JP2010286896A - Drawing processor and drawing processing method - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To quickly arrange, project and draw two-dimensional graphics in a three-dimensional virtual space, in equipment limited in memory capacity. <P>SOLUTION: The drawing processor includes: a coordinate calculation means for calculating coordinates of four vertexes of a bounding box of the two-dimensional graphics, based on arrangement information and a projection condition of the two-dimensional graphics in the three-dimensional virtual space; a parallelogram calculation means for calculating a parallelogram approximating to a rectangular shape constituted by the four vertexes calculated by the coordinate calculation means; a calculation means for calculating two-dimensional deformation information, based on the four vertexes calculated by the parallelogram calculation means; and a drawing means for deforming the shape of the two-dimensional graphics to be drawn, based on the two-dimensional deformation information calculated by the calculation means. <P>COPYRIGHT: (C)2011,JPO&INPIT

Description

  The present invention relates to a drawing processing apparatus and a drawing processing method.

  Conventionally, two-dimensional graphics are arranged and projected and drawn in a three-dimensional virtual space, a two-dimensional graphic is once drawn to obtain a bitmap, and the two-dimensional graphics are arranged and drawn in the three-dimensional virtual space. It was common. In addition, there is also a proposal for converting two-dimensional graphics shape information into polygons and rendering them as three-dimensional graphics instead of once converting them into a bitmap (for example, Patent Document 1). Furthermore, a proposal has been made to improve performance by rendering two-dimensional graphics instead of rendering three-dimensional graphics (for example, Patent Document 2).

JP 2002-260008 A JP-A-8-263690

  However, in the conventional general method, it is necessary to secure a memory capable of storing a bitmap having a size including two-dimensional graphics, and it is difficult to realize processing with a device having a small installed memory. . Further, according to the method of converting 2D graphics to polygons, there is a possibility that the processing load for converting complex 2D graphics to polygons may increase. Further, according to the method of holding the two-dimensional graphics, although the processing can be performed at high speed, there is a risk that the memory usage amount increases.

  The present invention has been made in view of such problems, and an object of the present invention is to project and draw two-dimensional graphics in a three-dimensional virtual space at high speed in an apparatus having a limited memory capacity.

  Therefore, the drawing processing apparatus of the present invention calculates coordinates at which the four vertices of the bounding box of the two-dimensional graphics are projected based on the arrangement information of the two-dimensional graphics in the three-dimensional virtual space and the projection conditions. A coordinate calculation means; a parallelogram calculation means for calculating a parallelogram approximated to a quadrangle constituted by four vertices calculated by the coordinate calculation means; and a parallelogram calculated by the parallelogram calculation means. Calculation means for calculating two-dimensional deformation information based on the four vertices; and drawing means for changing the shape of the two-dimensional graphics based on the two-dimensional deformation information calculated by the calculation means. .

  According to the present invention, two-dimensional graphics can be arranged and projected and drawn in a three-dimensional virtual space at high speed in a device having a limited memory capacity.

It is a figure which shows an example of the hardware constitutions of a drawing processing apparatus. 2 is a diagram illustrating an example of a functional configuration of a drawing processing apparatus according to Embodiment 1. FIG. 3 is a flowchart illustrating an example of processing in the drawing processing apparatus according to the first embodiment. It is a figure which shows an example of three-dimensional virtual space arrangement information. It is a figure for demonstrating a two-dimensional graphics drawing process command. FIG. 9 is a first diagram for explaining an example of a process in step 305; FIG. 10 is a second diagram illustrating an example of a process in step 305; It is a figure for demonstrating an example of the process of step 305 and step 306. FIG. It is a figure for demonstrating an example of the process of step 307. FIG. FIG. 6 is a diagram illustrating an example of a drawing result according to the first embodiment. It is a figure for demonstrating the other example of the process of step 306. FIG. FIG. 6 is a diagram illustrating an example of a functional configuration of a drawing processing apparatus according to a second embodiment. 10 is a flowchart illustrating an example of processing in the drawing processing apparatus according to the second embodiment. It is a figure which shows an example of a two-dimensional graphics drawing process command. 12 is a flowchart showing details of processing in step 1310. It is a figure for demonstrating an example of the process of step 1504. FIG. It is a figure which shows an example of the drawing result of Embodiment 2. It is a figure for demonstrating an example of the process of step 1306 and step 1307. FIG.

