DE3821322A1 - Method of controlling a graphic output device - Google Patents

Abstract

Method of controlling a graphic output device (PL, VS), in which surface data of at least one multiply curved surface is entered parametrically via an input device (E) into a control device (ST), and the surface data is prepared by differential geometry in such a way that output data for outstanding curves, i.e. contours, intersection lines and edges of the surface, is continuously generated, following their course, and stored in the curve file memory (DM), and continuously output, in relation to their visibility, as co-ordinate control signals (Ax, Ay; Su1, Sv1) and visibility control signals (DS, SZ). The requirement for memory and computer time is several times less, whereas the accuracy of the curves is high. The method is shown for different projection methods, for families of curves on the surface, and for several intersecting surfaces. <IMAGE>

Description

The invention relates to a method for controlling a graphic output device for displaying a by a predefined parameter representation in the room and, if necessary, by given edge data certain area after a given Projection type on a parametrically predetermined Projection plane, whereby from the area parameters, if necessary the Boundary data and the projection parameters coordinate parameters and visibility data of projection pixels from excellent curves, namely a contour, the edge and / or possibly a self-cut, generated and if necessary controlled for graphic display on a Output device are output.

In previously known methods, the data is multiple curved surfaces that are mapped on one plane and to be graphically represented by the data of many the area covering partial areas is described in a network-like manner. Becomes the mesh size chosen is narrow, so is a recovery of graphical output data from curves that are projective Area display taking into account visibility result with such an approximation to the mathematical theoretically exact curve data possible that the points of the resulting approximate polygon about the selected Line thickness. However, this generally results in very high scores to be saved and long processing times, since the coordinates of the points to be displayed are by means of tedious sorting process from the area network data are winning. For this reason i.a. a relatively large one Mesh size chosen, which is only a rough, visible polygonal approximation of the curves results.  

Another disadvantage is that the coordinates of the points to be displayed occur in such a sequence during data preparation, which does not correspond to the course of the individual lines to be displayed, but to the sequence of the points in each case the same distance from the image plane. A further sorting process of the coordinate data to be output is therefore required if an xy- controlled drawing device is to be activated in a time-saving sequence in a curve-shaped sequence during the output. A completely new time-consuming data preparation is also required for a display with a changed projection type.

The object of the invention is a method for controlling a to disclose graphical output device in which the Data storage of an area in the room and the description of Edges and lines on this surface are relatively few Space required and where the graphical output data of lines and curves of a projective area display immediately following the curves or lines and with their Visibility means generated and saved or controlled be issued.

The solution to the problem is that in one Search submethod based on one on the surface lying starting point, through a quasi-base point projection on one of the curves a first curve point and that of him associated type of curve by checking the occurrence of the Contour edge and / or self intersection associated suitably higher derivatives of the area parameters Visual indices and their associated ones Validity criteria is determined and its parameters be saved, starting from which in a first Subsequent sub-process by determining the respective curve corresponding  View index and observance of the associated validity criteria each for points incrementally adjacent on the surface Determination of another one belonging to the curve found Point and storing its parameters in an associated Continuous curve file takes place, after which this first Subsequent sub-processes cyclically in the direction of the through the given curve points previously found is repeated until a change in corresponding visual index determined in the course of the curve is carried out or by one Test procedure step determines that the to examining point at a given point Tolerance range around the first curve point is whereupon that Search submethod with changed initial conditions, namely in another direction or from one to the starting point spaced surface point as the starting point, performed again is, each time after finding a curve point it is checked whether this is within the tolerance range around one of the already determined curve points, whereupon the Search sub-procedure again with changed initial conditions is started and otherwise the first subsequent sub-procedure is running continuously and the search and first subsequent sub-procedures are carried out until found all excellent curves and as parameter sequences are saved, after which on the curve files one after the other a second subsequent sub-procedure is used in each case by whose cuts take into account the Tolerance range around the cutting curve can be determined and with regard to the intersections found in this way applicable visual index is determined, according to which based on a predefined starting number of views given starting point, each by intersection if necessary limited curve component by successive accumulation  the visual index found between the given one Starting point and the curve component associated with formed View number is assigned and saved, which is controlled in Connection with the associated curve data as Visibility data of the associated coordinate parameters in the Projection plane is output.

The subclaims contain advantageous refinements of the Procedure.

The areas to be displayed are represented by formula Parameter representations specified. You can be bordered closed and / or self-penetrating, and figurative There are no restrictions. The lines, are used for the projective representation of a surface Projection of the edge, the contour, the self - cut in the In the case of self-penetration, the intersection of the surface with other surfaces, e.g. Coordinate surfaces, and / or on the Area running, mathematically to be described lines. The innovative method immediately generates exact contour data as well as data of the area curves Continuous additions of further area curves at any time relatively little computational effort are possible. Let continue data for representations of an area different projection methods, e.g. Parallel projection or central perspective, following the same process steps generate with only modified parameter specifications.

