CN112255869A - Parameter-based three-dimensional graph dynamic projection implementation method - Google Patents
Parameter-based three-dimensional graph dynamic projection implementation method Download PDFInfo
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- CN112255869A CN112255869A CN202011207061.7A CN202011207061A CN112255869A CN 112255869 A CN112255869 A CN 112255869A CN 202011207061 A CN202011207061 A CN 202011207061A CN 112255869 A CN112255869 A CN 112255869A
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- G—PHYSICS
- G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
- G03B—APPARATUS OR ARRANGEMENTS FOR TAKING PHOTOGRAPHS OR FOR PROJECTING OR VIEWING THEM; APPARATUS OR ARRANGEMENTS EMPLOYING ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ACCESSORIES THEREFOR
- G03B21/00—Projectors or projection-type viewers; Accessories therefor
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- G—PHYSICS
- G09—EDUCATION; CRYPTOGRAPHY; DISPLAY; ADVERTISING; SEALS
- G09B—EDUCATIONAL OR DEMONSTRATION APPLIANCES; APPLIANCES FOR TEACHING, OR COMMUNICATING WITH, THE BLIND, DEAF OR MUTE; MODELS; PLANETARIA; GLOBES; MAPS; DIAGRAMS
- G09B5/00—Electrically-operated educational appliances
- G09B5/02—Electrically-operated educational appliances with visual presentation of the material to be studied, e.g. using film strip
Abstract
A method for realizing three-dimensional graph dynamic projection based on parameters comprises the following steps: s1 dividing the projected geometry into a plurality of discrete points Pi; s2, setting a projection surface and a light source; s3 displaying the projected geometric solid through the projection shape of the light source on the projection surface; the display process comprises the following change of a projection shape when the projection rate is changed; pi is a discrete point on the projected geometric solid, and Interi is a final projection point of the point Pi on the projected geometric solid on the projection plane through the light source; then the corresponding displayed middle projection point Ei = Pi + λ · ini as the projection ratio λ changes. The invention can dynamically display the middle process of projecting the three-dimensional graph to the designated plane according to the light propagation direction by continuously modifying the parameter assignment of the projection rate, thereby forming continuous projection animation. The projection transformation function and the application range of the three-dimensional dynamic geometric system are expanded.
Description
Technical Field
The invention relates to the technical field of education, teaching software, in particular to a parameter-based three-dimensional graph dynamic projection implementation method.
Background
The dynamic process of maintaining the constraint relation of the geometric figure is vividly obtained by dynamically dragging the free points in the geometric figure, so that the geometric properties contained in the geometric figure are better understood, and the system with the characteristics is called dynamic geometry. The dynamic geometry is an important application of geometric constraint solving, is widely applied to teaching assistance and is an educational informatization tool which goes deep into basic mathematics disciplines. Research shows that by utilizing the characteristic of interchangeability of dynamic geometric software, the cognitive load of students in the learning process can be effectively reduced, and the learning effect is improved.
The three-dimensional dynamic geometry software is education-oriented solid geometry teaching auxiliary software, and by using the three-dimensional dynamic geometry software for auxiliary teaching, real three-dimensional space and solid figures can be intuitively established, the thinking difficulty of students in the learning process is reduced, and the three-dimensional dynamic geometry software has a remarkable effect on the students establishing space figures and improving the space thinking capability. Three-dimensional dynamic geometry software commonly used at home and abroad comprises: an InRon drawing board (inRm 3D), a Cabri-3D, GeoGebra, a network drawing board (Netpad) and the like.
In the solid geometry, the projection of a three-dimensional figure to a designated plane or a curved surface is a teaching difficulty, and students can clearly and intuitively know the projection concept and the influence of projection surface change on the projection shape through dynamic continuous display. The existing geometric software can only provide a static result for projecting the three-dimensional geometric figure, and cannot dynamically display the projection process, so that the use effect and the application range of a three-dimensional dynamic geometric system are limited.
Disclosure of Invention
In order to overcome the defects in the prior art, the invention discloses a parameter-based dynamic projection implementation method for a three-dimensional graph.