  Hereinafter, embodiments of the present invention will be described with reference to the drawings.

<Embodiment 1>
The first embodiment will be described.
FIG. 1 is a diagram illustrating an example of a hardware configuration of a drawing processing apparatus (computer). In FIG. 1, a CPU 102 executes and controls each function provided in the drawing processing apparatus. The ROM 103 stores programs that do not need to be changed and various parameters. The RAM 104 is configured by SDRAM, DRAM, or the like, and temporarily stores programs and data supplied from an external device or the like. The drawing unit 105 outputs the graphics drawn by the CPU 102 based on the program. A bus 101 is a system bus, and connects the CPU 102, the ROM 103, the RAM 104, and the drawing unit 105.

Hereinafter, the flow of processing of the present embodiment will be described with reference to FIGS.
FIG. 2 is a diagram illustrating an example of a functional configuration of the drawing processing apparatus according to the first embodiment. FIG. 3 is a flowchart illustrating an example of processing in the drawing processing apparatus according to the first embodiment. FIG. 4 is a diagram illustrating an example of the three-dimensional virtual space arrangement information. FIG. 5 is a diagram for explaining a two-dimensional graphics drawing processing command. FIG. 6 is a diagram (part 1) for explaining an example of the processing in step 305. FIG. 7 is a diagram (part 2) for explaining an example of the process in step 305. FIG. 8 is a diagram for explaining an example of the processing in step 305 and step 306. FIG. 9 is a diagram for explaining an example of the processing in step 307. FIG. 10 is a diagram illustrating an example of a drawing result according to the first embodiment. FIG. 11 is a diagram for explaining another example of the processing in step 306.
First, the three-dimensional virtual space arrangement information receiving unit 203 receives arrangement information (three-dimensional virtual space arrangement information) in the three-dimensional virtual space from another device or the like (step 301). Here, it is assumed that the three-dimensional virtual space is expressed by an XYZ orthogonal coordinate system, and the three-dimensional virtual space arrangement information is expressed by a three-dimensional affine transformation matrix. The three-dimensional affine transformation is expressed as a general form by a matrix of 3 rows and 4 columns, and the 3D virtual space arrangement information input here is the matrix of 3 rows and 4 columns shown in FIG.

Next, the projection condition receiving unit 204 receives the projection condition from another device or the like (step 302). The projection conditions are composed of viewpoint coordinates, projection plane information, and a projection method. Here, it is assumed that the coordinates of the viewpoint are (0, 0, −ze) (where ze> 0), the information on the projection plane is the XY plane, and the projection method is one-point perspective projection.
Next, the two-dimensional graphics drawing processing command receiving unit 202 receives a two-dimensional graphics drawing processing command from another device or the like (step 303). It is assumed that the two-dimensional graphics drawing processing command is a command for drawing processing of a polygon 501 (shaded portion) surrounded by AOBCD as shown in FIG. Subsequently, the two-dimensional graphics bounding box calculation unit 205 calculates a two-dimensional graphics bounding box based on the received two-dimensional graphics drawing processing command (step 304). Here, it is assumed that the bounding box is the smallest rectangle that includes two-dimensional graphics and has sides parallel to the XY axes. The polygonal bounding box in FIG. 5 (a) is a rectangle 502 (dotted line portion) shown in FIG. 5 (b).

  Next, the bounding box projection processing unit 206 calculates (coordinates calculation) the projection destinations of the four vertices of the bounding box when the calculated bounding box is projected based on the three-dimensional virtual space arrangement information and the projection conditions ( Step 305). Here, the process of step 305 will be described in more detail with reference to the schematic diagram of FIG. First, assuming that the viewpoint is E, the coordinates of E are (0, 0, −ze) (where ze> 0). When there is an arbitrary point P (x, y, z) and this is projected onto the XY plane according to the projection conditions, the projection destination of the point P becomes the intersection Q between the straight line PE and the XY plane. The coordinates of the intersection point Q are obtained as (x * d / (z + ze), y * d / (z + ze), 0). The mapping from the point P to the point Q can be expressed by a 4 × 4 homogeneous transformation matrix as shown in FIG. In addition, a 3 × 4 matrix expressing the 3D virtual space arrangement information can also be expressed by a homogeneous transformation matrix, and can be expressed by a 4 × 4 matrix as shown in FIG. Further, an arbitrary point A (x, y, z) in the XYZ orthogonal coordinate system is also expressed by four-dimensional coordinates in the homogeneous coordinate system as shown in FIG.