The method is used for imaginary projection beams not the cutting coordinates with the one to be displayed Area determined, but only the number of penetrations of these areas through the respective projection beam. To The visibility of a surface point is determined in each case the number of penetrations of an imaginary visual ray,  that passes through this surface point, determines the between this surface point and the imagined observer lie. This number of penetrations is the following Visibility number called. In surface points that are on a contour, an edge or in a self-penetration changes the number of views during a transition to an infinitesimal neighboring point beyond the curve mentioned is an integer.

Such jumps in visibility, one for visibility characterize relevant line here as the Visible indexes. The different process steps rely on the knowledge of the reversal of the definition the visual indices, namely the number of views in one Area, except on an edge, on a contour or self-penetration is constant. You can find them excellent curves by determining Visual indices.

Starting from a point on the surface, its visibility is known and is predetermined in a first Sub-procedure with a search procedure step a point one Edge line, a contour line or a self-cut line determined. From this first point, it then becomes one the first follow-up process, the entire line found, whose Following the course, determined. In other corresponding ones Search sub procedure and with the following first Subsequent sub-procedures then gradually become the others area-relevant area curves determined and their data saved continuously.

From this complete database of spatial coordinates of the surface curves are on Requirement in a second sub-procedure, again one Subsequent sub-process, each data for a graphic output  regarding the visibility of the surface curves according to a predefined projection type and position of a projection plane determined by the curve data in follow this sub-procedure on their position to each other Determination of their number of views to be examined. One for one Point or a part of the curve determined "zero" leads then to an output of the projection line Representation in a visible line type, which generally means with full Line width happens. With other view numbers one controlled according to the respective specification according to different line types output, and the view number is assigned if necessary saved.

To a sub-process analogous to the second sub-process If necessary, you can specify other, any surface curves in relation to the database of coordinates and the Check view numbers for their visibility and for display bring their data as a point coordinate sequence to the second follow-up procedure.

The search submethod and the two named Subsequent sub-procedures include the ongoing review of the Curve data on the presence of jumps, that is from Visual indices, in the visual number. These visual indices one given area are each made from differential geometric Data determined why the second fundamental form and simple sizes of first differential order formed and to be analyzed.

In the first part of the process, based on the known Area point, in incremental steps, that in its sequence form a surface curve, progressing to the Area indices searched and thus a visibility limit and Contour lines, border lines or self-penetration lines  determined according to the area index. In the event that an edge or Self intersection points exist, these are analyzed of the visual indices i.e. determined.

The method is carried out using the differential geometry according to
(1) Walter, R. differential geometry. Bibl. Inst. Mannheim / Vienna / Zurich, BI-Wiss.-Verlag 1978, i-iii u. 1-278 p.
(2) Walter, R. Compact Hypersurfaces with a Constant Higher Mean Curvature Function. Math. Ann. 270 (1985) 125-145.

The following mathematical foundations are abbreviated. The area of the three-dimensional space R ³ under consideration is given locally by a parameter representation with the parameters x , u , v :

Here there is an open set in the two-dimensional space (parameter level) in R ² , the parameter area A. The differentiation class is C ; depending on the question, this can be reduced considerably. The mapping x is immersive, that is, it is rank 2 everywhere. The field of the normal unit vectors N is to be represented accordingly:

The orientation is such that

[ x _u , x _v , N] <0.

Square brackets [] designate here and in the following the determinant, angle brackets ⟨⟩ the dot product, a horizontal cross in a circle ⊗ the vector product in the given space R ³.

The first fundamental form I and the second fundamental form II of the surface are formed with the differentials of the location and normal vectors d x , d N

I: = ⟨d x d x ⊗⟩,

II: ⟨= d x ⊗ N⟩ d.

The evaluation of such shapes, for example the second fundamental shape II, at point p with the coordinates u and v , which is the element of the parameter range A , for two tangential vectors µ , ν of the parameter level R ² at point p is described in detail as II (p ; µ, ν ) or, abbreviated as II ( µ , ν ).

According to the theorema egregium, the expression for the Gaussian curvature K is :

where µ , ν are two linearly independent vectors at the relevant point p of the parameter level. The surface curve data for a parallel or central projection of a given surface are advantageously analyzed together, since the essential results do not differ between the two types of projection. If the center a and a surface point x are given for the central projection or if the constant direction vector denotes the visual ray a for the parallel projection , the surface points result for the direction vector L

In the case of the central projection, it is a prerequisite that no surface point falls into the center a , and also that the center a never lies on a line of sight between two surface points. As a common representation for the formula is to write as introducing a parameter δ as

L = a - δ x ,

where the parameter δ is zero for a parallel projection and one for a central projection. The position of the projection plane plays no role in the clarification of visibility.

On the area to be examined, a visual function σ is defined as a scalar R :

σ : = ⟨L , N⟩: A → R.

The zeros of this visual function provide those surface points whose visual rays are tangent to the surface. These surface points are the so-called contour points. The entire contour C of a surface is represented by the set of zeros of the view function σ in the parameter area:

C : = { (u, v) ∈ A | σ (u, v) = 0}.