The invention relates to a parameter-based dynamic projection implementation method of a three-dimensional graph, which is characterized by comprising the following steps of:
s1 dividing the projected geometry into a plurality of discrete points Pi;
s2, setting a projection surface and a light source;
s3 displaying the projected geometric solid through the projection shape of the light source on the projection surface;
the display process comprises the following change of a projection shape when the projection rate is changed; pi is a discrete point on the projected geometric solid, and Interi is a final projection point of the point Pi on the projected geometric solid on the projection plane through the light source;
when the projection ratio lambda is changed, the corresponding displayed middle projection point
Ei = Pi + λ·Interi。
Preferably: the light source is a point light source or a parallel light source;
when the light source is a point light source, constructing a ray ri from the point light source weight to Pi;
when the light source is a parallel light source, constructing a ray ri in the direction of the parallel light by taking Pi as a starting point;
calculating an intersection point of the ray ri and the projection plane, wherein the intersection point is used as a final projection point Interi and forms a directed line segment (Pi, Interi) with the corresponding Pi; the intermediate projection point Ei is located on the directed line segment.
Preferably: the projected geometry is a three-dimensional volumetric shape.
Preferably: the projection ratio λ is defined in the range of [0, 1 ].
Preferably: pi is the vertex of the projected geometry located at the surface or boundary.
Preferably, the change of the projection rate λ is controlled by a dragging bar on the display screen.
Preferably, the projection surface is a plane or a curved surface.
The invention can dynamically display the middle process of projecting the three-dimensional graph to the designated plane according to the light propagation direction by continuously modifying the parameter assignment of the projection rate, thereby forming continuous projection animation. The projection transformation function and the application range of the three-dimensional dynamic geometric system are expanded.
Drawings
FIG. 1 is a schematic view of an embodiment of the projection under a point source according to the present invention;
FIG. 2 is a schematic view of an embodiment of the present invention projected under a collimated light source.
Detailed Description
The present invention is further described below with reference to specific examples, which are only exemplary and do not limit the scope of the present invention in any way. It will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention, and that such changes and modifications may be made without departing from the spirit and scope of the invention.
The invention relates to a parameter-based three-dimensional graph dynamic projection implementation method, which comprises the following steps:
s1 dividing the projected geometry into a plurality of discrete points Pi;
the projected geometry is generally a plane or solid figure, such as a square, circle, cube, sphere, etc., but also can be a point, a straight line, a curve, a curved surface, and other polyhedrons.
When the projected geometric body is divided into discrete points, the boundary points of the plane figure and the surface points of the three-dimensional shape are generally taken, and for the regular geometric body, only the vertexes, for example, eight vertexes of the cube, may be taken for simplification of the operation.
The more the discrete points are, the more complete the projection graph is, but the calculation amount is increased, and the discrete point segmentation quantity can be set according to the teaching requirement and the system hardware condition.
S2, setting a projection surface and a light source;
the projection surface can be a plane or a curved surface; the light source can be a point light source or a parallel light source.
The setting processes of steps S1 and S2 are usually performed based on a three-axis coordinate system, and for simplification of calculation, it is preferable to set the projection plane as an XOY plane when the projection plane is a plane, and to set a perpendicular line between the point light source and the projection plane as a Z axis when the light source is the point light source.
For the projected geometric body, the projected geometric body is divided into a plurality of discrete points positioned on the boundary or the surface of the geometric body, the position of each discrete point and the point light source is determined by XYZ coordinates in a three-axis coordinate system, and for the parallel light source, the slope of the parallel light source in the three-axis coordinate system defines the light source direction.
S3 displaying the projected geometric solid through the projection shape of the light source on the projection surface;
the display process comprises the following change of a projection shape when the projection rate is changed; pi is a discrete point on the projected geometric solid, and Interi is a final projection point of the point Pi on the projected geometric solid on the projection plane through the light source;
when the projection ratio lambda is changed, the corresponding displayed middle projection point
Ei = Pi + λ·Interi。
Taking the example of the point light source shown in fig. 1 projecting a circle, the projection plane is a plane, the plane of the projection plane is used to establish the XOY coordinate, and the Z axis is set perpendicular to the plane of the projection plane.
The projected geometric volume of a circle is divided into a plurality of discrete points, for example, one discrete point may be taken every 1 degree central angle interval on the circumference, the circumference may be divided into 359 discrete points, and the central points may be 360 discrete points in total.
For each discrete point Pi, i =1, 2 … 360, a connecting line between the discrete point Pi and the point light source light is used for constructing a ray ri, the two rays extend to a projection plane, namely an XOY plane, and an intersection point is a final projection point Interi corresponding to the Pi point.
As shown in fig. 2, for the collimated light source, a ray ri is constructed in the direction of the collimated light passing through the discrete point Pi, and an intersection point of the ray ri and the projection plane is a final projection point ini corresponding to the Pi point;
in the invention, in order to better show the projection process, an intermediate projection point concept Ei = Pi + lambda. Interi is set, and the change process of the projection image in the intermediate area from the position of the projected geometric solid to the projection surface is displayed through the change of the parameter projection rate lambda.
