast alle Kipprichtungen in eine sich von der Kipprichtung unterscheidende Moiré-Bewegungsrichtung bewegt.In a second aspect, the invention includes a generic security element having a micro-optic moiré magnification arrangement for displaying a three-dimensional moiré image having a structure extending in the depth of the space, the image constituents to be displayed spaced apart in at least two, in a direction perpendicular to the moiré magnification arrangement Moiré image layers contains, with

• a motif image which contains two or more, arranged at different heights, periodic or at least locally periodic grid cell arrangements, each associated with a moire image plane and containing the micromotif image constituents for representing the image constituent of the associated moire image plane,
• a focusing element grid arranged at a distance from the motif image for moire-magnified viewing of the motif image, which contains a periodic or at least locally periodic arrangement of a plurality of grid cells, each with a microfocusing element,

wherein the magnified, three-dimensional Moire image moves when tilting the security element for almost all tilt directions in a different direction of the tilting moire movement direction.

In this aspect of the invention, the grid cell arrangements of the motif image preferably have the same lattice periods and the same lattice orientations, so that different moiré magnifications only result from the different height of the micromotif image components and thus a different distance of the micromotif image components and the focusing element grid. With particular advantage, the micromotif image components are available in a stamping layer in different embossing heights.

In both aspects, the security element according to the invention advantageously has an opaque cover layer for covering the moiré magnification arrangement in regions. Thus, no moire magnification effect occurs within the covered area, so that the optically variable effect can be combined with conventional information or with other effects. This cover layer is advantageously in the form of patterns, characters or codes before and / or has recesses in the form of patterns, characters or codes.

In all of the aforementioned variants of the invention, the motif image and the focusing element grid are preferably arranged on opposite surfaces of an optical spacer layer. The spacer layer may comprise, for example, a plastic film and / or a lacquer layer.

The arrangement of microfocusing elements can moreover be provided with a protective layer whose refractive index preferably deviates by at least 0.3 from the refractive index of the microfocusing elements, if refractive lenses serve as microfocusing elements. In this case, changes the focal length of the lenses through the protective layer, which in the dimensioning of the lens radii of curvature and / or Thickness of the spacer layer must be considered. In addition to protection against environmental influences, such a protective layer also prevents the microfocusing element arrangement from easily being molded for counterfeiting purposes.

The security element itself in both aspects of the invention preferably represents a security thread, a tear thread, a security tape, a security strip, a patch or a label for application to a security paper or document of value. In an advantageous embodiment, the security element can span a transparent or recessed area of a data medium. Different appearances can be realized on different sides of the data carrier.

• in einer Motivebene ein Motivbild erzeugt wird, das zwei oder mehr periodische oder zumindest lokal periodische Gitterzellen-Anordnungen mit unterschiedlichen Gitterperioden und/oder unterschiedlichen Gitterorientierungen enthält, die jeweils einer Moiré-Bildebene zugeordnet sind und die mit Mikromotiv-Bildbestandteilen zur Darstellung des Bildbestandteils der zugeordneten Moire-Bildebene versehen werden,
• ein Fokussierelementraster zur Moire-vergrößerten Betrachtung des Motivbilds mit einer periodischen oder zumindest lokal periodischen Anordnung einer Mehrzahl von Gitterzellen mit jeweils einem Mikrofokussierelement erzeugt und zum Motivbild beabstandet angeordnet wird,

wobei die Gitterzellen-Anordnungen der Motivebene, die Mikromotiv-Bildbestandteile und das Fokussierelementraster so aufeinander abgestimmt werden, dass sich das vergrößerte, dreidimensionale Moiré-Bild beim Kippen des Sicherheitselements für fast alle Kipprichtungen in eine sich von der Kipprichtung unterscheidende Moire-Bewegungsrichtung bewegt.The invention also includes a method for producing a security element having a micro-moire magnification optical arrangement for displaying a three-dimensional moiré image having a structure extending in the depth of the space, the image components to be displayed in at least two, spaced in a direction perpendicular to the moiré magnification arrangement Moire image planes contains, in which

• a motif image is generated in a motif plane, which contains two or more periodic or at least locally periodic lattice cell arrangements with different lattice periods and / or different lattice orientations, each associated with a moire image plane and those with micromotif image components for representing the image constituent of the associated Moire image plane,
• a focusing element grid is produced for moire-magnified viewing of the motif image with a periodic or at least locally periodic arrangement of a plurality of grid cells each having a microfocusing element and being arranged at a distance from the motif image,

wherein the grid level arrangements of the motif plane, the micromotif image constituents and the focusing element grid are coordinated so that the enlarged three-dimensional moiré image moves when tilting the security element for almost all tilt directions in a direction of tilting different Moire movement direction.

The image components of the three-dimensional moiré image to be displayed can be formed by individual pixels, a group of pixels, lines or patches, with the use of individual pixels being presented as image components to be displayed, especially in the case of more complex moiré images.

• ein Motivbild mit zwei oder mehr in unterschiedlicher Höhe angeordneten Motivebenen erzeugt wird, die jeweils eine periodische oder zumindest lokal periodische Gitterzellen-Anordnung enthalten, die einer Moiré-Bildebene zugeordnet ist und die mit Mikromotiv-Bildbestandteilen zur Darstellung des Bildbestandteils der zugeordneten Moiré-Bildebene versehen wird,
• ein Fokussierelementraster zur Moiré-vergrößerten Betrachtung des Motivbilds mit einer periodischen oder zumindest lokal periodischen Anordnung einer Mehrzahl von Gitterzellen mit jeweils einem Mikrofokussierelement erzeugt und zum Motivbild beabstandet angeordnet wird,

wobei die Gitterzellen-Anordnungen der Motivebenen, die Mikromotiv-Bildbestandteile und das Fokussierelementraster so aufeinander abgestimmt werden, dass sich das vergrößerte, dreidimensionale Moiré-Bild beim Kippen des Sicherheitselements für fast alle Kipprichtungen in eine sich von der Kipprichtung unterscheidende Moire-Bewegungsrichtung bewegt.According to another inventive method for producing a security element with a micro-optical moire magnification arrangement for displaying a three-dimensional moiré image with a structure extending into the depth of the space, the image components to be displayed in at least two moiré spaced apart in a direction perpendicular to the moiré magnification arrangement - contains image layers, it is provided that

• a motif image is generated with two or more arranged at different heights motif levels, each containing a periodic or at least locally periodic grid cell array, which is associated with a moire image plane and those with micromotive image components is provided to represent the image component of the associated moire image plane,
• a focusing element grid for moire-magnified viewing of the motif image is generated with a periodic or at least locally periodic arrangement of a plurality of grid cells, each with a microfocusing element and spaced from the motif image,

wherein the grid level arrangements of the motif planes, the micromotif image constituents and the focusing element grid are coordinated so that the magnified three-dimensional moiré image moves in tilting the security element for almost all tilt directions in a moire movement direction different from the tilt direction.

1. a) ein gewünschtes, bei Betrachtung zu sehendes dreidimensionales Moiré-Bild als Sollmotiv festgelegt wird,
2. b) eine periodische oder zumindest lokal periodische Anordnung von Mikrofokussierelementen als Fokussierelementraster festgelegt wird,
3. c) eine gewünschte Vergrößerung und eine gewünschte Bewegung des zu sehenden dreidimensionalen Moiré-Bilds beim seitlichen Kippen und beim vor-/rückwärtigen Kippen der Moiré-Vergrößerungsanordnung festgelegt wird,
4. d) für jeden darzustellenden Bildbestandteil aus dem Abstand der zugehörigen Moire-Bildebene von der Moiré-Vergrößerungsanordnung, dem festgelegten Vergrößerungs- und Bewegungsverhalten und dem Fokussierelementraster der zugehörige Mikromotiv-Bildbestandteil zur Darstellung dieses Bildbestandteils des dreidimensionalen Moire-Bilds, sowie die zugehörige Gitterzellen-Anordnung für die Anordnung der Mikromotiv-Bildbestandteile in der Motivebene berechnet werden, und
5. e) die für jeden darzustellenden Bildbestandteil berechneten Mikromotiv-Bildbestandteile entsprechend der zugehörigen Gitterzellen-Anordnung zu einem in der Motivebene anzuordnenden Motivbild zusammengesetzt werden.

Moreover, in a method of producing a security element having a micro-moiré moiré magnification arrangement for displaying a three-dimensional moiré image having a structure extending into the depth of the space, the image constituents to be displayed are in at least two moiré spaced in a direction perpendicular to the moire magnification arrangement - contains image layers, provided that

1. a) a desired three-dimensional moiré image, which is to be viewed, is set as the target motif,
2. b) a periodic or at least locally periodic arrangement of microfocusing elements is determined as focussing element grid,
3. c) a desired magnification and a desired movement of the three-dimensional moiré image to be seen during lateral tilting and is determined during forward / backward tilting of the moiré magnification device,
4. d) for each image component to be displayed from the distance of the associated moiré image plane from the moiré magnification arrangement, the specified magnification and movement behavior and the Fokussierelementraster the associated micromotive image component to represent this image component of the three-dimensional Moire image, and the associated grid cell arrangement for the arrangement of the micromotif image components in the motive plane, and
5. e) the micromotif image components calculated for each image component to be displayed are assembled according to the associated grid cell arrangement to form a motif image to be arranged in the motif plane.

In many, in particular in the case of complex moiré images, it is advantageous to start from individual pixels of the three-dimensional moiré image as image components to be displayed and, in step d), for each of these moiré pixels an associated micromotive pixel and a grid cell arrangement for repeated arrangement of the micromotif. To determine pixels in the motif layer. For a single moiré pixel, the distance of the associated moire image plane from the moiré magnification arrangement is simply given by the height of the moiré pixel above the magnification arrangement. Even if several or even many moiré pixels lie at the same height and thus in the same moiré image plane, it is generally easier and more favorable for the calculation of the motif image to determine the determination after step d) for each of these moiré pixels perform separately and the motif image in step e) then composed of the repeatedly arranged micromotifs pixels, as first summarize the lying in a moiré image plane moire pixels and then perform the determination after step d) for the combined pixel set.

In step c), a tilting direction γ in which the parallax is to be observed is further specified for a reference point of the three-dimensional moiré image, as well as a desired magnification and movement behavior for this reference point and the predetermined tilting direction. The moire magnification factors in step d) for the other points of the three-dimensional moiré image are then related to the predetermined magnification factor for the reference point and the predetermined tilt direction.

vorgegeben und der Vergrößerungsfaktor für den Bezugspunkt aus der Transformationsmatrix A und der Kipprichtung γ wird unter Verwendung der Beziehung $$\mathrm{v}=\sqrt{{{\mathrm{v}}_x}^2+{{\mathrm{v}}_y}^2}=\sqrt{{\left({\mathrm{a}}_{11}\cos \gamma +{\mathrm{a}}_{12}\sin \gamma \right)}^2+{\left({\mathrm{a}}_{21}\cos \gamma +{\mathrm{a}}_{22}\sin \gamma \right)}^2}$$

berechnet.The desired magnification and movement behavior for the reference point is preferably in the form of the matrix elements of a transformation matrix $\mathrm{A}=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}& {\mathrm{a}}_{\mathrm{12}}\\ {\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{22}}\end{array}\right)$

and the magnification factor for the reference point from the transformation matrix A and the tilt direction γ is determined using the relationship $$\mathrm{v}=\sqrt{{{\mathrm{v}}_x}^2+{{\mathrm{v}}_y}^2}=\sqrt{{\left({\mathrm{a}}_{11}\cos \gamma +{\mathrm{a}}_{12}\sin \gamma \right)}^2+{\left({\mathrm{a}}_{21}\cos \gamma +{\mathrm{a}}_{22}\sin \gamma \right)}^2}$$

calculated.

bzw. deren Umkehrung $$\frac{{\mathrm{v}}_{\mathrm{i}}}{\mathrm{v}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{i}}\\ {\mathrm{y}}_{\mathrm{i}}\\ \mathrm{e}\end{array}\right)=\frac{\mathrm{1}}{\left({\mathrm{a}}_{\mathrm{11}}{\mathrm{a}}_{\mathrm{22}}-{\mathrm{a}}_{\mathrm{12}}{\mathrm{a}}_{\mathrm{21}}\right)}\cdot \left(\begin{array}{ccc}{\mathrm{a}}_{\mathrm{22}}& -{\mathrm{a}}_{\mathrm{12}}& \mathrm{0}\\ -{\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{11}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{1}\end{array}\right)\cdot \left(\begin{array}{c}{\mathrm{X}}_{\mathrm{i}}\\ {\mathrm{Y}}_{\mathrm{i}}\\ {\mathrm{Z}}_{\mathrm{i}}\end{array}\right)$$

berechnet, wobei e den effektiven Abstand des Fokussierelementrasters von der Motivebene bezeichnet.Z _i) of the three-dimensional moire image, the magnification factors are advantageously employed in step d) for additional points (X _i, Y _i, v _i and its Point coordinates in the motive plane (x _i , y _i ) using the relationship $$\left(\begin{array}{c}{\mathrm{X}}_{\mathrm{i}}\\ {\mathrm{Y}}_{\mathrm{i}}\\ {\mathrm{Z}}_{\mathrm{i}}\end{array}\right)=\frac{{\mathrm{v}}_{\mathrm{i}}}{\mathrm{v}}\cdot \left(\begin{array}{ccc}{\mathrm{a}}_{\mathrm{11}}& {\mathrm{a}}_{\mathrm{12}}& \mathrm{0}\\ {\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{22}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{v}\end{array}\right)\cdot \left(\begin{array}{c}{\mathrm{x}}_{\mathrm{i}}\\ {\mathrm{y}}_{\mathrm{i}}\\ \mathrm{e}\end{array}\right)$$

or their reversal $$\frac{{\mathrm{v}}_{\mathrm{i}}}{\mathrm{v}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{i}}\\ {\mathrm{y}}_{\mathrm{i}}\\ \mathrm{e}\end{array}\right)=\frac{\mathrm{1}}{\left({\mathrm{a}}_{\mathrm{11}}{\mathrm{a}}_{\mathrm{22}}-{\mathrm{a}}_{\mathrm{12}}{\mathrm{a}}_{\mathrm{21}}\right)}\cdot \left(\begin{array}{ccc}{\mathrm{a}}_{\mathrm{22}}& -{\mathrm{a}}_{\mathrm{12}}& \mathrm{0}\\ -{\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{11}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{1}\end{array}\right)\cdot \left(\begin{array}{c}{\mathrm{X}}_{\mathrm{i}}\\ {\mathrm{Y}}_{\mathrm{i}}\\ {\mathrm{Z}}_{\mathrm{i}}\end{array}\right)$$

where e denotes the effective distance of the focus element grid from the motif plane.