A distinction must be made between the contour C and the set x (C) of the associated surface points, the so-called true contour, and their image in the collecting projection plane, the so-called apparent contour.

In the case of convex closed surfaces, their visible and invisible parts differ by the sign of the visual function σ for them, and the contour C separates these parts from one another. For more general areas, the situation is more complicated, but the visual function σ still plays an important role in characterization.

In the following, the direction vector of a visual ray is advantageously divided along the surface into a tangential and a normal portion, a vector field L with the tangential parameter range λ being represented as:

L = (d x) ( λ ) + σ N.

This decomposition is due to the injectivity of the differential dx by scalar multiplication with the normal vector N.

A surface curve of a running curve parameter x is given by:

(w) = x (u (w), v (w)) ,

where the parameter w runs through a real interval. A running point c (w) is represented by its coordinates u and v :

c (w) : = u (w), v (w)) .

It describes the corresponding curve in the parameter level. Its two component functions u (w) and v (w) are again of the class C ^oo . The restrictions of any area sizes, which are defined on parameter level A , on parameter points c (w) of the running curve are generally designated here by a dash, for example

(w) : = x (c (w)), (w) : = L (c (w)) etc.

A visual ray with the visual ray parameter t through the surface point x (u, v) thus has a parameter representation of a scalar R :

t → x (u, v) + t L (u, v) , t ∈ R.

The visibility test results from these considerations in the following way: A point of the visual ray, approximately x (u, v) + t ₁ L (u, v) , lies in front of another of its points approximately x (u, v) + t ₂ L (u, v) if t ₁ is greater than t ₂.

The following restrictive assumptions are made: The line of sight through a surface point x (u ₀, v ₀) has and only contains x (u ₀, v ₀) transverse cuts with the surface; ie whenever a visual ray parameter is greater or not equal to zero and an associated surface point belongs to parameter area A , the following applies:
x (u ₀, v ₀) + t ₁ L (u ₀, v ₁) = x (u ₁, v ₁), then the visual function σ is not equal to zero. Then:

• - There are only finitely many interfaces of this line of sight before and including at point x (u ₀, v ₀) with the surface, about n pieces.
• - There is an environment U from the point (u ₀, v ₀) in the parameter area A , so that for each pair of parameters (u, v) , the element of the environment U , the line of sight through the point x (u, v) is also accurate n has intersections before and including x (u, v) with the surface.

This clearly shows that the number of views does not change, as long as on and before a running point only transversal Cuts of his visual ray take place with the surface.

In order to determine where the stated requirement is met and where it is not fulfilled, namely because there is a contour, various visual indices D 1- D 6 are checked, which are shown below.

The existence of a contour point C (w ₀) with the running parameter w and with the parameter w ₀ is checked with formations of the second basic form II as follows:

• (4.1) If the visual functionσ (c (w₀)) is zero educated: II(c (w₀));(w₀),λ (c (w₀))) =Z 1, where  the Is tangential vector which is related to the value of the Vector fieldλ is evaluated. II(c (w₀):λ (c (w₀)),λ (c (w₀))) =N 1 caseZ 1 = 0 andN 1 = 0 becomes the first view indexD 1 educated. it has the value + or - one.
• (4.2) If the visual function σ is not equal to zero, then x (w ₀ ) + t ₀ L (w ₀ ) = x (u ₁, ν ₁ ) ; t ₀ is greater than zero and a scalar product and a basic form are formed: ⟨ (w ₀ ) , N (u ₁, v ₁ ) ⟩ = Z 2
  II ((u ₁, v ₁ ) ; λ (u ₁, v ₁ ) , λ (u ₁, v ₁ )) = N 2 If Z 2 = 0 and N 2 = 0, the second visual index D 2 is formed: which therefore has the value + or - two.

In cases (4.1: 4.2) in which one of the numerators ( Z 1, Z 2) or denominator ( N 1, N 2) is zero, a substitute test must be carried out, since more than two scalar equations apply to the relevant points , or it has to be changed to a slightly changed projection. The substitute test procedures are to be carried out analogously to the above using higher derivations. Such a substitute test is given below in the event that the denominator N 1 = 0 and the numerator Z 1 = 0:

• (4.3) Cod (c (w ₀ ) , λ (c (w ₀ )) , λ (c (w ₀ ) , λ (c (w ₀ ))) = N 3, which is the covariant derivative Cod of the second fundamental form , namely Cod: = ∇ II. In the event that the size N 3 ≠ 0, the substitute view index D 1 * is formed:

The visibility check of a contour component x (w) is carried out analogously to the above, with the special case that the counter Z 1 is zero. In addition, the denominator N 1 often disappears at individual points on the contour component, namely if the running line of sight touches the true contour. A change in projection then does not help.

• (5.1) For these contour components x (w) the visual function σ is zero. If the basic shape according to the denominator N 1 is equal to zero and the curvature K (c (w ₀ )) is not equal to zero, as well as the covariant derivative Cod, corresponding to the denominator N 3 not equal to zero, the third visual index D 3 with the dot product is closed form as: D 3 = -1 sign ⟨ ′ (w ₀ ) , (w ₀ ) ⟩, which thus has the value + or - one.