When the projection rate lambda is changed, corresponding to the displayed middle projection point
Ei = Pi + λ·Interi。
As shown in fig. 1, a plurality (two are given in fig. 1) of intermediate states of the projected image from the position of the projected geometry to the intermediate region of the projection surface are shown. The plurality of intermediate projection points are combined to form an intermediate state pattern.
The projection rate lambda definition range is preferably set in a closed interval [0, 1 ]; when the projection pattern is 0, the projection pattern is completely superposed with the projected geometric solid; when 1, the projected pattern is completely located in the projection plane.
The invention is generally displayed by a display and the like, as shown in fig. 1, in the teaching process, the change of the projection rate lambda can be controlled by a teacher dragging a dragging bar on the display screen, and a complete projection process is shown for students. It can be seen that for a point light source, as the projection ratio λ changes, the boundary and direction of the projected intermediate state pattern continuously change and gradually approach the final pattern finally located on the projection surface.
A more specific embodiment is:
the light source ray is constructed as follows:
if the point light source is perspective projection, recording the point light source as light, and recording a point set discretized by the projection graph as P = { Pi | i = 0, 1, 2, …, n }; then the point light ray is a ray ri constructed from weight to Pi; if the projection is parallel projection, constructing a ray ri according to the parallel light direction by taking Pi as a starting point;
calculating an intersection point Interi by the ray ri and the projection plane Projplane, and forming a directed line segment (Pi, Interi) by the intersection point Interi and the corresponding Pi;
combining the directional line segments obtained in the above steps into a directional line segment set:
S = {(Pi , Interi) | i = 0, 1, 2, … , n }；
the specific steps of calculating and obtaining the projection point set according to the projection rate lambda are as follows:
e = { Ei | Ei = Pi + λ · ini, i = 0, 1, 2, …, n }, with a corresponding set of projection points for each different projection rate λ.
The invention can dynamically display the middle process of projecting the three-dimensional graph to the designated plane according to the light propagation direction by continuously modifying the parameter assignment of the projection rate, thereby forming continuous projection animation. The projection transformation function and the application range of the three-dimensional dynamic geometric system are expanded.
The above is a description of one embodiment of the present invention in more detail and detail, but it should not be understood that the scope of the invention is limited thereby. It should be noted that, for a person skilled in the art, several variations and modifications can be made without departing from the spirit of the invention, which falls within the scope of the invention. Therefore, the protection scope of the present patent shall be subject to the appended claims.
Claims (7)
1. A method for realizing three-dimensional graph dynamic projection based on parameters is characterized by comprising the following steps:
s1 dividing the projected geometry into a plurality of discrete points Pi;
s2, setting a projection surface and a light source;
s3 displaying the projected geometric solid through the projection shape of the light source on the projection surface;
the display process comprises the following change of a projection shape when the projection rate is changed; pi is a discrete point on the projected geometric solid, and Interi is a final projection point of the point Pi on the projected geometric solid on the projection plane through the light source;
when the projection ratio lambda is changed, the corresponding displayed middle projection point
Ei = Pi + λ·Interi。
2. The method for realizing dynamic projection of three-dimensional graphics based on parameters as claimed in claim 1, wherein: the light source is a point light source or a parallel light source;
when the light source is a point light source, constructing a ray ri from the point light source weight to Pi;
when the light source is a parallel light source, constructing a ray ri in the direction of the parallel light by taking Pi as a starting point;
calculating an intersection point of the ray ri and the projection plane, wherein the intersection point is used as a final projection point Interi and forms a directed line segment (Pi, Interi) with the corresponding Pi; the intermediate projection point Ei is located on the directed line segment.
3. The method for realizing dynamic projection of three-dimensional graphics based on parameters as claimed in claim 1, wherein: the projected geometry is a three-dimensional volumetric shape.
4. The method for realizing dynamic projection of three-dimensional graphics based on parameters as claimed in claim 1, wherein: the projection ratio λ is defined in the range of [0, 1 ].
5. The method for realizing dynamic projection of three-dimensional graphics based on parameters as claimed in claim 1, wherein: pi is the vertex of the projected geometry located at the surface or boundary.
6. The method for realizing dynamic projection of three-dimensional graphics based on parameters as claimed in claim 1, wherein the change of the projection rate λ is controlled by a dragging bar on the display screen.
7. The method as claimed in claim 1, wherein the projection plane is a plane or a curved plane.
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