The bounding box projection processing unit 206 calculates the product of the perspective projection matrix derived from the three-dimensional affine transformation matrix and the projection condition as a projection destination where an arbitrary point A is projected by the three-dimensional virtual space arrangement information and the projection condition. The point A can be obtained by mapping the matrix. When expressed by an expression, it is as shown in FIG. Note that the Z coordinates of all four vertices of the bounding box are zero. Therefore, the bounding box projection processing unit 206 adds the four vertices A (−s, t, 0) and B (−s, B) of the bounding box shown in FIG. 5B to the expression shown in FIG. -T, 0), C (s, -t, 0), D (s, t, 0). This makes it possible to acquire four projection destination points.
The projection destinations of the four vertices A, B, C, and D of the bounding box calculated in step 305 are A ′, B ′, C ′, and D ′, respectively. FIG. 8A illustrates the projection destination points A ′, B ′, C ′, and D ′.

  Next, the parallelogram calculation unit 207 obtains a parallelogram that approximates a quadrangle surrounding the four projection points (step 306). Here, a method for deriving the parallelogram will be described. First, the parallelogram calculation unit 207 calculates the midpoint R of the line segment A′B ′, the midpoint S of the line segment B′C ′, the midpoint T of the line segment C′D ′, and the line segment D′ A ′. Each midpoint U is obtained. Next, the parallelogram calculation unit 207 calculates a parallelogram EFGH such that the points R, S, T, and U are midpoints of the four sides. The parallelogram calculation unit 207 may determine the parallelogram EFGH so that the sides FG and HE are parallel to the line segment RT so that the sides EF and GH are parallel to the line segment SU.

  Next, the two-dimensional deformation information acquisition unit 208 obtains two-dimensional deformation information from the parallelogram obtained in step 306 (step 307). In the parallelogram EFGH of FIG. 9A, the coordinates of the point F are (p, q). Further, the vector FE is (αe, βe), and the vector FG is (αg, βg). Here, it is assumed that a square E′F′G′H ′ having a length of 1 has a point F ′ that coincides with the origin and G ′ is arranged on the X axis, and the square E′F′G′H ′ Find the mapping to the parallelogram EFGH. What is obtained by expressing the mapping in the homogeneous coordinate system of the two-dimensional affine transformation is the matrix of 3 rows and 3 columns shown in FIG. Next, when a map for transforming the rectangular ABCD that is the shape of the bounding box of the two-dimensional graphics shown in FIG. 5B to E′F′G′H ′ in FIG. 9A is calculated, FIG. The 3 × 3 two-dimensional affine transformation matrix shown in c) is derived. Therefore, the mapping for transforming the rectangle ABCD into the parallelogram EFGH is obtained as the product of the matrix shown in FIG. 9B and the matrix shown in FIG. 9C.

Next, the two-dimensional graphics drawing unit 209 draws two-dimensional graphics based on the two-dimensional affine transformation matrix calculated in step 307 (step 308). As a result, a drawing result as shown in FIG. 10 is obtained. Thus, the drawing processing apparatus ends the processing shown in FIG.
Note that although the method of calculating the approximate parallelogram in step 306 has been described, other methods may be used as the calculation method. For example, the parallelogram calculation unit 207 calculates a parallelogram EFGH in which the sides that do not pass through the vertex having the largest corner among the inner angles of the rectangle A′B′C′D ′ as shown in FIG. It is good also as an approximate parallelogram. In the example of FIG. 11, the largest internal angle of the quadrangle A′B′C′D ′ is B ′, so that the parallelogram EFGH has two sides of the side D′ A ′ and the side C′D ′. Is obtained.