However, it is the basic form according to the denominatorN 1 not equal to zero and the line of sight hits through the contour component ( w₀) the Contour spatially in front of the contour component(w₀), that's how it is second view indexD 2 form.

For the edge check, it is determined whether the line of sight hits the point (w ₀) spatially in front of a curve (w ₀) to be examined continuously. For this purpose, the field of the tangential unit edge vectors R is defined with a given orientation, for example, "interior to the left". Then it is formed

(w ₀ ) + t ₀ (w ₀ ) = x (u ₁, v ₁ )

and checked whether t ₀ is greater than zero. The fourth view index D 4 then results from the determinant Z 4 using the edge vector R.

• (6.1) [ ′ (w ₀ ) , (w ₀ ) , R (u ₁, v ₁)] = Z 4 if Z 4 is non-zero, and also the scalar N 4, which is formed by means of the normal vector, ⟨ (w ₀ ) , N (u ₁, v ₁ ) ⟩ = N 4, is not equal to zero,

The procedure for the edge check is carried out on the condition that the parameter area A is a closed, axially parallel rectangle. Otherwise, D 4 is a factor for the specified fourth visual index

sign [ E (u ₁, v ₁ ) , R (u ₁, v ₁ ) , N (u ₁, v ₁)]

to accept.

To check the visibility of the edge, an edge piece (w) without corners is examined in a corresponding orientation. The line of sight can hit the contour or the edge at another point in front of it. In this case, the second visual index D 2 or the fourth visual index D 4 is to be formed according to (4.2) or (6.1). However, if the edge meets the contour itself, ie if the visual function σ (c (w ₀ ) is zero, then a fifth visual index D 5 is to be formed if the conditions that the numerator Z 1 and the denominator N 1 according to (4.1 ) and the scalar ⟨ (w ₀ ) , (w ₀ ) ⟩ are not equal to zero, according to

This view index can therefore have the value +1, -1 or 0.

To check for self-penetration points, the Section of the visual ray viewed with two surfaces. For this are the sizes that affect one or the other surface marked with a crossbar or a tilde.

Two scalars are formed and checked beforehand:  

• (7.1) ⟨ (w ₀ ) , Ñ (u ₁, v ₁ ) ⟩ = Z 7 if Z 7 is not equal to zero,
  ⟨ (W ₀ ) , Ñ (u ₁, v ₁ ) = N 7 if N 7 is not zero,
  the sixth visual index D 6 results

If the size Z 7 is zero, there is a point on the intersection curve. If the size N 7 is zero, there is a contour point of the surface which is identified by the label tilde.

To check the visibility of an intersection curve, assuming that the intersection is transverse, it is checked whether the line of sight hits a contour or an edge of one of the surfaces before the current point. For this purpose, according to (4.2) and (6.1), the second or fourth view index D 2, D 4 must be formed for the two surfaces. Furthermore, in the event that the current point itself is a contour point, that is, the intersection curve meets a contour there, the first visual index D 1 for the corresponding area must be formed according to (4.1).

For the determination of the view indices, the data is the Contours, possibly the edge and possibly the self-cuts represent kept in constant access in memory. The Border is defined as a field by its field data and thereby directly available. The data of the contour and the self-cuts, which are also fields in the room are first calculated and gradually built up in memory accordingly.

Set both the contour and the possible self-cut generally composed of curve pieces. Is a contour point or Self intersection has been found so  the subsequent sub-procedures applied and the other curve sections and determined their point coordinates. Your calculation is done according to the same principle: because acts mathematically in both cases it is a non-linear, undetermined System of equations of shape

For the contour, the maximum index is n = 2, and the area parameters u ₁, u ₂ correspond to the parameters u, v , and the function f ₁ is the visual function.

When intersecting two surfaces are the maximum indexn = 4, and the area parameters(u₁,u₂) correspond to the parameters u, v one area as well(u₃,u₄) correspond(,) the other area, and the functionsf _i are the differences the coordinates of the parameter representations:
f _i(u, v,,): =x ⁱ(u, v) - ⁱ(,),i ∈ {1, 2, 3}, where the Tilde marks the second surface.

The numerical solution of the system of equations (8.1) is expediently done using the base point or gradient method. Starting from an arbitrary point a = (a ₁,..., A _n ), this supplies a solution point a + h , h : = (h ₁,..., H _n ) that is as close as possible to this. The requirement f _i (a + h) = 0 for i ∈ {1,. . ., n - 1} is approximately equivalent to:

commas in the index indicate partial derivatives.  

One looks for the solutions h of the linear system of equations (8.2) for the square norm g

becomes minimal. The Lagrangian method for the extremum of the square norm g under the constraints (8.2) leads to the equations for the Lagrangian multiplications λ _k :

and the associated extreme point H _j :

If the output functions (8.1) have the maximum rank n - 1 at point a , the coefficient matrix of (8.4) is regular as a Gram matrix, and the unambiguous inversion of (8.4) leads to an exact solution with (8.5)

(8.6) a ^# : = a + h

from (8.2) and thus to an generally improved approximate solution from (8.1). The iterated application of this subsequent ^{process step} yields the points a, a ^# , a ^##,. . . and under favorable conditions in the Limes an exact solution a * from (8.1).