berechnet, wobei die Transformationsmatrizen A_i durch $${\mathrm{A}}_{\mathrm{i}}=\frac{{\mathrm{v}}_{\mathrm{i}}}{\mathrm{v}}\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}& {\mathrm{a}}_{\mathrm{12}}\\ {\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{22}}\end{array}\right)$$

gegeben sind, und A _i ^-1 die Umkehrmatrizen bezeichnet.The focusing element grid is advantageously specified in step b) by a raster matrix W. In step d), the points of the motif plane which belong to an enlargement v _{i are} advantageously combined to form a micromotif image constituent and for this micromotif image constituent a motif grid U _i for the periodic or at least locally periodic arrangement of this micromotif image constituent using the relationship $${\overleftrightarrow{U}}_i=\left(\overleftrightarrow{I}-{\overleftrightarrow{A}}_i^{-1}\right)\cdot \overleftrightarrow{W}$$

calculated, the transformation matrices A _i by $${\mathrm{A}}_{\mathrm{i}}=\frac{{\mathrm{v}}_{\mathrm{i}}}{\mathrm{v}}\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}& {\mathrm{a}}_{\mathrm{12}}\\ {\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{22}}\end{array}\right)$$

are given, and A _i ^-1 denotes the reversing matrices.

vorgegeben, wobei w_1i, w_2i die Komponenten der Gitterzellenvektoren w _i, mit i=1,2 darstellen.In a variant of the method, in step b) the focusing element grid is in the form of a two-dimensional Bravais grid with the raster matrix $\overleftrightarrow{W}=\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$

where w _1i , w _{2i are} the components of the lattice cell _vectors w _i , with i = 1.2.

vorgegeben, wobei D den Linsenabstand und φ die Orientierung der Zylinderlinsen bezeichnet.According to another variant of the method for producing a cylindrical lens 3D moire magnifier, in step b) a cylindrical lens grid is formed by the raster matrix $$W=\left(\begin{array}{cc}\cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{array}\right)\cdot \left(\begin{array}{cc}\mathrm{D}& 0\\ 0& \infty \end{array}\right)\mathrm{respectively}\mathrm{,}{W}^{-1}=\left(\begin{array}{cc}{}^1{/}_{\mathrm{<figure-callout\; id="1"\; label="D"\; filenames="imgf0002.png"\; state="\{\{state\}\}">D</figure-callout>}}& 0\\ 0& 0\end{array}\right)\cdot \left(\begin{array}{cc}\cos \phi & \sin \phi \\ -\sin \phi & \cos \phi \end{array}\right)$$

given, where D denotes the lens pitch and φ the orientation of the cylindrical lenses.

In all aspects of the invention, the lattice parameters of the Bravais lattices may be location independent. However, it is also possible to use the grid vectors of the motif grid grid cells u ₁ and u ₂ (resp. u ₁ ^{( i )} and u ₂ ^{( i )} for a plurality of motif grids U _i ) and the grating vectors of the focusing element grid w ₁ and w _{2 to be} spatially dependent, where the local period parameters | u ₁ |, | u ₂ |, ∠ ( u ₁ , u ₂ ) or | w ₁ |, | w ₂ |, ∠ ( w ₁ , w ₂ ) according to the invention change only slowly in relation to the periodicity length. This ensures that the arrangements can always be meaningfully described locally by Bravais lattice.

A security paper for the production of security or value documents, such as banknotes, checks, identity cards, documents, is equipped with a security element of the type described above. The security paper may in particular comprise a carrier substrate made of paper or plastic.

The invention also includes a data carrier, in particular a brand article, a document of value, a decorative article, such as a packaging or postcards, with a security element of the type described above. The security element can in particular be arranged in a window area, ie a transparent or recessed area of the data carrier be.

Further embodiments and advantages of the invention are explained below with reference to the figures. For better clarity, a scale and proportioned representation is omitted in the figures.

Fig. 1

eine schematische Darstellung einer Banknote mit einem eingebetteten Sicherheitsfaden und einem aufgeklebten Transferelement,

Fig. 2

schematisch den Schichtaufbau eines erfindungsgemäßen Sicherheitselements im Querschnitt,

Fig. 3

schematisch die Verhältnisse bei der Betrachtung einer Moiré-Vergrößerungsanordnung zur Definition der auftretenden Größen,

Fig. 4

weitere Definitionen auftretender Größen bei einer Moire-Vergrößerungsanordnung zur Darstellung eines einfachen dreidimensionalen Moiré-Bilds,

Fig. 5

schematisch die Verhältnisse bei der Betrachtung einer Moire-Vergrößerungsanordnung zur Veranschaulichung des Zustandekommens unterschiedlicher Vergrößerungen bei unterschiedlichen Motivrastern in der Motivebene,

Fig. 6

in (a) ein einfaches dreidimensionales Motiv in Form eines Buchstabens "P", in (b) eine Darstellung dieses Motivs durch nur zwei parallele Bildebenen, in (c) durch fünf parallele Bildebenen,

Fig. 7

in (a) ein erfindungsgemäß konstruiertes Motivbild und in (b) schematisch einen Ausschnitt des sich bei Betrachtung des Motivbilds von (a) mit einem passenden hexagonalen Linsenraster ergebenden dreidimensionalen Moiré-Bilds,

Fig. 8

in (a) ein erfindungsgemäß konstruiertes Motivbild mit orthoparallaktischem Bewegungsverhalten und in (b) schematisch einen Ausschnitt des sich bei Betrachtung des Motivbilds von (a) mit einem passenden rechtwinkligen Linsenraster ergebenden dreidimensionalen Moire-Bilds,

Fig. 9

in (a) ein erfindungsgemäß konstruiertes Motivbild mit schrägem Bewegungsverhalten und in (b) schematisch einen Ausschnitt des sich bei Betrachtung des Motivbilds von (a) mit einem passenden rechtwinkligen Linsenraster ergebenden dreidimensionalen Moire-Bilds, und

Fig. 10

schematisch die Verhältnisse bei der Betrachtung einer Moire-Vergrößerungsanordnung zur Veranschaulichung des Zustandekommens unterschiedlicher Vergrößerungen bei Motivebenen in unterschiedlichen Tiefen d₁, d₂.

Show it:

Fig. 1

a schematic representation of a banknote with an embedded security thread and a glued transfer element,

Fig. 2

schematically the layer structure of a security element according to the invention in cross-section,

Fig. 3

schematically the conditions when considering a moiré magnification arrangement for defining the occurring variables,

Fig. 4

further definitions of occurring magnitudes in a moiré magnification arrangement for the representation of a simple three-dimensional moiré image,

Fig. 5

FIG. 2 schematically shows the conditions when viewing a moiré magnification arrangement to illustrate the occurrence of different magnifications for different motif grids in the motif plane, FIG.

Fig. 6

in (a) a simple three-dimensional motif in the form of a letter "P", in (b) a representation of this motif by only two parallel image planes, in (c) by five parallel image planes,

Fig. 7

in (a) a motif image constructed according to the invention and in (b) schematically a section of the three-dimensional moiré image resulting from viewing the motif image of (a) with a suitable hexagonal lenticular grid,

Fig. 8

in (a) a motif image with orthoparallactic motion behavior constructed according to the invention and in (b) a schematic section of the three-dimensional moiré image resulting from viewing the motif image of (a) with a suitable rectangular lenticular grid,

Fig. 9

in (a) a motif image with oblique motion behavior constructed according to the invention and in (b) schematically a section of the three-dimensional moire image resulting from viewing the motif image of (a) with a suitable right-angled lenticular image, and

Fig. 10

schematically shows the circumstances in the consideration of a moiré magnification arrangement to illustrate the emergence of different magnifications at motif levels at different depths d ₁ , d ₂ .

The invention will now be explained using the example of a security element for a banknote. Fig. 1 shows a schematic representation of a banknote 10, which is provided with two security elements 12 and 16 according to embodiments of the invention. The first security element represents a security thread 12 that emerges in certain window areas 14 on the surface of the banknote 10, while it is embedded in the intervening areas inside the banknote 10. The second security element is formed by a glued transfer element 16 of any shape. The security element 16 can also be designed in the form of a cover film, which is arranged over a window area or a through opening of the banknote. The security element may be designed for viewing in supervision, review or viewing both in supervision and in review. Bilateral designs are also possible in which lenticular screens are arranged on both sides of a motif image.

Both the security thread 12 and the transfer element 16 may comprise a moire magnification arrangement according to an embodiment of the invention contain. The mode of operation and the production method according to the invention for such arrangements will be described in more detail below with reference to the transfer element 16.

Fig. 2 shows schematically the layer structure of the transfer element 16 in cross section, wherein only the parts of the layer structure required for the explanation of the principle of operation are shown. The transfer element 16 includes a carrier 20 in the form of a transparent plastic film, in the embodiment of an approximately 20 micron thick polyethylene terephthalate (PET) film.

The upper side of the carrier film 20 is provided with a grid-like arrangement of microlenses 22 which form on the surface of the carrier film a two-dimensional Bravais grid with a preselected symmetry. The Bravais lattice may, for example, have a hexagonal lattice symmetry, but because of the higher security against forgery, preferred are lower symmetries and thus more general shapes, in particular the symmetry of a parallelogram lattice.

The spacing of adjacent microlenses 22 is preferably chosen as small as possible in order to ensure the highest possible area coverage and thus a high-contrast representation. The spherically or aspherically configured microlenses 22 preferably have a diameter between 5 μm and 50 μm and in particular a diameter between only 10 μm and 35 μm and are therefore not visible to the naked eye. It is understood that in other designs, larger or smaller dimensions come into question. For example, in the case of Moiré-Magnifier structures, the microlenses can have a diameter of between 50 μm and 5 mm for decorative purposes, while Moire Magnifier structures can only be deciphered with a magnifying glass or a microscope should also dimensions below 5 microns can be used.

On the underside of the carrier film 20 is a motif layer 26 is arranged, which contains two or more likewise grid-shaped grid cell arrangements with different grating periods and / or different grating orientations. The grid cell arrangements are each formed from a plurality of grid cells 24, wherein in Fig. 2 for the sake of clarity, only one of these grid cell arrangements is shown. Designs having multiple grid cell arrays are exemplified in FIG Fig. 5 . 7 (a) . 8 (a) and 9 (a) shown.

As will be explained in more detail below, the moiré magnification arrangement of FIG Fig. 2 for the viewer, a three-dimensional moiré image, ie a moiré image containing image components in at least two, in a direction perpendicular to the moire magnification arrangement spaced moire image planes. For this purpose, each of the grid cell arrangements of the motif layer 26 is in each case assigned to one of the moiré image planes, and the grid cells 24 of this grid cell arrangement contain micromotif image components 28 for representing the image constituent of just this assigned moire image plane.

In addition to the lenticular grid, the motif lattices also form two-dimensional Bravais lattices with a preselected or calculated symmetry, again assuming a parallelogram lattice. As in Fig. 2 indicated by the offset of the grid cells 24 relative to the microlenses 22, the Bravais grid of the grid cells 24 differs slightly in its symmetry and / or in the size of its grid parameters from the Bravais grid of the microlenses 22 to the desired moire magnification effect to produce. The grating period and the diameter of the grid cells 24 are in the same order of magnitude as those of the microlenses 22, that is preferably in the range of 5 .mu.m to 50 .mu.m and in particular in the range of 10 .mu.m to 35 .mu.m, so that even the micromotif image components 28 themselves unrecognizable to the naked eye. In designs with the above-mentioned larger or smaller microlenses, of course, the grid cells 24 are correspondingly larger or smaller.

The optical thickness of the carrier film 20 and the focal length of the microlenses 22 are coordinated so that the motif layer 26 is located approximately at a distance of the lens focal length. The carrier foil 20 thus forms an optical spacer layer which ensures a desired, constant spacing of the microlenses 22 and the motif layer with the micromotif image constituents 28.

Due to the slightly different lattice parameters, the observer sees a slightly different subarea of the micromotif image constituents 28 when viewed from above through the microlenses 22, so that the large number of microlenses 22 as a whole produces an enlarged image of the micromotives. The resulting moiré magnification depends on the relative difference of the lattice parameters of the Bravais gratings used. If, for example, the grating periods of two hexagonal gratings differ by 1%, the result is a 100-fold moire magnification. For a more detailed representation of the operation and advantageous arrangements of the motif grid and the microlens grid is on the German patent application 10 2005 062132.5 and the post-published document WO 2007/076952 A2 directed.

The moiré magnification arrangements of the present application now not only produce planar objects floating in front of or behind the plane of the array but also produce three-dimensional moiré images with a structure extending into the depth of the space. These moiré magnification arrangements are therefore also referred to below as 3D moiré magnifier.

According to the invention, three-dimensional moiré images are shown which, when the moire magnification arrangement is tilted, move in a direction different from the tilting direction. As explained in detail below, in such designs, the visual spatial impression and the spatial experience due to the tilting movement are not in harmony with each other or even contradict each other, resulting in striking, sometimes dizzying, high attention and recognition effects for the viewer.

In addition, a mathematical approach will be presented, with which all variants of 3D Moiré Magnifiern can be described and modeled for production using a computer. Also, the three-dimensional Moire images generated by the 3D moiré Magnifiern should be able to be viewed without field limitations.

To explain the procedure according to the invention will therefore be first with reference to FIGS. 3 and 4 the required sizes are defined and briefly described. For a more detailed representation is in addition to the already mentioned German patent application 10 2005 062132.5 and the document WO 2007/076952 A2 directed.

FIGS. 3 and 4 schematically show a non-scaled moire magnification arrangement 30 with a motif plane 32 in which the motif image is arranged with the micromotif image components, and with a lens plane 34 in which the microlens grid is located. Moire magnification arrangement 30 produces two or more moiré image planes 36, 36 '(two are in FIG Fig. 3 in which the enlarged three-dimensional moiré image 40 (FIG. Fig. 4 ) is described.

The arrangement of the micromotif image constituents in the motif plane 32 is described by two or more two-dimensional Bravais lattices, their unit cells each by vectors u ₁ and u ₂ (with the components u ₁₁ , u ₂₁ and u ₁₂ , u ₂₂ ) can be represented. For the sake of clarity, is in Fig. 3 one of these unit cells singled out and presented.