<Embodiment 2>
Embodiment 2 will be described.
Since the hardware configuration is the same as that described in the first embodiment, the description thereof is omitted.
Hereinafter, the flow of processing according to the present embodiment will be described with reference to FIGS.
FIG. 12 is a diagram illustrating an example of a functional configuration of the drawing processing apparatus according to the second embodiment. FIG. 13 is a flowchart illustrating an example of processing in the drawing processing apparatus according to the second embodiment. FIG. 14 is a diagram illustrating an example of a two-dimensional graphics drawing processing command. FIG. 15 is a flowchart showing details of the processing in step 1310. FIG. 16 is a diagram for explaining an example of the processing in step 1504. FIG. 17 is a diagram illustrating an example of a drawing result according to the second embodiment. FIG. 18 is a diagram for explaining an example of the processing in step 1306 and step 1307.
First, the three-dimensional virtual space arrangement information receiving unit 1203 receives arrangement information in the three-dimensional virtual space from another device or the like (step 1301). Here, it is assumed that the three-dimensional virtual space is expressed by an XYZ orthogonal coordinate system, and the three-dimensional virtual space arrangement information is expressed by a three-dimensional affine transformation matrix. The three-dimensional affine transformation is represented by a matrix of 3 rows and 4 columns as a general form. The three-dimensional virtual space arrangement information input here is assumed to be the same matrix of 3 rows and 4 columns as shown in FIG.

Next, the projection condition receiving unit 1204 receives the projection condition from another device or the like (step 1302). The projection conditions are composed of viewpoint coordinates, projection plane information, and a projection method. Here, it is assumed that the coordinates of the viewpoint are (0, 0, −ze) (where ze> 0), the information on the projection plane is the XY plane, and the projection method is one-point perspective projection.
Next, the two-dimensional graphics drawing processing command receiving unit 1202 receives a two-dimensional graphics drawing processing command from another device or the like (step 1303). Next, the drawing processing determination unit 1205 determines whether to draw with two-dimensional graphics or three-dimensional graphics (step 1304). If the two-dimensional graphics drawing processing command is a text drawing processing command, the drawing processing determination unit 1205 determines that drawing with two-dimensional graphics is performed. Otherwise, drawing is performed with three-dimensional graphics. Is determined.
Here, the input two-dimensional graphics drawing processing command will be described with reference to the example of FIG. FIG. 14A is a schematic diagram for explaining a two-dimensional graphics processing instruction. The drawing processing apparatus executes processing based on the graphic drawing processing command 1401 and then executes processing based on the text drawing processing command 1402. FIG. 14B shows a drawing result obtained when the figure drawing processing command 1401 and the text drawing processing command 1402 are drawn on the same XY plane.

First, the flow of processing for the graphic drawing processing instruction 1401 will be described. In the case of the graphic drawing processing instruction 1401, it is determined in step 1304 that it is not a text drawing processing instruction, so the three-dimensional graphics drawing processing unit 1211 draws as three-dimensional graphics (step 1310). The process of step 1310 will be described in more detail with reference to the flowchart of FIG.
First, the three-dimensional graphics drawing processing unit 1211 acquires the size of the bounding box of the two-dimensional graphics included in the figure drawing processing command 1401 (step 1501). The width s and height t of the acquired bounding box. Next, the three-dimensional graphics rendering processing unit 1211 secures a memory area in the RAM 104 that can store rectangular pixel data having a width s and a height t (step 1502).

Next, the three-dimensional graphics drawing processing unit 1211 outputs the drawing result of the figure drawing processing instruction 1401 to the secured memory area (step 1503). This processing is called so-called rasterization processing, and the three-dimensional graphics drawing processing unit 1211 converts the drawing processing command expressed in the vector format into data in the raster image format. Next, the 3D graphics rendering processing unit 1211 uses the stored raster image format data to calculate a 3D affine transformation matrix based on the 3D virtual space layout information and the projection conditions (step 1504). ). FIG. 16A represents the projection condition, and FIG. 16B represents the three-dimensional virtual space arrangement information as a 4 × 4 matrix. A transformation matrix T related to the graphic drawing processing instruction 1401 is represented as shown in FIG.
Subsequently, the three-dimensional graphics drawing processing unit 1211 projects and draws raster image format data based on the three-dimensional affine transformation matrix (step 1505). Here, the image interpolation calculation algorithm is not particularly limited. The nearest neighbor method or the bilinear method may be used. FIG. 17A shows a drawing result at the stage when the drawing process of the figure drawing processing instruction 1401 is completed.