Although this gradient method often does not provide the exact solution point to point a , it is useful. This is because it is advantageously assumed that the assignment a → a * does this and it is referred to as a quasi-base point projection, in short: QF projection.

In any case, a first solution point of (8.1) can be found. A second, neighboring solution point arises by continuing from a * in the direction of a weak change in the f _i and subsequent QF projection. If you only have two neighboring solution points, the chord is extended by a suitable factor beyond the newer point and the further point thus reached is subjected to the QF projection again. This process step is carried out iteratively, so that an entire solution component is gradually calculated and stored.

It happens relatively often that the solution components form closed curves. To recognize this and the To cause computers not to iterate endlessly, is advantageous in each case an auxiliary process step, the Tube processing step carried out. This one Parallel set (tube) around the beginning of the solution curve considered and the occurrence of the last generated Point in this tube tested tangentially.

Irregularities or non-convergence that occur are at the Follow-up process step not disruptive; then you break and goes back to the search part procedure. To be sure be that all contours and contour parts, these are the Solution components will be found, the Search procedure part repeated from a variety of different ones Starting points applied from. This is useful here Dot successively on two by a given Coordinate axes starting point, including baselines called, pushed in relatively large steps and then in further search procedure repetitions the distance of the Points already used halved or after one similar rule reduced and searched from there.  

For every further solution point found, the Tube process step based on those already found Solution points applied. In the event that the point already was known and is in or on the tube, the search continued. However, if there is a new point, then the iterative follow-up step is applied again. Bumps the further consequence of the solution component again on one known solution or possibly on an edge, which is also due to the Tube procedure step is checked based on the edge each from the starting point of this solution component in reverse direction iteratively with the subsequent process step went on and then again to search for a new one Starting point off, i.e. with changed starting conditions passed over.

The density of the search points can be specified based on experience will. It can also stop searching in the event be specified that a previously known maximum number of Solution components were found completely. Furthermore can the generation of search points is ended, e.g. if at two successive halving of the starting point density no other new solution point found outside of a tube or if the starting point density is the pixel density or Line width of an intended graphic representation of the Curves reached or undercut by a predetermined amount Has.  

When creating the self-cut, as with the Contour generation process, however, the dimension is higher is. The points on the diagonal of the corresponding four-dimensional space are excluded. For the Representation of the surfaces and desired surface curves Functions specified and required in a known manner if supplemented with warping parameters so that they are in the process Are available.

For predefined and generated curves, it is necessary to provide or calculate such a dense number of their point coordinates that a quasi-continuous display results from an output. It has been shown in practice that a resolution of 1000 points for a curve generally leads to sufficient accuracy. The tube radius with a resolution that is a hundred times coarser has proven to be sufficient. Since these are space curves, their points are generated as sequence elements in the parameter area. Their projection is then formed from them if necessary by the mapping equations on the image plane. The data type "curve" is expediently defined for this figure, which forms a string of the parameters u , v on the surface of real numbers. A corresponding data type can of course also be formed in three-dimensional space from a triplet of numbers per point.

An advantageous simplification of data storage and Execution of the procedure results from the Dimensional reduction, which is reflected in the Projection plane results. In cases where one Resolution of a borderline or intersection due to the location multiple surface points in a projection beam is clearly possible, as already mentioned above described a slight change in the inclination of the projection  or a one-level view only remedy by the projection beam.

The point density should be chosen so that the Quasi-infinitesimal calculations with sufficient accuracy are feasible.

The following files are created:

The file of the start and start points a of type 2 array, real. The result file of the search procedure for this, with the point coordinates of the type 2 array found by the QF projection, real. If there is no convergence or if you exit the parameter area, an improper value is entered in the file. In the incremental follow-up method, a 2-array, real, of the "curve" type is generated, which begins with a point coordinate pair found in the search method step, to which the further point coordinate pairs follow. In addition, an integer is useful as an array length pointer. If the contour is followed one after the other in both directions, the two contour sections should then be appropriately rearranged in a sequence. The array length pointer serves as a limit for subsequent indexed processing. Usually a curve is covered by a few hundred to a thousand points.

During the repetitions of the search steps, each at a certain density of the search points, the Array length pointer of the individual curve components in one 1 array of type Integer continuously written. The length of this array is an integer Curve component pointer carried.

To control the search procedure step, the Distance halving number performed in a counter, if necessary with  a predetermined maximum halving number in each case is compared, whereby the search end when reaching the Maximum number is controlled. As previously described, is also search end control based on other criteria feasible.

Area intersection curves are created analog to the contours and stored in curve files.