In compact notation, the unit cell of the motif grid can also be represented in matrix form by a motif grid matrix U (hereinafter often simply called motif grid): $$\overleftrightarrow{U}=\left({\stackrel{\rightharpoonup }{u}}_1,{\stackrel{\rightharpoonup }{u}}_2\right)=\left(\begin{array}{cc}{u}_{11}& u_{12}\\ u_{21}& {\mathrm{<figure-callout\; id="22"\; label="u"\; filenames="imgf0001.png"\; state="\{\{state\}\}">u</figure-callout>}}_{22}\end{array}\right)$$

In the case of two or more motif rasters in the motif plane, the associated motif raster matrices are distinguished below by their indices U ₁ , U ₂ ,....

The arrangement of microlenses in the lens plane 34 is also described by a two-dimensional Bravais lattice whose unit cell is represented by the vectors w ₁ and w ₂ (with the components w ₁₁ , w ₂₁ and w ₁₂ , w ₂₂ ) is given.

With the vectors t ₁ and t ₂ (with the components t ₁₁ , t ₂₁ and t ₁₂ , t ₂₂ ), the unit cell in one of the moire image planes 36, 36 'is described. In the case of the three-dimensional moiré images, in addition to the two-dimensional position of the point in one of the image planes, a complete description of a moiré pixel also requires the specification in which moiré image plane a pixel lies. This is done in the context of this description by the indication of the Z component of the moire pixel, ie the perceived flying height of the pixel above or below the plane of the moire magnification arrangement, as in FIG 3 and 4 shown.

ist nachfolgend ein allgemeiner Punkt der Motivebene 32 bezeichnet, mit ${\overrightarrow{R}}^{3D}=\left(\begin{array}{c}X\\ Y\\ Z\end{array}\right)$

ein allgemeiner Moiré-Bildpunkt in einer der Moiré-Bildebenen 36, 36'. Innerhalb jeder (zweidimensionalen) Moiré-Bildebene 36 können die Bildpunkte durch die zweidimensionalen Koordinaten $\overrightarrow{R}=\left(\begin{array}{c}X\\ Y\end{array}\right)$

beschrieben werden.With $\overrightarrow{r}=\left(\begin{array}{c}x\\ y\end{array}\right)$

is hereinafter a general point of the motif level 32 denotes, with ${\overrightarrow{R}}^{3D}=\left(\begin{array}{c}X\\ Y\\ Z\end{array}\right)$

a general moiré pixel in one of the moiré image planes 36, 36 '. Within each (two-dimensional) moire image plane 36, the pixels may pass through the two-dimensional coordinates $\overrightarrow{R}=\left(\begin{array}{c}X\\ Y\end{array}\right)$

to be discribed.

in der Motivebene 32 angegeben wird. Analog zur Motivrastermatrix werden zur kompakten Beschreibung des Linsenrasters und des Bildrasters die Matrizen $\overleftrightarrow{W}=\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& w_{22}\end{array}\right)$

(als Linsenrastermatrix oder einfach Linsenraster bezeichnet) und $\overleftrightarrow{T}=\left(\begin{array}{cc}{t}_{11}& t_{12}\\ t_{21}& t_{22}\end{array}\right)$

verwendet.In addition to vertical viewing (viewing direction 35) to be able to describe non-perpendicular viewing directions of the moiré magnification arrangement, such as the general direction 35 ', an additional shift between lens plane 34 and motive plane 32 is permitted by a displacement vector ${\overrightarrow{r}}_0=\left(\begin{array}{c}{x}_0\\ y_0\end{array}\right)$

is indicated in the motif level 32. Analogous to the motif grid matrix, the matrices are used to compactly describe the lens raster and the image raster $\overleftrightarrow{W}=\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& w_{22}\end{array}\right)$

(called lenticular matrix or simply lenticular grid) and $\overleftrightarrow{T}=\left(\begin{array}{cc}{t}_{11}& t_{12}\\ t_{21}& {\mathrm{<figure-callout\; id="22"\; label="t"\; filenames="imgf0001.png"\; state="\{\{state\}\}">t</figure-callout>}}_{22}\end{array}\right)$

used.

In the lens plane 34, instead of lenses 22, it is also possible, for example, to use pinholes on the principle of the pinhole camera. Also all other types of lenses and imaging systems, such as aspherical lenses, cylindrical lenses, slit apertures, apertured apertured or slit apertures, Fresnel lenses, GRIN lenses ( G radient R efraction I ndex), zone plates (diffractive lenses), holographic lenses, concave mirrors, Fresnel mirrors , Zonenspiegel and other elements with focussing or blanking effect, can be used as Mikrofokussierelemente in Fokussierelementraster.

In principle, in addition to elements with focussing effect elements with ausblendender effect (hole or slit, even mirror surfaces behind hole or slit) can be used as Mikrofokussierelemente in Fokussierelementraster.

When using a concave mirror array and in other inventively used reflective focusing grids, the viewer looks through the partially transparent in this case motif image on the mirror array behind it and sees the individual small mirror as light or dark spots from which builds the image to be displayed. The motif image is generally so finely structured that it can only be seen as a veil. The described formulas for the relationships between the moiré image to be displayed and the motif image apply, even if this is not mentioned in detail, not only for lenticular raster but also for mirror raster. It is understood that in the inventive use of concave mirrors at the location of the lens focal length, the mirror focal length occurs.

In the inventive application of a mirror array instead of a lens array is in Fig. 2 to think of the viewing direction from below, and in Fig. 3 In the mirror array arrangement, the levels 32 and 34 are interchanged. The further description of the invention is based on lens grids, which are representative of all other inventively used Fokussierelementraster.

und die Bildpunkte innerhalb der Moire-Bildebene 36 können mithilfe der Beziehung $$\overleftrightarrow{R}=\overleftrightarrow{W}\cdot {\left(\overleftrightarrow{W}-\overleftrightarrow{U}\right)}^{-1}\cdot \left(\overrightarrow{r}-{\overrightarrow{r}}_0\right)$$

aus den Bildpunkten der Motivebene 32 bestimmt werden. Umgekehrt ergeben sich die Gittervektoren der Motivebene 32 aus dem Linsenraster und dem gewünschten Moiré-Bild-Gitter einer Motivebene 36 durch $$\overleftrightarrow{U}=\overleftrightarrow{W}\cdot {\left(\overleftrightarrow{T}+\overleftrightarrow{W}\right)}^{-1}\cdot \overleftrightarrow{T}$$

und $$\overleftrightarrow{r}=\overleftrightarrow{W}\cdot {\left(\overleftrightarrow{T}+\overleftrightarrow{W}\right)}^{-1}\cdot \overrightarrow{R}+{\overrightarrow{r}}_0\mathrm{.}$$

Every motif grid U Thus, each of the various grid level arrangements of the motif level 32 is associated with exactly one of the moire image levels 36, 36 '. The Moire Picture Grid T this associated moire image plane 36 results from the grating vectors of the motif plane 32 and the lens plane 34 $$\overleftrightarrow{T}=\overleftrightarrow{W}\cdot {\left(\overleftrightarrow{W}-\overleftrightarrow{U}\right)}^{-1}\cdot \overleftrightarrow{U}$$

and the pixels within moire image plane 36 can be determined using the relationship $$\overleftrightarrow{R}=\overleftrightarrow{W}\cdot {\left(\overleftrightarrow{W}-\overleftrightarrow{U}\right)}^{-1}\cdot \left(\overrightarrow{r}-{\overrightarrow{r}}_0\right)$$

be determined from the pixels of the motif level 32. Conversely, the grid vectors of the motif plane 32 result from the lenticular grid and the desired moiré image grid of a motif plane 36 $$\overleftrightarrow{U}=\overleftrightarrow{W}\cdot {\left(\overleftrightarrow{T}+\overleftrightarrow{W}\right)}^{-1}\cdot \overleftrightarrow{T}$$

and $$\overleftrightarrow{r}=\overleftrightarrow{W}\cdot {\left(\overleftrightarrow{T}+\overleftrightarrow{W}\right)}^{-1}\cdot \overrightarrow{R}+{\overrightarrow{r}}_0\mathrm{,}$$

so können aus jeweils zwei der vier Matrizen U , W , T , A die beiden anderen berechnet werden. Insbesondere gilt: $$\overleftrightarrow{T}=\overleftrightarrow{A}\cdot \overleftrightarrow{U}=\overleftrightarrow{W}\cdot {\left(\overleftrightarrow{W}-\overleftrightarrow{U}\right)}^{-1}\cdot \overleftrightarrow{U}=\left(\overleftrightarrow{A}-\overleftrightarrow{I}\right)\cdot \overleftrightarrow{W}$$

$$\overleftrightarrow{U}=\overleftrightarrow{W}\cdot {\left(\overleftrightarrow{T}+\overleftrightarrow{W}\right)}^{-1}\cdot \overleftrightarrow{T}={\overleftrightarrow{A}}^{-1}\cdot \overleftrightarrow{T}=\left(\overleftrightarrow{I}-{\overleftrightarrow{A}}^{-1}\right)\cdot \overleftrightarrow{W}$$

$$\overleftrightarrow{W}=\overleftrightarrow{U}\cdot {\left(\overleftrightarrow{T}-\overleftrightarrow{U}\right)}^{-1}\cdot \overleftrightarrow{T}={\left(\overleftrightarrow{A}-\overleftrightarrow{I}\right)}^{-1}\cdot \overleftrightarrow{T}={\left(\overleftrightarrow{A}-\overleftrightarrow{I}\right)}^{-1}\cdot \overleftrightarrow{A}\cdot \overleftrightarrow{U}$$

$$\overleftrightarrow{A}=\overleftrightarrow{W}\cdot {\left(\overleftrightarrow{W}-\overleftrightarrow{U}\right)}^{-1}=\left(\overleftrightarrow{T}+\overleftrightarrow{W}\right)\cdot {\overleftrightarrow{W}}^{-1}=\overleftrightarrow{T}\cdot {\overleftrightarrow{U}}^{-1}$$

wobei I die Einheitsmatrix bezeichnet.Define the transformation matrix A = W · (W - U ) ^-1 , which converts the coordinates of the points of the motif plane 32 and the points of the moire image plane 36 into each other, $$\overrightarrow{R}=\overleftrightarrow{A}\cdot \left(\overrightarrow{r}-{\overrightarrow{r}}_0\right).\mathrm{respectively}\mathrm{,}\overrightarrow{r}={\overleftrightarrow{A}}^{-1}\cdot \overrightarrow{R}+{\overrightarrow{r}}_0.$$

so can each have two of the four matrices U . W . T . A the other two are calculated. In particular: $$\overleftrightarrow{T}=\overleftrightarrow{A}\cdot \overleftrightarrow{U}=\overleftrightarrow{W}\cdot {\left(\overleftrightarrow{W}-\overleftrightarrow{U}\right)}^{-1}\cdot \overleftrightarrow{U}=\left(\overleftrightarrow{A}-\overleftrightarrow{I}\right)\cdot \overleftrightarrow{W}$$

$$\overleftrightarrow{U}=\overleftrightarrow{W}\cdot {\left(\overleftrightarrow{T}+\overleftrightarrow{W}\right)}^{-1}\cdot \overleftrightarrow{T}={\overleftrightarrow{A}}^{-1}\cdot \overleftrightarrow{T}=\left(\overleftrightarrow{I}-{\overleftrightarrow{A}}^{-1}\right)\cdot \overleftrightarrow{W}$$

$$\overleftrightarrow{W}=\overleftrightarrow{U}\cdot {\left(\overleftrightarrow{T}-\overleftrightarrow{U}\right)}^{-1}\cdot \overleftrightarrow{T}={\left(\overleftrightarrow{A}-\overleftrightarrow{I}\right)}^{-1}\cdot \overleftrightarrow{T}={\left(\overleftrightarrow{A}-\overleftrightarrow{I}\right)}^{-1}\cdot \overleftrightarrow{A}\cdot \overleftrightarrow{U}$$

$$\overleftrightarrow{A}=\overleftrightarrow{W}\cdot {\left(\overleftrightarrow{W}-\overleftrightarrow{U}\right)}^{-1}=\left(\overleftrightarrow{T}+\overleftrightarrow{W}\right)\cdot {\overleftrightarrow{W}}^{-1}=\overleftrightarrow{T}\cdot {\overleftrightarrow{U}}^{-1}$$

in which I denotes the unit matrix.

Like in the German patent application 10 2005 062 132.5 and the document WO 2007/076952 A2 described in detail, describes the transformation matrix A both the moiré magnification and the resulting movement of the enlarged moiré image upon movement of moire-forming assembly 30 resulting from the shift of the motif plane 32 towards the lens plane 34.

The raster matrices T, U, W, the unit matrix I and the transformation matrix A are often subsequently also written without a double arrow, if it is clear from the context that these are matrices.

As noted, the three-dimensional extent of the depicted moire image 40, in addition to these two-dimensional relationships, is accounted for by the indication of an additional coordinate indicating the distance in which a moiré pixel is above or below the plane of the moiré magnification arrangement seems to float. If v denotes the moiré magnification and e is an effective distance of the lens plane 34 from the motif plane 32, in which not only the physical distance d but also the lens data and the refractive index of the medium between lenticular and motif grid are considered heuristically, the Z component is one Moiré pixel given by $$\mathrm{Z}=\mathrm{v}*\mathrm{e}\mathrm{,}$$

A three-dimensional moiré image 40, ie an image with different Z values, can now be generated in two different ways according to equation (1). On the one hand, the moire magnification v can be left constant and different values of e can be realized in the Moire Magnifier, or one can produce different moire magnifications with a uniform effective distance e by means of different motif grids. The former approach is related below Fig. 10 described in more detail, the latter is the following description of the FIGS. 3 to 9 based.

ergeben. Fig. 4 Fig. 12 shows a representation of a simple three-dimensional moiré image 40 and its decomposition into image constituents 42, 44 in only two spaced moire image planes 36, 36 ', which is sufficient to explain the essential design features of the invention. In particular, a moire magnification v _{1 is} realized for the image constituents in the image plane 36 (upper side 42 of the letter "P") by a suitably selected motif grid U ₁ , and for the image constituents in the image plane 36 '(lower side 44 of the letter "P"). ) is realized by a suitably selected motif grid U ₂ a moire magnification v ₂ , so that at constant effective distance e two image planes 36, 36 'with different Z values $${\mathrm{Z}}_{}$$$${\mathrm{}}_{\mathrm{1}}={\mathrm{<figure-callout\; id="1"\; label="v"\; filenames="imgf0002.png"\; state="\{\{state\}\}">v</figure-callout>}}_{\mathrm{1}}*\mathrm{e}.{\mathrm{Z}}_{\mathrm{2}}={\mathrm{v}}_{\mathrm{2}}*\mathrm{e}.$$

result.