Next, the text drawing processing instruction 1402 will be described. Returning to the processing of step 1303, the two-dimensional graphics drawing processing command receiving unit 1202 receives the text drawing processing command 1402. The drawing process determination unit 1205 determines whether to draw two-dimensionally in accordance with the received text drawing process command 1402 (step 1304). As described above, in the case of a text drawing processing command, the drawing processing determination unit 1205 determines to draw with two-dimensional graphics. Subsequently, the two-dimensional graphics bounding box calculation unit 1206 calculates the bounding box of the two-dimensional graphics (step 1305). The bounding box acquired here is assumed to be a rectangle ABCD in FIG.
Next, the bounding box projection processing unit 1207 calculates (coordinates calculation) the projection destinations of the four vertices of the bounding box when the calculated bounding box is projected based on the three-dimensional virtual space arrangement information and the projection conditions ( Step 1306). Since the detailed processing in step 1306 is the same as that in step 305 using FIG. 7 in the first embodiment, the description thereof is omitted here.

The projection destinations of the four vertices A, B, C, and D of the bounding box obtained by calculation in step 1306 are A ′, B ′, C ′, and D ′, respectively. FIG. 18A illustrates the projection destination points A ′, B ′, C ′, and D ′. Next, the parallelogram calculation unit 1208 obtains a parallelogram that approximates a quadrangle surrounding the four projection points (step 1307). Here, a parallelogram derivation method will be described with reference to FIG. In FIG. 18B, the parallelogram calculation unit 1208 obtains midpoints R, S, T, and U of each side of the quadrangle A′B′C′D ′. Next, the parallelogram calculation unit 1208 selects a straight line having an intersection with one of the sides B′C ′ and C′D ′ among the straight lines parallel to the line segment RT and the line segment SU as a point A ′. Pull from. The drawn straight line is a dotted line 1801 in FIG. The parallelogram calculation unit 1208 draws straight lines for other vertices in the same manner, and a dotted line 1802, a dotted line 1803, and a dotted line 1804 are added. The parallelogram calculation unit 1208 can derive the parallelogram EFGH included in the quadrangle A′B′C′D ′ by obtaining the intersections E, F, G, and H of these four dotted lines. it can.
Next, the two-dimensional deformation information acquisition unit 1209 calculates a two-dimensional affine transformation matrix as two-dimensional deformation information from the parallelogram EFGH (step 1308). Since the calculation method here is the same as the description using FIG. 9 in step 307 of the first embodiment, the description is omitted. Next, the two-dimensional graphics drawing unit 1210 executes a text drawing processing instruction 1402 using the acquired two-dimensional affine transformation matrix (step 1309). When the two-dimensional graphics drawing unit 1210 combines the drawing result by the text drawing processing command 1402 and the drawing result by the graphic drawing processing command 1401, a final drawing result as shown in FIG. 17B is obtained.

  In step 1304 of the present embodiment, it has been described that whether or not to draw with two-dimensional graphics is based on whether or not the two-dimensional graphics drawing processing command is a text drawing processing command. You may make it use. For example, the drawing processing determination unit 1205 determines that the influence on the output result is slight when the ratio of the two-dimensional graphics drawing processing command to the whole is lower than a specified threshold, and determines the two-dimensional graphics. It may be determined to perform the process according to the above. Alternatively, the drawing processing determination unit 1205 performs processing by two-dimensional graphics when it is predicted that the drawing time per frame will exceed a threshold value in repeatedly executing the two-dimensional graphics processing command in the animation processing. You may judge that.

<Other embodiments>
The object of the above-described embodiment is achieved by the following. That is, a storage medium (or recording medium) in which a program code of software that realizes the functions of the above-described embodiments is recorded is supplied to the system or apparatus. Then, the central processing means (CPU or MPU) of the system or apparatus reads and executes the program code stored in the storage medium. In this case, the program code itself read from the storage medium realizes the functions of the above-described embodiment, and the storage medium on which the program code is recorded constitutes the above-described embodiment.

  Further, by executing the program code read by the central processing means of the system or apparatus, an operating system (OS) or the like operating on the system or apparatus is part of the actual processing based on the instruction of the program code. Or do everything. The case where the functions of the above-described embodiment are realized by this processing is also included in the configuration of the embodiment.

  Furthermore, it is assumed that the program code read from the storage medium is written in a memory provided in a function expansion card inserted into the system or apparatus or a connected function expansion unit. Thereafter, based on the instruction of the program code, the CPU or the like provided in the function expansion card or the function expansion unit performs part or all of the actual processing, and the function of the above-described embodiment is realized by this processing. Included in the configuration.