To clarify the visibility, the data of the curves are under Control called up according to the pointer values and successively the occurrence of the described visual indexes was examined. This creates two files, namely a 1 array, real, the respective point designation of the start and end point and the points at which a change in visibility occurs, and a 1-array, integer that is assigned to the respective view number contains the points mentioned, the starting point being his View number is specified. When generating the view number file are, as already described above, the Visual index checks performed on the following cases: "The Curve meets the contour ";" The line of sight meets the contour the current point ";" The line of sight hits the edge in front of the current point ";" The line of sight hits the self-cut the current point ". This second follow-up step is used for the different curves, the contour, the edge and the self-cut, if available. The resulting files are with others Visibility analysis of predefined curves can be used.

In the same way, namely any other lines given on the surface for their visibility checked and saved in a corresponding file or brought directly to the output, the output control depending on the number of views.  

The view number N ₀ at the beginning of each curve section is determined on the basis of the view number of a known starting point, preferably the start search point of the search method section, along the base lines, whereby the associated view number file is created, via which the view number of the respective starting point of a curve part, which is from there was found out is given.

If the viewpoint of the starting point is originally unknown, so it can after the relative determination of all view numbers the excellent surface curves, the edge, the contour and the self-cut with respect to a fictitious initial value be normalized by using the lowest found number Is set to zero and the rest are related to it be entered.

To reduce the number of operations to be performed As already mentioned, the procedural steps become the investigation on finding the cuts with the curve components of the excellent curves in the two-dimensional Projection level made.

To further reduce the number of operations required A sectional view shows the entire projection plane as into similar subfields, e.g. Grid squares, divided considered. Accordingly, the files of the curves become such Subfields assigned divided into subfiles. For one Sectional view of a line or curve with the then only the files will be marked taking into account the tube process step on a possible cut, into which the projection of the examining line or curve falls. In practice it has a projection surface division into approximately 5 × 7 sub-fields proven. The division of the curve files into  the subfield files are expediently created when the Curve files. Shows browsing the affected files a possible cut in the tube area, the Section in detail according to those described above Process steps clarified.

Both the subfield formation and the application of the Tube process step bring a significant reduction the number of steps of the procedures, since they allow using relatively large increments in the search and Follow the procedural steps, and only in the event that a Cut in the roughly specified range is likely further steps with smaller increments are available successively carried out until a cut case clarification is present. The rough step size can be about half Correspond to the tube size, and iteratively, e.g. by respective Halving, the narrowest step size can be achieved predetermined fraction, e.g. half the pixel size correspond. With a given computer, these provide Process designs a significant gain in computing time.

The procedure shown can be expanded in a similar way also apply to other coordinate systems and projections. You can also use given areas and lines several projections, e.g. a parallel projection for a imaginary lighting and a central projection for one Apply observer. From the overlay of the two Projections of the corresponding excellent curves in the Projection level can be seen from the visual numbers of the two Projections the light and shadow conditions and their Determine visibility by overlay.

Examples of graphic outputs of areas are given in FIGS. 1 to 4, and FIG. 5 shows a block diagram of an apparatus for carrying out the method.

Fig. 1 shows the visible contours of a toroid in parallel projection obliquely from above;

Fig. 2 shows the toroid with applied in different line widths visible lines;

Fig. 3 shows the edges and contours of a torus section;

Fig. 4 shows the torus section with an applied line network of visible lines.

Fig. 5 is a block diagram showing an apparatus for generating and outputting procedural graphical control data.

Since the bevel of the projection of the toroid in FIG. 1 allows a view through the inner free space, a total of three contour parts are visible, namely the outer contour KA , which is self-contained, and two contour lines KL 1 , KL 2 , which are divided into two points Cut KP 1 , KP 2 in which the lower contour line KL 2 becomes invisible. Further invisible contour parts extend in the continuations of the lower contour line KL 1 and between their invisible end points and the end points KE 1 , KE 2 of the upper contour line KL 2 . The torus has no other excellent curves.

In Fig. 2 the torus is covered with a medium-sized group of lines LP 0 , LP 1 , LP 2 , which extend parallel to the inner circumferential axis of the torus at equal angular intervals of 20 degrees. As long as they do not run through in the visible surface area, their visibility ends transversally at the outer contour KA or the upper contour line KL 1 . Another set of lines LR 0 , LR 1 , LR 2 , which is shown thin, revolves the torus radially to the inner circumferential axis at equal intervals of 20 degrees of the central angle. With these lines, too, the limit of visibility is given in a cut with one of the contour components. Another helical circumferential line L 1 is shown in heavy line width. It rotates 90 degrees with a progress of 20 degrees, so that it is self-contained. All the lines shown were continuously examined for their interfaces with the contour lines and, provided they entered an area with zero visibility, continuously output for the drawing until a next intersection with a contour component was found. Further data management of these lines was therefore not necessary. The line widths of the lines were assigned.

To illustrate the search method step for generating the contours, the starting point a was selected as the starting point, for example , and the coordinate lines LP 0 , LR 0 were taken as the base lines. From this starting point a , the contours KA and KL 2 were found and then followed continuously or on both sides. The other contour components KL 1 and the two invisible contour parts were only found in further search procedure steps with the starting points shifted on the baseline LP 0 and then completed in each case. The intersections of the line sets provide a certain illustration of the position of the starting points on the surface. The storage of the visible and the invisible contour components and their visibility requires only a few thousand memory cells, so that processing can be carried out according to the method with today's average desktop computer.