To explain the basic effect, the special case of transformation matrices A is first considered, which describe a pure magnification, ie no rotation or distortion. $${\mathrm{A}}_{\mathrm{i}}={\mathrm{v}}_{\mathrm{i}}\cdot \mathrm{I}={\mathrm{v}}_{\mathrm{i}}\left(\begin{array}{cc}\mathrm{1}& \mathrm{0}\\ \mathrm{0}& \mathrm{1}\end{array}\right).\mathrm{with\; i}=\mathrm{1}.\mathrm{Second}$$

und $$U_2=\left(\begin{array}{cc}{{\mathrm{u}}^{\left(2\right)}}_{\mathrm{11}}& {{\mathrm{u}}^{\left(2\right)}}_{\mathrm{12}}\\ {{\mathrm{u}}^{\left(2\right)}}_{\mathrm{21}}& {{\mathrm{u}}^{\left(2\right)}}_{\mathrm{22}}\end{array}\right)=\left(\mathrm{1}-\frac{\mathrm{1}}{{\mathrm{v}}_2}\right)\cdot \left(\begin{array}{cc}{\mathrm{w}}_{\mathrm{11}}& {\mathrm{w}}_{\mathrm{12}}\\ {\mathrm{w}}_{\mathrm{21}}& {\mathrm{w}}_{\mathrm{22}}\end{array}\right)\mathrm{.}$$

For a given lens raster W, the motif rasters U ₁ and U _{2 are obtained} using relationship (M2): $${\mathrm{<figure-callout\; id="1"\; label="U"\; filenames="imgf0002.png"\; state="\{\{state\}\}">U</figure-callout>}}_1=\left(\begin{array}{cc}{{\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="imgf0002.png"\; state="\{\{state\}\}">u</figure-callout>}}^{\left(\mathrm{1}\right)}}_{\mathrm{11}}& {{\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="imgf0002.png"\; state="\{\{state\}\}">u</figure-callout>}}^{\left(\mathrm{1}\right)}}_{\mathrm{12}}\\ {{\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="imgf0002.png"\; state="\{\{state\}\}">u</figure-callout>}}^{\left(\mathrm{1}\right)}}_{\mathrm{21}}& {{\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="imgf0002.png"\; state="\{\{state\}\}">u</figure-callout>}}^{\left(\mathrm{1}\right)}}_{\mathrm{22}}\end{array}\right)=\left(\mathrm{1}-\frac{\mathrm{1}}{{\mathrm{<figure-callout\; id="1"\; label="v"\; filenames="imgf0002.png"\; state="\{\{state\}\}">v</figure-callout>}}_{\mathrm{1}}}\right)\cdot \left(\begin{array}{cc}{\mathrm{w}}_{\mathrm{11}}& {\mathrm{w}}_{\mathrm{12}}\\ {\mathrm{w}}_{\mathrm{21}}& {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{\mathrm{22}}\end{array}\right)$$

and $$U_2=\left(\begin{array}{cc}{{\mathrm{u}}^{\left(2\right)}}_{\mathrm{11}}& {{\mathrm{u}}^{\left(2\right)}}_{\mathrm{12}}\\ {{\mathrm{u}}^{\left(2\right)}}_{\mathrm{21}}& {{\mathrm{u}}^{\left(2\right)}}_{\mathrm{22}}\end{array}\right)=\left(\mathrm{1}-\frac{\mathrm{1}}{{\mathrm{v}}_2}\right)\cdot \left(\begin{array}{cc}{\mathrm{w}}_{\mathrm{11}}& {\mathrm{w}}_{\mathrm{12}}\\ {\mathrm{w}}_{\mathrm{21}}& {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{\mathrm{22}}\end{array}\right)\mathrm{,}$$

The realization of the different magnifications is in Fig. 5 3, which shows in the motif plane 32 as first micromotif elements dashed arrows 50 which are arranged in a first motif raster U ₁ with a grating period p ₁ and which shows solid arrows 52, drawn as second micromotif elements, at the same effective distance d from the lens plane 34 in FIG a second motif grid U _{2 are arranged} with a slightly larger grating period p ₂ .

The resulting magnified moiré images 54 and 56, respectively, for the observer 38 float at different heights Z ₁ , Z ₂ above the plane of the moiré magnification arrangement due to the different grating periods and the resulting different magnification factors v ₁ and v ₂ according to equation (1) , Of course, the different magnification factors must also be taken into account when designing the micromotif elements 50, 52. If, for example, the enlarged arrow images 54 and 56 appear to be of equal length, the dashed arrows 50 in the motif plane 32 must be correspondingly reduced in comparison with the solid arrows 52 in order to compensate for the higher magnification factor in the moiré image.

The presentation of the Fig. 5 in which the moiré images hover over the magnification arrangement, applies to negative magnification factors; with positive magnification factors, the moire images appear to float for the viewer correspondingly below the plane of the moiré magnification arrangement.

In general, the transformation matrices A _i in a 3-D moiré magnifier contain a respective matching component A ¹ , which describes twists and distortions, as well as the respective different magnification factors v _i for the image planes: $${\mathrm{A}}_{\mathrm{i}}={\mathrm{v}}_{\mathrm{i}}\cdot \mathrm{A\; \text{'}}={\mathrm{v}}_{\mathrm{i}}\left(\begin{array}{cc}{\mathrm{a}}_{11}^{\mathrm{\text{'}}}& {\mathrm{a}}_{12}^{\mathrm{\text{'}}}\\ {\mathrm{a}}_{21}^{\mathrm{\text{'}}}& {\mathrm{a}}_{22}^{\mathrm{\text{'}}}\end{array}\right)\mathrm{,}$$

beziehungsweise umgekehrt $${\mathrm{v}}_{\mathrm{i}}\cdot \left(\begin{array}{c}{\mathrm{x}}_{\mathrm{i}}\\ {\mathrm{y}}_{\mathrm{i}}\\ \mathrm{e}\end{array}\right)=\frac{\mathrm{1}}{\left({\mathrm{a}}_{11}^{\mathrm{ʹ}}{\mathrm{a}}_{22}^{\mathrm{ʹ}}-{\mathrm{a}}_{12}^{\mathrm{ʹ}}{\mathrm{a}}_{21}^{\mathrm{ʹ}}\right)}\cdot \left(\begin{array}{ccc}{\mathrm{a}}_{22}^{\mathrm{ʹ}}& -{\mathrm{a}}_{12}^{\mathrm{ʹ}}& \mathrm{0}\\ -{\mathrm{a}}_{21}^{\mathrm{ʹ}}& {\mathrm{a}}_{11}^{\mathrm{ʹ}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{1}\end{array}\right)\cdot \left(\begin{array}{c}{\mathrm{X}}_{\mathrm{i}}\\ {\mathrm{Y}}_{\mathrm{i}}\\ {\mathrm{Z}}_{\mathrm{i}}\end{array}\right)\mathrm{.}$$

The basic equations of the 3D Moire Magnifier now connect the points R ^3D in the moiré image planes 36, 36 'with the coordinates r the points of the motif level 32 over $${\overrightarrow{R}}_i^{3D}=\left(\begin{array}{c}{\mathrm{X}}_{\mathrm{i}}\\ {\mathrm{Y}}_{\mathrm{i}}\\ {\mathrm{Z}}_{\mathrm{i}}\end{array}\right)={\mathrm{v}}_{\mathrm{i}}\cdot \left(\begin{array}{ccc}{\mathrm{a}}_{11}^{\mathrm{\text{'}}}& {\mathrm{a}}_{12}^{\mathrm{\text{'}}}& \mathrm{0}\\ {\mathrm{a}}_{21}^{\mathrm{\text{'}}}& {\mathrm{a}}_{22}^{\mathrm{\text{'}}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& 1\end{array}\right)\cdot \left(\begin{array}{c}{\mathrm{x}}_{\mathrm{i}}\\ {\mathrm{y}}_{\mathrm{i}}\\ \mathrm{e}\end{array}\right)\mathrm{,}$$

or vice versa $${\mathrm{v}}_{\mathrm{i}}\cdot \left(\begin{array}{c}{\mathrm{x}}_{\mathrm{i}}\\ {\mathrm{y}}_{\mathrm{i}}\\ \mathrm{e}\end{array}\right)=\frac{\mathrm{1}}{\left({\mathrm{a}}_{11}^{\mathrm{\text{'}}}{\mathrm{a}}_{22}^{\mathrm{\text{'}}}-{\mathrm{a}}_{12}^{\mathrm{\text{'}}}{\mathrm{a}}_{21}^{\mathrm{\text{'}}}\right)}\cdot \left(\begin{array}{ccc}{\mathrm{a}}_{22}^{\mathrm{\text{'}}}& -{\mathrm{a}}_{12}^{\mathrm{\text{'}}}& \mathrm{0}\\ -{\mathrm{a}}_{21}^{\mathrm{\text{'}}}& {\mathrm{a}}_{11}^{\mathrm{\text{'}}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{1}\end{array}\right)\cdot \left(\begin{array}{c}{\mathrm{X}}_{\mathrm{i}}\\ {\mathrm{Y}}_{\mathrm{i}}\\ {\mathrm{Z}}_{\mathrm{i}}\end{array}\right)\mathrm{,}$$

The special case of a pure magnification without distortion or distortion described at the beginning results as a special case of equation (2a) $${\overrightarrow{R}}_i^{3D}=\left(\begin{array}{c}{\mathrm{X}}_{\mathrm{i}}\\ {\mathrm{Y}}_{\mathrm{i}}\\ {\mathrm{Z}}_{\mathrm{i}}\end{array}\right)=\left(\begin{array}{ccc}{\mathrm{v}}_{\mathrm{i}}& 0& \mathrm{0}\\ 0& {\mathrm{v}}_{\mathrm{i}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& {\mathrm{v}}_{\mathrm{i}}\end{array}\right)\cdot \left(\begin{array}{c}{\mathrm{x}}_{\mathrm{i}}\\ {\mathrm{y}}_{\mathrm{i}}\\ \mathrm{e}\end{array}\right)\mathrm{,}$$

Starting from the three-dimensional moiré image motif to be represented, which is given by a set of points (X, Y, Z), and a desired movement behavior of the moiré image, which is indicated by the matrix A 'in more detail below, one can using the relationship (2b) calculate the associated pixels (x, y) in the motif plane and the associated magnification factor v. The associated motif grid U is determined according to relationship (7), as indicated below.

In this case, the points of the three-dimensional moiré image motif to be displayed, which are to be at the same height Z above or below the magnification arrangement, can be combined, because for these points, because of Z = v * e, the same magnification factors v and thus the same motif grid matrices belong. In other words, the parallel cuts Z _i can be arranged in-Moire motif motif corresponding pixels in respective uniformly to create design grids U _i.

In particular, two effects contribute to a three-dimensional image effect for a viewer, which are referred to as "two-eyed vision" or "movement behavior".

After the effect of binocular vision, the magnified moiré image appears to be deep-shaded in binocular viewing, provided that the moiré magnifier is designed in such a way that lateral tilting of the arrangement leads to a lateral displacement of the pixels. Because of the lateral "tilt angle" of about 15 ° between the eyes at a normal viewing distance of about 25 cm, seen in the eyes laterally displaced pixels are interpreted by the brain as if the pixels were in front of or behind the direction of lateral displacement actual substrate level, depending on the size of the shift more or less high or low.

By the effect of "motion behavior" is meant that when tilting a moiré magnifier, which is designed so that a lateral tilting of the arrangement leads to a shift of the pixels, previously hidden back parts of the subject can be visible and thus the subject in three dimensions can be detected.

A consistent three-dimensional image impression results when the two effects have the same effect as with ordinary spatial vision.

In the case of the special 3D moiré magnifiers, which are designed in accordance with the special case of equation (2c), both effects are actually similar, as shown below. Such 3D moiré magnifiers therefore provide the viewer with a conventional, consistent three-dimensional image effect.

However, in general 3D Moire magnifiers constructed not according to the special case (2c) but according to the general equations (2a) and (2b), the two-eyesight and motion behaviors may be different or even different contradictory visual impressions, creating striking and dizzying effects for the viewer with high attention and recognition value.

In order to achieve such visual effects, it is important to know and to influence the movement behavior of the moiré image when tilting the moiré magnification arrangements.