  When the embodiment described above is applied to a storage medium, program codes corresponding to the flowcharts described above are stored in the storage medium.

  As described above, according to each of the above-described embodiments, it is possible to project and draw two-dimensional graphics in a three-dimensional virtual space at high speed in a device having a limited memory capacity.

  The preferred embodiments of the present invention have been described in detail above, but the present invention is not limited to such specific embodiments, and various modifications can be made within the scope of the gist of the present invention described in the claims.・ Change is possible.

102 CPU

Claims (12)

Coordinate calculation means for calculating coordinates on which the four vertices of the bounding box of the two-dimensional graphics are projected based on the arrangement information of the two-dimensional graphics in the three-dimensional virtual space and the projection condition;
Parallelogram calculation means for calculating a parallelogram approximated to a quadrangle constituted by four vertices calculated by the coordinate calculation means;
Calculation means for calculating two-dimensional deformation information based on the four vertices of the parallelogram calculated by the parallelogram calculation means;
Based on the two-dimensional deformation information calculated by the calculation means, a drawing means for deforming and drawing the shape of the two-dimensional graphics;
A drawing processing apparatus.   The drawing processing apparatus according to claim 1, further comprising a projection condition receiving unit that receives the projection condition.   The drawing processing apparatus according to claim 1, further comprising arrangement information receiving means for receiving the arrangement information. A drawing processing command receiving means for receiving a drawing processing command;
A bounding box calculating means for calculating a bounding box of two-dimensional graphics based on the drawing processing command received by the image processing command receiving means;
The drawing processing apparatus according to claim 1, further comprising: The arrangement information received by the arrangement information receiving means is represented by a three-dimensional affine transformation matrix,
The drawing processing apparatus according to claim 3, wherein the coordinate calculating unit calculates coordinates at which four vertices of a bounding box of the two-dimensional graphics are projected based on the three-dimensional affine transformation matrix and a projection condition.   The calculation means calculates two-dimensional deformation information represented by a two-dimensional affine transformation matrix based on the four vertices of the parallelogram calculated by the parallelogram calculation means. The drawing processing apparatus according to item.   7. The parallelogram calculation unit according to claim 1, wherein the parallelogram calculation unit calculates a parallelogram passing through a midpoint of each side of a quadrangle constituted by four vertices calculated by the coordinate calculation unit. Drawing processing device.   The drawing processing apparatus according to claim 1, wherein the parallelogram calculation unit calculates a parallelogram included in a quadrangle constituted by four vertices calculated by the coordinate calculation unit. In accordance with the drawing processing command received by the drawing processing command receiving means, the image processing apparatus further includes a determining unit that determines whether to draw with two-dimensional graphics or three-dimensional graphics,
The bounding box calculation unit calculates a bounding box of the two-dimensional graphics based on the drawing processing command received by the image processing command receiving unit when the determination unit determines to draw with two-dimensional graphics. The drawing processing apparatus according to claim 4. A coordinate calculation step of calculating coordinates at which the four vertices of the bounding box of the two-dimensional graphics are projected based on the arrangement information of the two-dimensional graphics in the three-dimensional virtual space and the projection condition;
A parallelogram calculation step for calculating a parallelogram approximated to a quadrangle constituted by the four vertices calculated in the coordinate calculation step;
A calculation step of calculating two-dimensional deformation information based on the four vertices of the parallelogram calculated in the parallelogram calculation step;
A drawing step of deforming and drawing the shape of the two-dimensional graphics based on the two-dimensional deformation information calculated in the calculating step;
A drawing processing method in a computer, comprising: Computer
Coordinate calculation means for calculating coordinates on which the four vertices of the bounding box of the two-dimensional graphics are projected based on the arrangement information of the two-dimensional graphics in the three-dimensional virtual space and the projection condition;
Parallelogram calculation means for calculating a parallelogram approximated to a quadrangle constituted by four vertices calculated by the coordinate calculation means;
Calculation means for calculating two-dimensional deformation information based on the four vertices of the parallelogram calculated by the parallelogram calculation means;
Based on the two-dimensional deformation information calculated by the calculation means, a drawing means for deforming and drawing the shape of the two-dimensional graphics;
A program that makes it work.   A computer-readable storage medium storing the program according to claim 11.

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