In Fig. 1, an image field divided into sub-areas is shown, illustrating the distribution of contour data in each of these sub-areas associated sub-files. It can be seen that only a greatly reduced amount of data has to be checked to examine the visibility of the contour components and the lines, namely the data of the points in the fields in which the respective contour component or line runs.

For a representation of the torus on a screen with dynamic line and image deflection with usual Line resolution are the image data in an image memory transferred transformed, with about 100 fields per field times 100 pixels memory cells for control variables for the Brightness and / or coloring are provided. These will As is known, read cyclically and the picture tube control fed. If a point-controllable quasi-static Image display device, e.g. a liquid crystal If there is a display field, click on the point field transformed curve coordinate and visibility data directly to the control electronics of the pixels to hand over.

Fig. 3 shows a bounded tunnel-shaped torus section. The edges R 1 - R 4 are easy to specify as parameterized curves in surface coordinates. The contour consists of three contour components KA 1 , KL 11 , KL 3 . The inner contour component KL 3 continues invisibly up to the upper contour line KL 11 . The edges R 1 , R 2 , R 4 also have invisible edge components.

FIG. 4 shows the torus section according to FIG. 3 with lines of lines applied in the radial and parallel directions. In particular, the obliquely cutting course of the contour to the lines reveals which dense sequence of spatial cuts with corresponding data storage would be necessary according to the previously known method in order to achieve an approximately equivalent step-free contour reproduction. This clearly shows the advantages of the new method.

FIG. 5 shows a block diagram of a control device ST which is suitable for carrying out the method depending on the data specified via the input device E and for continuously writing the correspondingly generated curve and visibility files into the file memory DM in an orderly manner accessible according to the subfields. A drawing device PL is connected to the control device ST via a drawing device output interface AP . Via this interface AP , the coordinate data (u ₁, v ₁ ) of the pixels in the projection plane are continuously supplied as control variables to the coordinate drives Mx, My , the respective number of views SZ being output to a line type control device MZ . The processing of the number of views for line modification or selection of crayons is carried out in a known manner by a control circuit on the drawing device side or a control subroutine in the control device in advance.

Furthermore, a graphic screen output device VS is connected to the control device ST . For this purpose, an output circuit AD is provided for the memory coordinate control signals Ax, Ay of an image memory MB and for the associated visibility data DS for data writing. A cyclic image output control device BZ continuously takes the stored visibility data cyclically from the image memory MB and outputs them with periodically added line synchronization signals ZS and image synchronization signals BS as brightness signals HS or corresponding color signals to the picture tube device VS , which is controlled periodically in line and in alternation.

The coordinates can be determined by known transformations and handle visibility data and share it with others Output requirements adjusted.

Claims (15)