The columns of the transformation matrix A can be interpreted as vectors: $$\mathrm{A}=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}& {\mathrm{a}}_{\mathrm{12}}\\ {\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{22}}\end{array}\right).{\overrightarrow{a}}_1=\left(\begin{array}{c}{\mathrm{a}}_{\mathrm{11}}\\ {\mathrm{a}}_{\mathrm{21}}\end{array}\right).{\overrightarrow{a}}_2=\left(\begin{array}{c}{\mathrm{a}}_{12}\\ {\mathrm{a}}_{22}\end{array}\right)\mathrm{,}$$

gibt an, in welcher Richtung sich das entstehende Moiré-Bild bewegt, wenn man die Anordnung aus Motivraster und Linsenraster seitlich kippt. Der Vektor ${\overrightarrow{a}}_2=\left(\begin{array}{c}{\mathrm{a}}_{12}\\ {\mathrm{a}}_{22}\end{array}\right)$

gibt an, in welcher Richtung sich das entstehende Moire-Bild bewegt, wenn man die Anordnung aus Motivraster und Linsenraster vor-/rückwärts kippt. Dabei wird die Bewegungsrichtung folgendermaßen festgelegt:The vector ${\overrightarrow{a}}_1=\left(\begin{array}{c}{\mathrm{a}}_{\mathrm{11}}\\ {\mathrm{a}}_{\mathrm{21}}\end{array}\right)$

indicates the direction in which the resulting moiré image moves when tilting the arrangement of motif grid and lenticular laterally. The vector ${\overrightarrow{a}}_2=\left(\begin{array}{c}{\mathrm{a}}_{12}\\ {\mathrm{a}}_{22}\end{array}\right)$

indicates the direction in which the resulting Moire image moves, if the arrangement of motif grid and lens raster tilts forward / backward. The direction of movement is defined as follows:

The angle β ₁ in which the moiré image moves relative to the horizontal when the assembly is tilted sideways is given by $$\mathrm{<figure-callout\; id="1"\; label="tan\; \beta "\; filenames="imgf0002.png"\; state="\{\{state\}\}">tan\; </figure-callout>}{\beta }_{}{}_1=\frac{{\mathrm{a}}_{\mathrm{21}}}{{\mathrm{a}}_{\mathrm{11}}}\mathrm{,}$$

The angle β ₂ in which the moiré image moves relative to the horizontal when the assembly is tilted forward / backward is given by $$\tan {\beta }_2=\frac{{\mathrm{a}}_{22}}{{\mathrm{a}}_{12}}\mathrm{,}$$

mit dem sich das dreidimensionale Moiré-Bild 40 relativ zu einer Bezugsrichtung, beispielsweise der Waagrechten W, bewegt, wenn die Anordnung nicht in einer der Vorzugsrichtungen seitlich (0°) oder vor-/rückwärts (90°) bewegt, sondern in einer allgemeinen, durch einen Winkel γ zur Bezugsrichtung W angegebenen Richtung k gekippt wird, ist gegeben durch $$\overrightarrow{v}=\left(\begin{array}{c}{v}_x\\ v_y\end{array}\right)=\left(\begin{array}{cc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}\end{array}\right)\cdot \left(\begin{array}{c}\cos \gamma \\ \sin \gamma \end{array}\right)=\left(\begin{array}{c}{\mathrm{a}}_{11}\cos \gamma +{\mathrm{a}}_{12}\sin \gamma \\ a_{21}\cos \gamma +{\mathrm{a}}_{22}\sin \gamma \end{array}\right)\mathrm{.}$$

Coming back to the presentation of the Fig. 4 is the motion vector $\overrightarrow{v}=\left(\begin{array}{c}{v}_x\\ v_y\end{array}\right).$

with which the three-dimensional moiré image 40 moves relative to a reference direction, for example the horizontal W, if the arrangement does not move laterally (0 °) or forwards / backwards (90 °) in one of the preferred directions, but in a general, by an angle γ to the reference direction W direction k is given by $$\overrightarrow{v}=\left(\begin{array}{c}{v}_x\\ v_y\end{array}\right)=\left(\begin{array}{cc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}\end{array}\right)\cdot \left(\begin{array}{c}\cos \gamma \\ \sin \gamma \end{array}\right)=\left(\begin{array}{c}{\mathrm{a}}_{11}\cos \gamma +{\mathrm{a}}_{12}\sin \gamma \\ a_{21}\cos \gamma +{\mathrm{a}}_{22}\sin \gamma \end{array}\right)\mathrm{,}$$

Thus, the angle β ₃ at which the moiré image 40 moves with respect to the reference direction W when the moiré magnification device is tilted in the general direction γ is given by $$\tan {\beta }_3=\frac{{\mathrm{a}}_{21}\cos \gamma +{\mathrm{a}}_{22}\sin \gamma }{{\mathrm{a}}_{11}\cos \gamma +{\mathrm{a}}_{12}\sin \gamma }\mathrm{,}$$

The distance of a pair of points lying in the direction of γ in the motif plane 32 therefore extends in the moiré image plane 36 in the direction β ₃ , increased by the factor $$\mathrm{v}=\sqrt{{{\mathrm{v}}_x}^2+{{\mathrm{v}}_y}^2}=\sqrt{{\left({\mathrm{a}}_{11}\cos \gamma +{\mathrm{a}}_{12}\sin \gamma \right)}^2+{\left({\mathrm{a}}_{21}\cos \gamma +{\mathrm{a}}_{22}\sin \gamma \right)}^2}\mathrm{,}$$

oberhalb bzw. unterhalb der Substratebene zu schweben ("Bewegungseffekt").Therefore, according to equation (1), the moire image 40 shown in the case of a 3D moire magnifier constructed with the transformation matrix A with the effective distance e between the motif plane 32 and the lens plane 34 by the parallax when tilting the device in the direction .gamma . Depth $${\mathrm{Z}}_{\mathrm{movement}}=\mathrm{v}\cdot \mathrm{e}=\mathrm{e}\cdot \sqrt{{\left({\mathrm{a}}_{11}\cos \gamma +{\mathrm{a}}_{12}\sin \gamma \right)}^2+{\left({\mathrm{a}}_{21}\cos \gamma +{\mathrm{a}}_{22}\sin \gamma \right)}^2}$$

float above or below the substrate plane ("movement effect").

On the other hand, in binocular viewing with an eye distance direction that is not in the direction of γ, only the component in the direction of eye distance is effective for the moiré magnification. For example, if the two eyes are next to each other in the x-direction, the result is a sense of depth $${\mathrm{Z}}_{\mathrm{binocular}}={\mathrm{v}}_{\mathrm{x}}\cdot \mathrm{e}=\mathrm{e}\cdot \left({\mathrm{a}}_{11}\cos \gamma +{\mathrm{a}}_{12}\sin \gamma \right)\mathrm{,}$$

The depth impression due to the movement effect, Z _movement and the deep impression through binocular vision, Z _binocular , therefore differ for almost all eye distance directions. The moiré image 40 therefore appears to be _binocular when tilted in the direction γ for the eyes at a different depth, namely at depth Z, than the depth Z _movement , which suggests parallax when tilted.

also a₁₁ = a₂₂ = v und a₂₁ = a₁₂ = 0 fallen die Werte für Z_binocular und Z_movement zusammen, so dass dort das beidäugige Sehen und die Parallaxe beim Kippen zum gleichem Tiefeneindruck und damit zu einer konsistenten dreidimensionalen Bildwahrnehmung führt.In the special case mentioned above $$\mathrm{A}=\mathrm{v}\cdot \mathrm{I}=\mathrm{<figure-callout\; id="1"\; label="v"\; filenames="imgf0002.png"\; state="\{\{state\}\}">v</figure-callout>}\left(\begin{array}{cc}\mathrm{1}& \mathrm{0}\\ \mathrm{0}& \mathrm{1}\end{array}\right).$$

therefore, a ₁₁ = a ₂₂ = v and a ₂₁ = a ₁₂ = 0, the values for Z _binocular and Z _movement coincide, so that binaural vision and parallax lead to the same depth impression and thus to a consistent three-dimensional image perception.

bezogen werden: $${\mathrm{A}}_{\mathrm{1}}=\frac{{\mathrm{v}}_{\mathrm{1}}}{\mathrm{v}}\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}& {\mathrm{a}}_{\mathrm{12}}\\ {\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{22}}\end{array}\right),{\mathrm{A}}_{\mathrm{2}}=\frac{{\mathrm{v}}_{\mathrm{2}}}{\mathrm{v}}\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}& {\mathrm{a}}_{\mathrm{12}}\\ {\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{22}}\end{array}\right),\&\mathrm{c}\mathrm{.}$$

wobei $${\mathrm{Z}}_{\mathrm{1}}={\mathrm{v}}_{\mathrm{1}}\cdot \mathrm{e},{\mathrm{Z}}_{\mathrm{2}}={\mathrm{v}}_{\mathrm{2}}\cdot \mathrm{e},\&\mathrm{c}\mathrm{.}$$

The above statements initially relate to the relationships for a motif point, a motif point set or a motif part with a single depth component Z. In order to realize motif points or subject parts in different depths Z ₁ , Z ₂ ...., the motif points or motif parts provided for different depths are used arranged according to the invention in the screen plane in modified screen rulings with modified transformation matrix A ₁ , A ₂ . The magnification factor v _{i of} the different parts of the subject can in each case be based on the magnification factor v in the tilt direction according to equation (3c) and the original transformation matrix $\mathrm{A}=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}& {\mathrm{a}}_{\mathrm{12}}\\ {\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{22}}\end{array}\right)$

be obtained: $${\mathrm{A}}_{\mathrm{1}}=\frac{{\mathrm{v}}_{\mathrm{1}}}{\mathrm{v}}\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}& {\mathrm{a}}_{\mathrm{12}}\\ {\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{22}}\end{array}\right).{\mathrm{A}}_{\mathrm{2}}=\frac{{\mathrm{v}}_{\mathrm{2}}}{\mathrm{v}}\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}& {\mathrm{a}}_{\mathrm{12}}\\ {\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{22}}\end{array}\right).\&\mathrm{c}\mathrm{,}$$

in which $${\mathrm{<figure-callout\; id="1"\; label="Z"\; filenames="imgf0002.png"\; state="\{\{state\}\}">Z</figure-callout>}}_{\mathrm{1}}={\mathrm{<figure-callout\; id="1"\; label="v"\; filenames="imgf0002.png"\; state="\{\{state\}\}">v</figure-callout>}}_{\mathrm{1}}\cdot \mathrm{e}.{\mathrm{Z}}_{\mathrm{2}}={\mathrm{v}}_{\mathrm{2}}\cdot \mathrm{e}.\&\mathrm{c}\mathrm{,}$$

$$\left(\begin{array}{c}{\mathrm{X}}_2\\ {\mathrm{Y}}_2\\ {\mathrm{Z}}_2\end{array}\right)=\frac{{\mathrm{v}}_2}{\mathrm{v}}\cdot \left(\begin{array}{ccc}{\mathrm{a}}_{\mathrm{11}}& {\mathrm{a}}_{\mathrm{12}}& \mathrm{0}\\ {\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{22}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{v}\end{array}\right)\cdot \left(\begin{array}{c}{\mathrm{x}}_2\\ {\mathrm{y}}_2\\ \mathrm{e}\end{array}\right),\&\mathrm{c}$$

bzw. über $$\frac{{\mathrm{v}}_1}{\mathrm{v}}\left(\begin{array}{c}{\mathrm{x}}_1\\ {\mathrm{y}}_1\\ \mathrm{e}\end{array}\right)=\frac{\mathrm{1}}{\left({\mathrm{a}}_{\mathrm{11}}{\mathrm{a}}_{\mathrm{22}}-{\mathrm{a}}_{\mathrm{12}}{\mathrm{a}}_{\mathrm{21}}\right)}\cdot \left(\begin{array}{ccc}{\mathrm{a}}_{\mathrm{22}}& -{\mathrm{a}}_{\mathrm{12}}& \mathrm{0}\\ -{\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{11}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{1}\end{array}\right)\cdot \left(\begin{array}{c}{\mathrm{X}}_1\\ {\mathrm{Y}}_1\\ {\mathrm{Z}}_1\end{array}\right)$$

$$\frac{{\mathrm{v}}_{\mathrm{2}}}{\mathrm{v}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{2}}\\ {\mathrm{y}}_{\mathrm{2}}\\ \mathrm{e}\end{array}\right)=\frac{\mathrm{1}}{\left({\mathrm{a}}_{\mathrm{11}}{\mathrm{a}}_{\mathrm{22}}-{\mathrm{a}}_{\mathrm{12}}{\mathrm{a}}_{\mathrm{21}}\right)}\cdot \left(\begin{array}{ccc}{\mathrm{a}}_{\mathrm{22}}& -{\mathrm{a}}_{\mathrm{12}}& \mathrm{0}\\ -{\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{11}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{1}\end{array}\right)\cdot \left(\begin{array}{c}{\mathrm{X}}_{\mathrm{2}}\\ {\mathrm{Y}}_{\mathrm{2}}\\ {\mathrm{Z}}_{\mathrm{2}}\end{array}\right),\&\mathrm{c}\mathrm{.}$$

In the terminology already used above, A _i = v _i A ', with a matching proportion A', then A '= A / v. The points in the moiré image planes 36, 36 'and the motif plane 32 are connected in analogy to equations (4a), (4b) $$\left(\begin{array}{c}{\mathrm{X}}_1\\ {\mathrm{Y}}_1\\ {\mathrm{<figure-callout\; id="1"\; label="Z"\; filenames="imgf0002.png"\; state="\{\{state\}\}">Z</figure-callout>}}_1\end{array}\right)=\frac{{\mathrm{v}}_1}{\mathrm{v}}\cdot \left(\begin{array}{ccc}{\mathrm{a}}_{\mathrm{11}}& {\mathrm{a}}_{\mathrm{12}}& \mathrm{0}\\ {\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{22}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{v}\end{array}\right)\cdot \left(\begin{array}{c}{\mathrm{x}}_1\\ {\mathrm{y}}_1\\ \mathrm{e}\end{array}\right)$$

$$\left(\begin{array}{c}{\mathrm{X}}_2\\ {\mathrm{Y}}_2\\ {\mathrm{Z}}_2\end{array}\right)=\frac{{\mathrm{v}}_2}{\mathrm{v}}\cdot \left(\begin{array}{ccc}{\mathrm{a}}_{\mathrm{11}}& {\mathrm{a}}_{\mathrm{12}}& \mathrm{0}\\ {\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{22}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{v}\end{array}\right)\cdot \left(\begin{array}{c}{\mathrm{x}}_2\\ {\mathrm{y}}_2\\ \mathrm{e}\end{array}\right).\&\mathrm{c}$$

or over $$\frac{{\mathrm{v}}_1}{\mathrm{v}}\left(\begin{array}{c}{\mathrm{x}}_1\\ {\mathrm{y}}_1\\ \mathrm{e}\end{array}\right)=\frac{\mathrm{1}}{\left({\mathrm{a}}_{\mathrm{11}}{\mathrm{a}}_{\mathrm{22}}-{\mathrm{a}}_{\mathrm{12}}{\mathrm{a}}_{\mathrm{21}}\right)}\cdot \left(\begin{array}{ccc}{\mathrm{a}}_{\mathrm{22}}& -{\mathrm{a}}_{\mathrm{12}}& \mathrm{0}\\ -{\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{11}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{1}\end{array}\right)\cdot \left(\begin{array}{c}{\mathrm{X}}_1\\ {\mathrm{Y}}_1\\ {\mathrm{Z}}_1\end{array}\right)$$

$$\frac{{\mathrm{v}}_{\mathrm{2}}}{\mathrm{v}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{2}}\\ {\mathrm{y}}_{\mathrm{2}}\\ \mathrm{e}\end{array}\right)=\frac{\mathrm{1}}{\left({\mathrm{a}}_{\mathrm{11}}{\mathrm{a}}_{\mathrm{22}}-{\mathrm{a}}_{\mathrm{12}}{\mathrm{a}}_{\mathrm{21}}\right)}\cdot \left(\begin{array}{ccc}{\mathrm{a}}_{\mathrm{22}}& -{\mathrm{a}}_{\mathrm{12}}& \mathrm{0}\\ -{\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{11}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{1}\end{array}\right)\cdot \left(\begin{array}{c}{\mathrm{X}}_{\mathrm{2}}\\ {\mathrm{Y}}_{\mathrm{2}}\\ {\mathrm{Z}}_{\mathrm{2}}\end{array}\right).\&\mathrm{c}\mathrm{,}$$

$${\mathrm{U}}_{\mathrm{2}}=\left(\begin{array}{cc}{{\mathrm{u}}^{\left(\mathrm{2}\right)}}_{\mathrm{11}}& {{\mathrm{u}}^{\left(\mathrm{2}\right)}}_{\mathrm{12}}\\ {{\mathrm{u}}^{\left(\mathrm{2}\right)}}_{\mathrm{21}}& {{\mathrm{u}}^{\left(\mathrm{2}\right)}}_{\mathrm{22}}\end{array}\right)=\left(\left(\begin{array}{cc}\mathrm{1}& \mathrm{0}\\ \mathrm{0}& \mathrm{1}\end{array}\right)-\frac{\mathrm{v}}{{\mathrm{v}}_{\mathrm{2}}}{\mathrm{A}}^{-\mathrm{1}}\right)\cdot \left(\begin{array}{cc}{\mathrm{w}}_{\mathrm{11}}& {\mathrm{w}}_{\mathrm{12}}\\ {\mathrm{w}}_{\mathrm{21}}& {\mathrm{w}}_{\mathrm{22}}\end{array}\right),\&\mathrm{c}\mathrm{.}$$