1.Method for controlling a graphic output device for displaying a surface determined by a predetermined parameter representation in space and possibly by predetermined boundary data according to a predetermined projection type on a parametrically predetermined projection plane, coordinate parameters and visibility data from the surface parameters, possibly the boundary data and the projection parameters projection pixels of excellent curves, namely a contour, the edge and / or possibly a self-cut, are generated and, if necessary, are output in a controlled manner for graphic display on an output device, characterized in that, in a search part method, starting from a starting point lying on the surface ( a ), by a quasi-base point projection onto one of the curves, a first curve point ( a ) and the type of curve associated with it by checking for the occurrence of the contour edge and / or self-intersection points belonging to higher Ab lines of area parameters suitably formed, visual indices ( D 1- D 6) and their respective validity criteria are determined and their parameters ( u ₁ , v ₁ ) are stored, starting from the first sub-part process by determining the visual index corresponding to the respective curve ( D 1- D 6) and observance of the associated validity criteria for points incrementally adjacent on the surface, the determination of a further point belonging to the curve found and the storage of its parameters in an associated continuous curve file are carried out, after which this first subsequent part method is cyclically repeated as often as in the direction of the previously found curve points given curve shape is carried out repeatedly until a change in the corresponding visual index ( D 1- D 6) in the course of the curve is determined or it is determined by a test method step carried out in each case that the point to be examined is in a respective The specified tolerance range (tube) lies around the first curve point (a) , whereupon the search part method is carried out again with changed initial conditions, namely in a different direction or from a surface point spaced from the start point ( a ) as the start point, checking each time a curve point is found whether this is within the tolerance range around one of the curve points already determined, whereupon the search sub-procedure is started again with changed initial conditions and otherwise the first subsequent sub-procedure is carried out continuously and, likewise, the search and first sub-sub-procedure are carried out until all those that have been awarded are awarded Curves are found and stored as parameter sequences, according to which a second subsequent sub-method is applied to the curve files one after the other, in that their cuts are determined taking into account the tolerance range around the curve to be cut and related ch of the intersection points found in this way, the applicable view index ( D 1- D 6) is determined, according to which, starting from a predefined starting point number of a predefined starting point, each curve component, which may be delimited by intersection points, contains the successive accumulation of the found indexes between the predefined starting point and the curve component associated with the curve component is assigned and stored, which is output in a controlled manner in connection with the associated curve data as visibility data of the associated coordinate parameters in the projection plane. 2. A method for controlling a graphic output device according to claim 1, characterized in that the first subsequent sub-process from the first curve point ( a ) is carried out successively in each of the two curve directions, after which the search sub-process is returned to. 3. A method for controlling a graphic output device according to claim 1 or 2, characterized in that the variation of the initial conditions by formation of starting coordinates according to a successive shift of the starting point to baseline coordinates parallel to the starting point ( a ), with a predetermined relatively large distance in each case with one another and in further variation sequences with a smaller, for example halved, distance. 4. Method of controlling a graphic Output device according to claim 3, characterized in that that the cyclical sequence of the search sub-procedures then ends when a predetermined number of distance reductions, e.g. Halved, done or after a given number of distance reductions none of the other Curve components were found or a predetermined number of curve components of the different curves found and were determined. 5. A method for controlling a graphic output device according to one of the preceding claims, characterized in that the incremental spacing of adjacent points to be checked on one of the visual indices ( D 1- D 6) is selected such that their spacing is a predetermined fraction of grid spacings or about equal to a line width of the output devices to be controlled and that the corresponding the incremental intervals incremental sizes to a differential approximated processing for forming the visible indices (D 1 - D 6) are sufficiently found. 6. Method of controlling a graphic Dispenser according to one of the preceding claims, characterized in that the tolerance range around a curve is specified such that its width is one Multiples, e.g. 1000 times, the incremental Point spacing is. 7. A method for controlling a graphical output device according to one of the preceding claims, characterized in that predetermined lines or surface intersection curves according to the formation of their cuts with the excellent curves of the surface by using the second subsequent part method on the line data of a line point in incremental direction with respect to the formation of their cuts The curve files are continuously examined for their visibility taking into account the tolerance range and the formation of the respective number of views for the line part from the view indices ( D 1- D 6). Controlled according to the visibility, the corresponding point data in the projection level in a data sequence of a line file stored or output as such on the output device. 8. Method of controlling a graphic Dispenser according to one of the preceding claims, characterized in that the storage of the excellent curve files and line files, if applicable that the curve coordinates of the respective Curve point mapping in the projection plane as Coordinate pair is saved.   9. Method of controlling a graphic Output device according to claim 8, characterized in that the second subsequent process step by means of the Curve files related to the projection level. 10. Method of controlling a graphic Dispenser according to one of the preceding claims, characterized in that the view numbers each assigned to the start coordinates of a curve component be kept continuously and the entirety of the View numbers normalized to the lowest view number to zero becomes. 11. Method of controlling a graphic Dispenser according to one of the preceding claims, characterized in that several represent themselves cutting, possibly multi-curved surfaces and / or Determination of images in different projections a projection plane, especially for the representation of Light-shadow boundaries and visibility, each to the Surfaces and projections belong to the excellent curves Search sub-procedure and the first subsequent sub-procedure according to can be determined separately and also, if necessary, surface intersection curves be determined and the respective visibility or Light-shadow boundary through accumulation of the visual indices of the respective cuts from mutual intersection of the Curve data determined according to the second sub-procedure and are saved and the curve data from the visibility and / or light-shadow boundary data are output in a controlled manner. 12. Method of controlling a graphic Dispenser according to one of the preceding claims, characterized in that the storage of the Curve files according to a division of the projection level into  Subfields divided into partial files and the Section formations after the second subsequent sub-procedure in each case done only with the applicable members. 13. Method of controlling a graphic Dispenser according to one of the preceding claims, characterized in that assigned to the curve files or the part files the number of coordinate pairs are saved and these numbers of access control serve. 14. Device for using the method according to one of claims 1 to 12, characterized in that it consists of a program-controlled control device ( ST) which is connected on the input side to an input device ( E ), through which it with the area parameters to be specified, the projection - and projection level parameters, possibly edge data and predetermined start dates and end criteria can be loaded, and with a file memory (DM) in which the curve files to be generated according to the method are to be saved from the surface as sequences of point coordinate pairs and the associated view numbers are to be saved, which are the indicates the respective number of points of the area which lie between the projection-oriented observer position and the curve point, and that the control device (ST) is connected on the output side to a graphic output device (PL, VS) which controls the coordinate pairs under the control of the output of the respective associated visual count l are to be fed continuously. 15. The apparatus for using the method according to claim 14, characterized in that the graphic output device (VS) is connected via a pixel memory (MB) to the control device (ST) and the output coordinate pairs are used to select a coordinate of the pixel memory locations, in each case according to the assigned number of views, a brightness or color code is to be stored, and that the pixel memory (MB) is continuously addressed and readable by a cyclic image control device (BZ) and the read brightness or color codes, mixed with synchronization signals (BS, ZS) , one line-controlled and image change-controlled video tube are supplied.