The respective motif grids U ₁ , U ₂ ,... Result from the lenticular grid W and the transformation matrices A ₁ , A ₂ ... By means of relationship (M ₂ ) $${\mathrm{<figure-callout\; id="1"\; label="U"\; filenames="imgf0002.png"\; state="\{\{state\}\}">U</figure-callout>}}_{\mathrm{1}}=\left(\begin{array}{cc}{{\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="imgf0002.png"\; state="\{\{state\}\}">u</figure-callout>}}^{\left(\mathrm{1}\right)}}_{\mathrm{11}}& {{\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="imgf0002.png"\; state="\{\{state\}\}">u</figure-callout>}}^{\left(\mathrm{1}\right)}}_{\mathrm{12}}\\ {{\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="imgf0002.png"\; state="\{\{state\}\}">u</figure-callout>}}^{\left(\mathrm{1}\right)}}_{\mathrm{21}}& {{\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="imgf0002.png"\; state="\{\{state\}\}">u</figure-callout>}}^{\left(\mathrm{1}\right)}}_{\mathrm{22}}\end{array}\right)=\left(\left(\begin{array}{cc}\mathrm{1}& \mathrm{0}\\ \mathrm{0}& \mathrm{1}\end{array}\right)-\frac{\mathrm{v}}{{\mathrm{v}}_{\mathrm{1}}}{\mathrm{A}}^{-\mathrm{1}}\right)\cdot \left(\begin{array}{cc}{\mathrm{w}}_{\mathrm{11}}& {\mathrm{w}}_{\mathrm{12}}\\ {\mathrm{w}}_{\mathrm{21}}& {\mathrm{w}}_{\mathrm{22}}\end{array}\right)$$

$${\mathrm{U}}_{\mathrm{2}}=\left(\begin{array}{cc}{{\mathrm{u}}^{\left(\mathrm{2}\right)}}_{\mathrm{11}}& {{\mathrm{u}}^{\left(\mathrm{2}\right)}}_{\mathrm{12}}\\ {{\mathrm{u}}^{\left(\mathrm{2}\right)}}_{\mathrm{21}}& {{\mathrm{u}}^{\left(\mathrm{2}\right)}}_{\mathrm{22}}\end{array}\right)=\left(\left(\begin{array}{cc}\mathrm{1}& \mathrm{0}\\ \mathrm{0}& \mathrm{1}\end{array}\right)-\frac{\mathrm{v}}{{\mathrm{v}}_{\mathrm{2}}}{\mathrm{A}}^{-\mathrm{1}}\right)\cdot \left(\begin{array}{cc}{\mathrm{w}}_{\mathrm{11}}& {\mathrm{w}}_{\mathrm{12}}\\ {\mathrm{w}}_{\mathrm{21}}& {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{\mathrm{22}}\end{array}\right).\&\mathrm{c}\mathrm{,}$$

In order to construct a motif image for a given three-dimensional moiré image, it is therefore possible according to the invention to proceed as follows:

In addition to the lenticular grid W, for a reference point X, Y, Z of the desired three-dimensional moiré image, the transformation matrix A and a tilting direction γ are provided, under which the parallax is to be observed.

For these specifications, calculate a magnification factor v using equation (3c). For other points of the moiré image, such as a general point X _i, Y _i, Z _i, is then determined according to formula (6b), the magnification factor v _i for the Z-component Z _i and the point coordinates in the image plane x _i, y _i , and according to formula (7) from the given lenticular grid W, the transformation matrix A and the magnification factor v _i the associated grid arrangement U _i .

Here, since occur depending on the position of X _i, Y _i, Z _i v _i different magnifications, it can happen that parts of the image does not fit into a grid cell of the motif grid U _i. In this case, according to the teaching of the published German patent application entitled "security element", DE 10 2007 029 203.3 proceeded, which relates to the division of a given motif element on multiple grid cells.

In particular, in this case, to generate a micro-optical moiré magnification arrangement for displaying a moiré image with one or more moiré picture elements, a motif image is formed in one motif plane periodic or at least locally periodic arrangement of a plurality of grid cells with micromotif images generated parts and a Fokussierelementraster moire-magnified viewing the motif image with a periodic or at least locally periodic arrangement of a plurality of grid cells each with a Mikrofokussierelement and arranged spaced from the motif image. In this case, the micromotif image parts are formed in such a way that the micromotif image parts of a plurality of spaced grid cells of the motif image taken together form in each case a micromotif element that corresponds to one of the moiré image elements of the enlarged moire image and whose extent is greater than a grid cell of the motif image.

In the document WO 2007-076952 Moire magnifiers are described with a cylindrical lens grid and / or with randomly extended in one direction motifs. Moire magnifiers of this kind can also be implemented as a 3D Moire Magnifier.

wobei D der Zylinderlinsenabstand und φ der Neigungswinkel der Zylinderlinsen und u_ij die Matrixelemente der Motivrastermatrix sind.As described in the document WO 2007-076952 A2 in the cylinder lens 3D moiré magnifier for the sub-matrix (a _ij ) in formula (6a), the relationship holds: $$\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}& {\mathrm{a}}_{\mathrm{12}}\\ {\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{22}}\end{array}\right)=\frac{1}{D-u_{11}\cos \phi -u_{21}\sin \phi }\left(\begin{array}{cc}D-u_{21}\sin \phi & u_{11}\sin \phi \\ u_{21}\cos \phi & D-u_{11}\cos \phi \end{array}\right).$$

where D is the cylinder lens pitch and φ is the tilt angle of the cylinder lenses and u _{ij are} the matrix elements of the motif grid matrix.

wobei (u₁₁, u₂₁) der Translationsvektor für das ausgedehnte Motiv ist.In the 3D moire magnifier with extended motifs, the sub-matrix (a _ij ) in formula (6a) takes the form: $$\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}& {\mathrm{a}}_{\mathrm{12}}\\ {\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{22}}\end{array}\right)=\frac{\mathrm{1}}{\mathrm{Det}\left(\mathrm{W}-\mathrm{U}\right)}\left(\begin{array}{cc}\mathrm{Det}\mathrm{W}+{\mathrm{u}}_{\mathrm{21}}{\mathrm{w}}_{\mathrm{12}}& -{\mathrm{u}}_{\mathrm{11}}{\mathrm{w}}_{\mathrm{12}}\\ {\mathrm{u}}_{\mathrm{21}}{\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{\mathrm{22}}& \mathrm{Det}\mathrm{W}-{\mathrm{u}}_{\mathrm{11}}{\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{\mathrm{22}}\end{array}\right)\mathit{With}U=\left(\begin{array}{cc}{\mathrm{u}}_{\mathrm{11}}& 0\\ {\mathrm{u}}_{21}& 0\end{array}\right)$$

where (u ₁₁ , u ₂₁ ) is the translation motif for the extended motif.

Examples

To illustrate the approach of the invention, some concrete exemplary designs will now be described. In addition shows Fig. 6 (a) a simple three-dimensional motif 60 in the form of a cut out of a plate letter "P". Fig. 6 (b) shows a representation of this motif through only two parallel image planes containing the top 62 and the bottom 64 of the three-dimensional letter motif, Fig. 6 (c) shows the representation of the motif by five parallel cutting planes and with five sectional images 66 of the letter motif.

Since it is already possible to explain in a vivid manner all essential inventive method steps based on a three-dimensional motif shown in only two image planes, the following examples are corresponding to such motifs Fig. 6 (b) designed. However, it is not difficult for a person skilled in the art to extend the method to a larger number of image planes, such as Fig. 6 (c) or quasi-continuously Fig. 6 (a) perform. Particularly in the case of complex moiré images, it is in most cases advantageous to start not from patches but from individual pixels of the three-dimensional moiré image as the image components to be displayed and, as described above in the description of the equations (6a), (6b) and (7) in general, to determine for each of these moiré pixels an associated micromotif pixel and a grid cell array for repeating the micromotif pixel in the motif plane. In practice, the number of image planes used or the number of pixels to be displayed will also depend in particular on the complexity of the desired three-dimensional subject.

Example 1:

Fig. 7 shows an embodiment for which a hexagonal lenticular grid W is specified. As a three-dimensional motif to be displayed, an O-shaped ring is selected, which, as in Fig. 6 (b) is described in two image planes by a Letter Top and Letter Bottom.

vorgegeben, die eine reine Vergrößerung beschreiben, wobei der Vergrößerungsfaktor für die Oberseitenflächen v₁ = 16 und der Vergrößerungsfaktor für die Unterseitenflächen v₂ =19 betragen soll.As transformation matrices A _i are the matrices $${\mathrm{A}}_{\mathrm{i}}={\mathrm{v}}_{\mathrm{i}}\cdot \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$$

given, which describe a pure magnification, wherein the magnification factor for the top surfaces v ₁ = 16 and the magnification factor for the bottom surfaces v ₂ = 19 should be.

With a desired motif size of 50 mm, an effective lens image width of e = 4 mm and a lens pitch of 5 mm in the hexagonal lenticular, one obtains the motif size for the top surfaces using the above-described relationships (6b) and (7) for the motif size Value of 50mm / 16 = 3.1mm and for the bottom surfaces a value of 50mm / 19 = 2.63mm.

The grid spacing of the motif grid for the top surfaces (1 - 1/16) * 5 mm = 4.69 mm and for the bottom surfaces (1 -1/19) * 5 mm = 4.74 mm. The perceived thickness of the three-dimensional moiré image is (19 -16) * 4 mm = 12 mm.

Fig. 7 (a) shows the thus constructed motif image 70, in which the different screen widths of the two micromotif elements "ring top" and "ring bottom" are clearly visible. Will the motif image 70 of Fig. 7 (a) With the said hexagonal lenticular grid, the result is a three-dimensional moiré image 72 hovering below the moiré magnification arrangement, of which FIG Fig. 7 (b) a section is shown schematically.

In the moiré image 72, a plurality of juxtaposed rings 74, 76 can be seen. If you look at the arrangement from the front, you can see the middle ring 74 from the front and the surrounding rings 76 obliquely from the corresponding side. If you tilt the arrangement, you can see the middle ring 74 obliquely from the side, the adjacent rings 76 change according to their perspective.

Example 2:

Fig. 8 shows an embodiment with orthoparallaktischer movement, for which a rectangular lens grid W is selected. As a three-dimensional motif to be displayed is a sawn from a plate letter "P", as in Fig. 6 shown.

vorgegeben, die neben einer Vergrößerung um einen Faktor v_i ein orthoparallaktisches Bewegungsverhalten beim Kippen der Moire-Vergrößerungsanordnung beschreiben.As transformation matrices A _i are the matrices $${\mathrm{A}}_{\mathrm{i}}={\mathrm{v}}_{\mathrm{i}}\cdot \left(\begin{array}{cc}\mathrm{0}& \mathrm{1}\\ \mathrm{1}& \mathrm{0}\end{array}\right)$$

predetermined, in addition to an increase by a factor v _{i describe} a orthoparallaktisches movement behavior when tilting the moiré magnification arrangement.

dar, Gleichung (7) in der Form $${\mathrm{U}}_{\mathrm{i}}=\left(\begin{array}{cc}{{\mathrm{u}}^{\left(\mathrm{i}\right)}}_{\mathrm{11}}& {{\mathrm{u}}^{\left(\mathrm{i}\right)}}_{\mathrm{12}}\\ {{\mathrm{u}}^{\left(\mathrm{i}\right)}}_{\mathrm{21}}& {{\mathrm{u}}^{\left(\mathrm{i}\right)}}_{\mathrm{22}}\end{array}\right)=\left(\mathrm{I}-{{\mathrm{A}}_{\mathrm{i}}}^{-\mathrm{1}}\right)\cdot \left(\begin{array}{cc}{\mathrm{w}}_{\mathrm{11}}& {\mathrm{w}}_{\mathrm{12}}\\ {\mathrm{w}}_{\mathrm{21}}& {\mathrm{w}}_{\mathrm{22}}\end{array}\right)\mathrm{mit}{{\mathrm{A}}_{\mathrm{i}}}^{-\mathrm{1}}=\frac{\mathrm{1}}{{\mathrm{v}}_{\mathrm{i}}}\cdot \left(\begin{array}{cc}\mathrm{0}& \mathrm{1}\\ \mathrm{1}& \mathrm{0}\end{array}\right)\mathrm{.}$$

Equation (6a) then arises in the form $$\left(\begin{array}{c}{\mathrm{X}}_{\mathrm{i}}\\ {\mathrm{Y}}_{\mathrm{i}}\\ {\mathrm{Z}}_{\mathrm{i}}\end{array}\right)=\left(\begin{array}{ccc}\mathrm{0}& {\mathrm{v}}_{\mathrm{i}}& \mathrm{0}\\ {\mathrm{v}}_{\mathrm{i}}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& {\mathrm{v}}_{\mathrm{i}}\end{array}\right)\cdot \left(\begin{array}{c}{\mathrm{x}}_{\mathrm{i}}\\ {\mathrm{y}}_{\mathrm{i}}\\ \mathrm{e}\end{array}\right)$$

and equation (7) in the form $${\mathrm{U}}_{\mathrm{i}}=\left(\begin{array}{cc}{{\mathrm{u}}^{\left(\mathrm{i}\right)}}_{\mathrm{11}}& {{\mathrm{u}}^{\left(\mathrm{i}\right)}}_{\mathrm{12}}\\ {{\mathrm{u}}^{\left(\mathrm{i}\right)}}_{\mathrm{21}}& {{\mathrm{u}}^{\left(\mathrm{i}\right)}}_{\mathrm{22}}\end{array}\right)=\left(\mathrm{I}-{{\mathrm{A}}_{\mathrm{i}}}^{-\mathrm{1}}\right)\cdot \left(\begin{array}{cc}{\mathrm{w}}_{\mathrm{11}}& {\mathrm{w}}_{\mathrm{12}}\\ {\mathrm{w}}_{\mathrm{21}}& {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{\mathrm{22}}\end{array}\right)\mathrm{With}{{\mathrm{A}}_{\mathrm{i}}}^{-\mathrm{1}}=\frac{\mathrm{1}}{{\mathrm{v}}_{\mathrm{i}}}\cdot \left(\begin{array}{cc}\mathrm{0}& \mathrm{1}\\ \mathrm{1}& \mathrm{0}\end{array}\right)\mathrm{,}$$

In this embodiment, the magnification factor for the top surfaces v ₁ = 8 and the magnification factor for the bottom surfaces v ₂ = 10 should be. The desired motif size (letter height) is 35 mm, the effective lens image size again e = 4 mm and the lens spacing in the right-angled lenticular grid should be 5 mm.

Using the relationships (6b) and (7), the motif size for the motif surface has a value of 35 mm / 8 = 4.375 mm for the motif surface and 35 mm / 10 = 3.5 mm for the underside surfaces.

das Motivraster U₂ für die Unterseitenflächen zu $${\mathrm{U}}_2=\left(\begin{array}{cc}5& -0,5\\ -0,5& 5\end{array}\right)\mathrm{.}$$

The motif grid U ₁ for the top surfaces results to $${\mathrm{<figure-callout\; id="1"\; label="U"\; filenames="imgf0002.png"\; state="\{\{state\}\}">U</figure-callout>}}_1=\left(\begin{array}{cc}5& -0.625\\ -0.625& 5\end{array}\right).$$

the motif grid U ₂ for the underside surfaces too $${\mathrm{U}}_2=\left(\begin{array}{cc}5& -0.5\\ -0.5& 5\end{array}\right)\mathrm{,}$$

The motif elements that are created in these grids are twisted and mirrored as usual by the transformation A ^-1 compared to the desired target motif. The perceived thickness of the three-dimensional moiré image is (10-8) * 4 mm = 8 mm.

Fig. 8 (a) shows the thus constructed motif image 80, in which the two different motif grid U ₁ , U _{2 of} the two micromotif elements "letter top" and "letter bottom" are clearly visible. Will the motif image 80 of Fig. 8 (a) With the said right-angled lenticular grid, the result is a three-dimensional moiré image 82 hovering above the moiré magnification arrangement, of which in Fig. 8 (b) a section is shown schematically.

If you tilt the moiré magnification arrangement horizontally (tilting direction 84), you see the motif from above or from below, tilting the arrangement vertically (tilting direction 86), you see the motif on the side, so that the impression is created, the motif be spatially extended and lie in the depth.

Through binocular vision, however, this impression of depth is not confirmed, since there is no x-component for lateral movement, the motif remains in the substrate plane. This perceptual contradiction is extreme striking and thus has a high attention and recognition value for the viewer.

Example 3:

The embodiment of Fig. 9 goes like the embodiment of the Fig. 8 from a letter "P" cut from a plate as the three-dimensional motif to be displayed. This motif should move obliquely in this embodiment when tilting the moiré magnification arrangement.

vorgegeben, die neben einer Vergrößerung um den Faktor v_i ein schräges Bewegungsverhalten beim Kippen der Moire-Vergrößerungsanordnung beschreiben.As transformation matrices A _i are the matrices $${\mathrm{A}}_{\mathrm{i}}={\mathrm{v}}_{\mathrm{i}}\cdot \left(\begin{array}{cc}1& 0\\ 1& 1\end{array}\right).$$

predetermined, in addition to an increase by the factor v _{i describe} an oblique movement behavior when tilting the moiré magnification arrangement.

dar, Gleichung (7) in der Form $${\mathrm{U}}_{\mathrm{i}}=\left(\begin{array}{cc}{{\mathrm{u}}^{\left(\mathrm{i}\right)}}_{\mathrm{11}}& {{\mathrm{u}}^{\left(\mathrm{i}\right)}}_{\mathrm{12}}\\ {{\mathrm{u}}^{\left(\mathrm{i}\right)}}_{\mathrm{21}}& {{\mathrm{u}}^{\left(\mathrm{i}\right)}}_{\mathrm{22}}\end{array}\right)=\left(\mathrm{I}-{{\mathrm{A}}_{\mathrm{i}}}^{-\mathrm{1}}\right)\cdot \left(\begin{array}{cc}{\mathrm{w}}_{\mathrm{11}}& {\mathrm{w}}_{\mathrm{12}}\\ {\mathrm{w}}_{\mathrm{21}}& {\mathrm{w}}_{\mathrm{22}}\end{array}\right)\mathrm{mit}{{\mathrm{A}}_{\mathrm{i}}}^{-\mathrm{1}}=\frac{\mathrm{1}}{{\mathrm{v}}_{\mathrm{i}}}\cdot \left(\begin{array}{cc}1& 0\\ -\mathrm{1}& 1\end{array}\right)\mathrm{.}$$

Equation (6a) then arises in the form $$\left(\begin{array}{c}{\mathrm{X}}_{\mathrm{i}}\\ {\mathrm{Y}}_{\mathrm{i}}\\ {\mathrm{Z}}_{\mathrm{i}}\end{array}\right)=\left(\begin{array}{ccc}{\mathrm{v}}_{\mathrm{i}}& 0& \mathrm{0}\\ {\mathrm{v}}_{\mathrm{i}}& {\mathrm{v}}_{\mathrm{i}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& {\mathrm{v}}_{\mathrm{i}}\end{array}\right)\cdot \left(\begin{array}{c}{\mathrm{x}}_{\mathrm{i}}\\ {\mathrm{y}}_{\mathrm{i}}\\ \mathrm{e}\end{array}\right)$$

and equation (7) in the form $${\mathrm{U}}_{\mathrm{i}}=\left(\begin{array}{cc}{{\mathrm{u}}^{\left(\mathrm{i}\right)}}_{\mathrm{11}}& {{\mathrm{u}}^{\left(\mathrm{i}\right)}}_{\mathrm{12}}\\ {{\mathrm{u}}^{\left(\mathrm{i}\right)}}_{\mathrm{21}}& {{\mathrm{u}}^{\left(\mathrm{i}\right)}}_{\mathrm{22}}\end{array}\right)=\left(\mathrm{I}-{{\mathrm{A}}_{\mathrm{i}}}^{-\mathrm{1}}\right)\cdot \left(\begin{array}{cc}{\mathrm{w}}_{\mathrm{11}}& {\mathrm{w}}_{\mathrm{12}}\\ {\mathrm{w}}_{\mathrm{21}}& {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{\mathrm{22}}\end{array}\right)\mathrm{With}{{\mathrm{A}}_{\mathrm{i}}}^{-\mathrm{1}}=\frac{\mathrm{1}}{{\mathrm{v}}_{\mathrm{i}}}\cdot \left(\begin{array}{cc}1& 0\\ -\mathrm{1}& 1\end{array}\right)\mathrm{,}$$

Also in this embodiment, the magnification factor for the top surfaces v ₁ = 8 and the magnification factor for the bottom surfaces v ₂ = 10, the desired motif size (letter height) should be 35 mm, the effective lens image width e = 4 mm and the lens distance in the also right-angled lenticular grid 5 mm.

Using the relationships (6b) and (7), the motif size for the motif surface has a value of 35 mm / 8 = 4.375 mm for the motif pattern and 35 mm / 10 = 3.5 mm for the underside surfaces.

das Motivraster U₂ für die Unterseitenflächen zu $${\mathrm{U}}_2=\left(\begin{array}{cc}4,5& 0\\ 0,5& 4,5\end{array}\right)\mathrm{.}$$

The motif grid U ₁ for the top surfaces results to $${\mathrm{<figure-callout\; id="1"\; label="U"\; filenames="imgf0002.png"\; state="\{\{state\}\}">U</figure-callout>}}_1=\left(\begin{array}{cc}4.375& 0\\ 0.625& 4.375\end{array}\right).$$

the motif grid U ₂ for the underside surfaces too $${\mathrm{U}}_2=\left(\begin{array}{cc}4.5& 0\\ 0.5& 4.5\end{array}\right)\mathrm{,}$$

verzehrt. Die wahrgenommene Dicke des dreidimensionalen Moiré-Bilds beträgt (10-8) * 4 mm = 8 mm.The motif elements that are created in these grids are compared to the desired target motif as usual by the transformation ${{\mathrm{A}}_{\mathrm{i}}}^{-\mathrm{1}}=\frac{\mathrm{1}}{{\mathrm{v}}_{\mathrm{i}}}\cdot \left(\begin{array}{cc}\mathrm{1}& \mathrm{0}\\ -\mathrm{1}& \mathrm{1}\end{array}\right)$

consumed. The perceived thickness of the three-dimensional moiré image is (10-8) * 4 mm = 8 mm.

Fig. 9 (a) shows the thus constructed motif image 90, in which the two different motif grid U ₁ , U _{2 of} the two micromotif elements "letter top" and "letter bottom" and the distortion of the motif elements are clearly visible.

Will the motif picture 90 of the Fig. 9 (a) With the said right-angled lenticular grid, the result is a three-dimensional moiré image 92 hovering below the moiré magnification arrangement, of which FIG Fig. 9 (b) a section is shown schematically.

If you tilt the Moire magnification arrangement horizontally, you can see diagonally at an angle of 45 ° to the subject. If one tilts the arrangement vertically, one sees from above or below on the motive, so that the impression arises, the motive is spatially extended and lies in the depth. Through binocular vision, the depth impression is not fully confirmed. The subject is not as deep as the tilting effect pretends to this depth impression, because for the impression of depth in binaural vision, only the x-component of the oblique motion acts.

Example 4:

Example 4 is a modification of Example 3 and is dimensioned to be particularly suitable for security threads of banknotes.

The moire image used (letter "P") and the transformation matrices A _i correspond to those of example 3. In this embodiment, however, the magnification factors for the top surfaces are v ₁ = 80 and for the bottom surfaces v ₂ = 100, the motif size (letter height ) should be 3 mm. E = 0.04 mm is chosen as the effective lens image width and 0.04 mm as the lens pitch in the rectangular lenticular grid.

Again, using relationships (6b) and (7), the motif size for the top surface areas in the motif grid is 3 mm / 80 = 0.0375 mm and the bottom surface is 3 mm / 100 = 0 , 03 mm.

das Motivraster U₂ für die Unterseitenflächen zu $${\mathrm{U}}_2=\left(\begin{array}{cc}0,0396& 0\\ 0,0004& 0,0396\end{array}\right)\mathrm{.}$$

The motif grid U ₁ for the top surfaces results to $${\mathrm{<figure-callout\; id="1"\; label="U"\; filenames="imgf0002.png"\; state="\{\{state\}\}">U</figure-callout>}}_1=\left(\begin{array}{cc}0.0395& 0\\ 0.0005& 0.0395\end{array}\right).$$

the motif grid U ₂ for the underside surfaces too $${\mathrm{U}}_2=\left(\begin{array}{cc}0.0396& 0\\ 0.0004& 0.0396\end{array}\right)\mathrm{,}$$

verzerrt.The motif elements that are created in these grids are also compared to the desired target motif by the transformation ${{\mathrm{A}}_{\mathrm{i}}}^{-\mathrm{1}}=\frac{\mathrm{1}}{{\mathrm{v}}_{\mathrm{i}}}\cdot \left(\begin{array}{cc}\mathrm{1}& \mathrm{0}\\ -\mathrm{1}& \mathrm{1}\end{array}\right)$

distorted.

The perceived thickness of the three-dimensional moiré image is (100-80) * 0.04 mm = 0.8 mm.

If the user tilts a banknote horizontally with a correspondingly equipped security thread, he sees it at an angle of 45 ° to the motif. If he tilts the arrangement vertically, he sees from above or below on the subject, so that the impression arises that the motif is spatially extended and lying in the depth. Through binocular vision, the depth impression is not fully confirmed. The subject is not as deep as the tilting effect pretends to this depth impression, because for the impression of depth in binaural vision, only the x-component of the oblique motion acts.

This contradiction of depth perception is extremely striking and thus has a high attention and recognition value for the viewer.

As with the description of Fig. 4 As already mentioned, different Z values in the case of a three-dimensional moiré image can also be achieved by realizing different values for the effective distance e between the lens plane and the motif plane at constant moiré magnification.

The coming of different magnifications is in Fig. 10 illustrating two motif planes 32, 32 'provided at different depths d ₁ , d _{2 of} the moiré magnification arrangement. Dashed arrows 50 are shown in the motif plane 32 as the first micromotif elements, and arrows 52 in the lower-lying motif plane 32 'are drawn through as second micromotif elements. Both the first and the second micromotif elements 50, 52 are arranged in the same motif grid U with grating period u.

The resulting enlarged moiré images 54 and 56, respectively, therefore appear to the viewer 38 at the same magnification factor v due to the coincident grating periods, so that the arrows 50, 52 are made equally long for enlarged arrow images 54 and 56 of equal length.

The different flying height Z ₁ , or Z ₂ above the plane of the moiré magnification arrangement results in this embodiment from the different distance d ₁ , d ₂ and thus also a different effective distance e ₁ , e ₂ between the lens plane 34 and the motif plane 32 or 32 ': $${\mathrm{<figure-callout\; id="1"\; label="Z"\; filenames="imgf0002.png"\; state="\{\{state\}\}">Z</figure-callout>}}_{\mathrm{1}}=\mathrm{v}*{\mathrm{<figure-callout\; id="1"\; label="e"\; filenames="imgf0002.png"\; state="\{\{state\}\}">e</figure-callout>}}_{\mathrm{1}}.{\mathrm{Z}}_{\mathrm{2}}=\mathrm{v}*{\mathrm{e}}_{\mathrm{2}}\mathrm{,}$$

Such a design can be realized with motif elements 50, 52 at different depths, for example by embossing the corresponding structures in a lacquer layer. The effective distances e ₁ , e ₂ effective for the flying height Z can in each case be determined from the physical distances d ₁ , d ₂ , the refractive index of the optical distance layer and the lens material and the lens focal length.

Analogous to Fig. 5 applies the representation of Fig. 10 in which the moiré images hover over the magnification arrangement, for negative magnification factors, at positive magnification factors the moire images appear to float for the viewer below the plane of the moiré magnification arrangement.

Claims (22)

1. A security element for security papers, value documents and the like, having a microoptical moire magnification arrangement (30) for depicting a three-dimensional moiré image (40) that includes, in at least two moiré image planes spaced apart in a direction normal to the moiré magnification arrangement, image components (42,44) to be depicted, having

   - a motif image that includes two or more periodic or at least locally periodic lattice cell arrangements having different lattice periods (p₁, p₂) and/or different lattice orientations that are each allocated to one moiré image plane (36, 36') and that include micromotif image components for depicting the image component (42,44) of the allocated moiré image plane (36,36'),

   - for the moire-magnified viewing of the motif image, a focusing element grid that is arranged spaced apart from the motif image and that includes a periodic or at least locally periodic arrangement of a plurality of lattice cells having one microfocusing element each,

   wherein, for almost all tilt directions ( k ), upon tilting the security element, the magnified, three-dimensional moiré image (40) moves in a moiré movement direction ( v ) that differs from the tilt direction.
2. The security element according to claim 1, characterized in that, for the viewer, due to the parallax upon tilting the security element, the three-dimensional moiré image appears floating at a first height or depth above or below the plane of the security element, and due to the eye separation in binocular vision, at a second height or depth above or below the plane of the security element, the first and second height or depth differing for almost all viewing directions.
3. The security element according to claim 1 or 2, characterized in that both the lattice cell arrangements of the motif image and the lattice cells of the focusing element grid are arranged periodically, or that, locally, both the lattice cell arrangements of the motif image and the lattice cells of the focusing element grid are arranged periodically, the local period parameters changing only slowly in relation to the periodicity length.
4. The security element according to at least one of claims 1 to 3, characterized in that the lattice cell arrangements of the motif image and the lattice cells of the focusing element grid form, at least locally, one two-dimensional Bravais lattice each.
5. The security element according to at least one of claims 1 to 4, characterized in that the microfocusing elements are formed by non-cylindrical microlenses or concave microreflectors, especially by microlenses or concave microreflectors having a circular or polygonally delimited base area, or that the microfocusing elements are formed by elongated cylindrical lenses or concave cylindrical reflectors whose dimension in the longitudinal direction measures more than 250 µm, preferably more than 300 µm, particularly preferably more than 500 µm and especially more than 1 mm.
6. The security element according to at least one of claims 1 to 5, characterized in that the micromotif image components are present in a printing layer.
7. A security element for security papers, value documents and the like, having a microoptical moiré magnification arrangement (30) for depicting a three-dimensional moiré image (40) that includes, in at least two moire image planes (36, 36') spaced apart in a direction normal to the moiré magnification arrangement, image components (42, 44) to be depicted, having

   - a motif image that includes, arranged at different heights periods (d₁, d₂), two or more periodic or at least locally periodic lattice cell arrangements that are each allocated to one moiré image plane (36, 36') and that include micromotif image components for depicting the image component (42,44) of the allocated moiré image plane (36, 36'),

   - for the moiré-magnified viewing of the motif image, a focusing element grid that is arranged spaced apart from the motif image and that includes a periodic or at least locally periodic arrangement of a plurality of lattice cells having one microfocusing element each,

   wherein, for almost all tilt directions ( k ), upon tilting the security element, the magnified, three-dimensional moiré image (40) moves in a moiré movement direction ( v ) that differs from the tilt direction.
8. The security element according to claim 7, characterized in that the lattice cell arrangements of the motif image exhibit identical lattice periods and identical lattice orientations, and/or the micromotif image components are present in an embossing layer at different embossing heights.
9. The security element according to at least one of claims 1 to 8, characterized in that the security element exhibits an opaque cover layer to cover the moiré magnification arrangement in some regions.
10. A method for manufacturing a security element having a microoptical moiré magnification arrangement (30) for depicting a three-dimensional moiré image (40) that includes, in at least two moiré image planes (36, 36') spaced apart in a direction normal to the moiré magnification arrangement, image components (42, 44) to be depicted, in which

    - in a motif plane (32), a motif image is produced that includes two or more periodic or at least locally periodic lattice cell arrangements having different lattice periods (p₁, p₂) and/or different lattice orientations that are each allocated to one moiré image plane (36, 36') and that are provided with micromotif image components for depicting the image component (42,44) of the allocated moiré image plane (36, 36'),

    - a focusing element grid for the moiré-magnified viewing of the motif image, having a periodic or at least locally periodic arrangement of a plurality of lattice cells having one microfocusing element each, is produced and arranged spaced apart from the motif image,

    the lattice cell arrangements of the motif plane (32), the micromotif image components and the focusing element grid being coordinated such that, for almost all tilt directions ( k ), upon tilting the security element, the magnified, three-dimensional moiré image (40) moves in a moiré movement direction ( v ) that differs from the tilt direction.
11. A method for manufacturing a security element having a microoptical moiré magnification arrangement (30) for depicting a three-dimensional moiré image (40) that includes, in at least two moiré image planes (36, 36') spaced apart in a direction normal to the moiré magnification arrangement, image components (42, 44) to be depicted, in which

    - a motif image is produced having, arranged at different heights (d₁, d₂), two or more motif planes (32, 32') that each include a periodic or at least locally periodic lattice cell arrangement that is allocated to one moiré image plane (36, 36') and that is provided with micromotif image components for depicting the image component (42,44) of the allocated moiré image plane (36, 36'),

    - a focusing element grid for the moiré-magnified viewing of the motif image, having a periodic or at least locally periodic arrangement of a plurality of lattice cells having one microfocusing element each, is produced and arranged spaced apart from the motif image,

    the lattice cell arrangements of the motif planes (32, 32'), the micromotif image components and the focusing element grid being coordinated such that, for almost all tilt directions ( k ), upon tilting the security element, the magnified, three-dimensional moiré image (40) moves in a moiré movement direction ( v ) that differs from the tilt direction.
12. The method according to claim 11, characterized in that the lattice cell arrangements of the motif planes are produced having identical lattice periods and identical lattice orientations, and/or that the motif image is embossed to produce micromotif image components at different embossing heights.
13. A method for manufacturing a security element having a microoptical moiré magnification arrangement (30) for depicting a three-dimensional moiré image (40) that includes, in at least two moiré image planes (36, 36') spaced apart in a direction normal to the moiré magnification arrangement, image components (42, 44) to be depicted, in which

    a) a desired three-dimensional moiré image (40) that is visible when viewed is defined as the target motif,

    b) a periodic or at least locally periodic arrangement of microfocusing elements is defined as the focusing element grid,

    c) a desired magnification and a desired movement of the visible three-dimensional moiré image when the moiré magnification arrangement (30) is tilted laterally and when tilted forward/backward is defined,

    d) for each image component (42, 44) to be depicted, the associated micromotif image component for depicting this image component (42, 44) of the three-dimensional moiré image (40), as well as the associated lattice cell arrangement for the arrangement of the micromotif image components in the motif plane (32), are calculated from the spacing of the associated moiré image plane (36, 36') from the moiré magnification arrangement (30), the defined magnification and movement behavior, and the focusing element grid, and

    e) the micromotif image components calculated for each image component (42,44) to be depicted are composed to form a motif image that is to be arranged in the motif plane (32) according to the associated lattice cell arrangement.
14. The method according to claim 13, characterized in that, in step c), further, for a reference point of the three-dimensional moiré image, a tilt direction γ is specified in which the parallax is to be viewed, and a desired magnification and movement behavior for this reference point and the specified tilt direction, and in that, for the other points of the three-dimensional moiré image, the moire magnification factors in step d) are based on the specified magnification factor for the reference point and the specified tilt direction.
15. The method according to claim 14, characterized in that the desired magnification and movement behavior for the reference point is specified in the form of the matrix elements of a transformation matrix $\mathrm{A}=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}& {\mathrm{a}}_{\mathrm{12}}\\ {\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{22}}\end{array}\right)$

    

    and the magnification factor for the reference point is calculated from the transformation matrix A and the tilt direction γ using the relationship $$\mathrm{v}=\sqrt{{{\mathrm{v}}_x}^{\mathrm{2}}+{{\mathrm{v}}_y}^{\mathrm{2}}}=\sqrt{{\left({\mathrm{a}}_{\mathrm{11}}\cos \gamma +{\mathrm{a}}_{\mathrm{12}}\sin \gamma \right)}^2+{\left({\mathrm{a}}_{21}\cos \gamma +{\mathrm{a}}_{22}\sin \gamma \right)}^2}\mathrm{.}$$

    
16. The method according to claim 15, characterized in that, in step d), for further points (X_i, Y_i, Z_i) of the three-dimensional moiré image, the magnification factors v_i and the associated point coordinates in the motif plane (x_i, y_i) are calculated using the relationship $$\left(\begin{array}{c}{\mathrm{X}}_{\mathrm{i}}\\ {\mathrm{Y}}_{\mathrm{i}}\\ {\mathrm{Z}}_{\mathrm{i}}\end{array}\right)=\frac{{\mathrm{v}}_{\mathrm{i}}}{\mathrm{v}}\cdot \left(\begin{array}{ccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{\mathrm{12}}& \mathrm{0}\\ {\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{22}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{v}\end{array}\right)\cdot \left(\begin{array}{c}{\mathrm{x}}_{\mathrm{i}}\\ {\mathrm{y}}_{\mathrm{i}}\\ \mathrm{e}\end{array}\right)$$

    

    or its inverse $$\frac{{\mathrm{v}}_{\mathrm{i}}}{\mathrm{v}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{i}}\\ {\mathrm{y}}_{\mathrm{i}}\\ \mathrm{e}\end{array}\right)=\frac{\mathrm{1}}{\left({\mathrm{a}}_{\mathrm{11}}{\mathrm{a}}_{\mathrm{22}}-{\mathrm{a}}_{\mathrm{12}}{\mathrm{a}}_{\mathrm{21}}\right)}\cdot \left(\begin{array}{ccc}{\mathrm{a}}_{\mathrm{22}}& -{\mathrm{a}}_{\mathrm{12}}& \mathrm{0}\\ -{\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{11}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{1}\end{array}\right)\cdot \left(\begin{array}{c}{\mathrm{X}}_{\mathrm{i}}\\ {\mathrm{Y}}_{\mathrm{i}}\\ {\mathrm{Z}}_{\mathrm{i}}\end{array}\right)$$

    

    where e denotes the effective distance of the focusing element grid from the motif plane.
17. The method according to claim 16, characterized in that the focusing element grid in step b) is specified by a grid matrix W, and in step d), the points of the motif plane belonging to a magnification v_i are each combined to form a micromotif image component, and for this micromotif image component, a motif grid U_i is calculated for arranging this micromotif image component periodically or at least locally periodically using the relationship $${\overleftrightarrow{U}}_i=\left(\overleftrightarrow{I}-{\overleftrightarrow{A}}_i^{-1}\right)\cdot \overleftrightarrow{W}$$

    

    the transformation matrices A_i being given by $${\mathrm{A}}_{\mathrm{i}}=\frac{{\mathrm{v}}_{\mathrm{i}}}{\mathrm{v}}\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}& {\mathrm{a}}_{\mathrm{12}}\\ {\mathrm{a}}_{\mathrm{21}}& {\mathrm{a}}_{\mathrm{22}}\end{array}\right)$$

    

    and A _i ^-1 denoting the inverse matrices.
18. The method according to claim 17, characterized in that the focusing element grid in step b) is specified in the form of a two-dimensional Bravais lattice having the grid matrix $\overleftrightarrow{W}=\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& w_{22}\end{array}\right),$

    

    representing the components of the lattice cell vectors w _i , where i=1,2, or that for manufacturing a cylindrical lens 3D moiré magnifier, in step b), a cylindrical lens grid is specified by the grid matrix $$W=\left(\begin{array}{cc}\cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{array}\right)\cdot \left(\begin{array}{cc}\mathrm{D}& 0\\ 0& \infty \end{array}\right)\mathrm{or}{W}^{-1}=\left(\begin{array}{cc}{}^1{/}_D& 0\\ 0& 0\end{array}\right)\cdot \left(\begin{array}{cc}\cos \phi & \sin \phi \\ -\sin \phi & \cos \phi \end{array}\right)$$

    

    where D denotes the lens spacing and φ the orientation of the cylindrical lenses.
19. The method according to at least one of claims 10 to 18, characterized in that the motif grid lattice cells and the focusing element grid lattice cells are described by vectors u ₁ and u ₂, or u ₁ ⁽ⁱ⁾ and u ₂ ⁽ⁱ⁾ in the case of multiple motif grids U_i, and w ₁ and w ₂ , and are modulated location dependently, the local period parameters | u ₁|, | u ₂|, ∠( u ₁, u ₂) and | w ₁|, | w ₂|, ∠( w ₁, w ₂) changing only slowly in relation to the periodicity length.
20. The method according to at least one of claims 10 to 19, characterized in that the motif image is printed on a substrate, the micromotif elements formed from the micromotif image portions constituting microcharacters or micropatterns, and/or that the security element is further provided with an opaque cover layer to cover the moire magnification arrangement in some regions, and/or that the image components of the three-dimensional moiré image to be depicted are formed by individual image points, a group of image points, lines or areal sections.
21. A security paper for manufacturing security or value documents (10), such as banknotes, checks, identification cards or certificates, that are furnished with a security element according to at least one of claims 1 to 9, wherein the security paper preferably comprises a carrier substrate composed of paper or plastic.
22. A data carrier (10), especially a branded article, value document or decorative article, having a security element according to one of claims 1 to 9, wherein the security element is preferably arranged in a window region of the data carrier.

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