JPH1011590A - Method for geometrically dividing recessed polygon - Google Patents

Abstract

PROBLEM TO BE SOLVED: To prevent the generation of an error in advance by equalizing the number of projecting polygons with the number of triangles obtained by a triangle dividing method at the worst and to mostly make it less than the number of the triangles. SOLUTION: Whether each vertex of a recessed polygon is recessed or projected is judged, all the recessed vertexes are stored (SP 1) in a recessed vertex list and whether a recessed vertex exists in the recessed vertex list is judged (SP 2). When the recessed vertex exists, the recessed vertex is fetched (SP 3) from the head of the recessed vertex list, three vertexes consecutive in a prescribed direction including the fetched recessed vertex are noticed (SP 4) and whether a triangle formed by these three vertexes can be cut off or not is judged (SP 5). Then when it is judged that the triangle can be cut off, whether an N-polygon can be cut off not is judged. (SP 6). When it is possible, the N-polygon is cut off (SP 7) but when it is impossible on the contrary, the triangle is cut off (SP 8).

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for geometrically dividing a concave polygon (a polygon having an interior angle of any vertex exceeding 180 ° or a polygon having an interior angle of any vertex of 180 ° or more). The present invention relates to a novel method for dividing a concave polygon into a plurality of polygons having the largest number of vertices within a range not exceeding a preset number of vertices.

Here, the definition of a concave polygon is defined as described above, depending on the computer graphics system, when a polygon whose interior angle at any vertex exceeds 180 ° is a concave polygon, This is because a polygon having an inner angle of 180 ° or more at that vertex may be a concave polygon.

[0003]

2. Description of the Related Art Conventionally, in order to achieve high-speed drawing, a concave polygon is formed as a convex polygon (a polygon having an inner angle of all vertices of 180 ° or less, or a polygon having an inner angle of all vertices of less than 180 °). Has been proposed. As such a concave polygon division method,
(1) A method of dividing a concave polygon into a plurality of triangles (a so-called triangulation method, see JP-A-63-184882 and JP-A-3-50686), and (2)
A method of dividing a concave polygon into a plurality of trapezoids (a so-called trapezoidal division method, see Japanese Patent Application Laid-Open No. Sho 62-36690) has been proposed.

In the triangulation method, three adjacent vertices are selected from all vertices of a concave polygon with reference to concave vertices (vertexes having an internal angle exceeding 180 ° or vertices having an internal angle of 180 ° or more).
This is a method in which one vertex is selected, a triangle is cut out, and this process is repeated to make the original concave vertex a convex vertex (a vertex having an internal angle of 180 ° or less or a vertex having an internal angle of less than 180 °). Of course, even after the convex polygon is formed, the triangle is cut out and the original concave polygon is finally divided into a plurality of triangles.

The trapezoid division method is a method in which a concave polygon is divided by a scan line for each vertex to form a trapezoid, and each obtained trapezoid is divided into triangles.

[0006]

The triangulation method comprises:
A simple process can divide a concave polygon into multiple triangles.However, since all the vertices of the concave polygon are used to cut the triangle, the number of triangles after division may increase significantly depending on the shape of the concave polygon. As a result, there is an inconvenience that the amount of polygon data handled in the post-processing shadow calculation processing unit is significantly increased.

In the trapezoidal division method, a concave polygon is divided by a scan line for each vertex to form a trapezoid. Therefore, except for a case where a pair of vertices are located on the same scan line, one of the vertexes is one. It is necessary to set a vertex to be paired with the vertex of. And, since this vertex is not the vertex given in the original concave polygon, an error due to calculation accuracy or the like is inevitably caused by setting a new vertex, and due to this error, When the post-processing shadow calculation processing is performed, the drawing quality based on the shadow calculation result is deteriorated (specifically, the discontinuity of the brightness value, the perforation of the shape, and the like occur). There is.

[0008]

SUMMARY OF THE INVENTION The present invention has been made in view of the above problems, and does not require the setting of new vertices and makes the number of convex polygons after division equal to or less than that of the triangulation method. It is an object of the present invention to provide a method of geometrically dividing concave polygons that can be performed.

[0009]

According to a first aspect of the present invention, there is provided a method of geometrically dividing a concave polygon, wherein a concave vertex is detected from geometric data representing a concave polygon to be divided, and a concave vertex list is created. One concave vertex is selected, and a maximum number of vertices that can form a convex polygon is selected based on the selected concave vertex, and the selected vertex is formed by the selected concave vertex and the selected maximum number of vertices. This is a method in which a convex polygon is cut from the concave polygon and if the remaining polygons need to be divided, the above series of processing is repeated.

According to a second aspect of the present invention, a concave vertex is detected from geometric data representing a concave polygon to be divided, a concave vertex list is created, and one concave vertex list is created from the concave vertex list.
Three concave vertices are selected, two vertices in a predetermined direction are selected based on the selected concave vertices, it is determined whether or not a triangle formed by these three selected vertices can be cut, and the triangle is cut. If it is determined that it is possible, increase the number of vertices selected based on the concave vertex, and determine whether the convex polygon formed by all the selected vertices including the concave vertex can be cut out. In the range in which the number of selected vertices does not exceed a preset number, it is determined whether or not the convex polygon formed by all the selected vertices can be cut off, and the number of selected vertices is increased. The determination is repeated, and the convex polygon having the largest number of vertices is cut out from the concave polygon to be divided, and whether the remaining polygon is a concave polygon,
Or repeat the above series of processing on condition that it is a convex polygon having a larger number of vertices than the number of vertices set in advance,
If it is determined that the triangle is not cuttable, another one of the concave vertices is selected from the concave vertex list, and the series of processes is repeated.

[0011]

According to the first aspect of the present invention, a concave vertex is detected from geometric data representing a concave polygon to be divided, a concave vertex list is created, and one concave vertex list is formed from the concave vertex list. Is selected, and based on the selected concave vertices, the maximum number of vertices that can form the convex polygon is selected, and the convex polygon composed of the selected concave vertices and the selected maximum number of vertices is selected. From the concave polygon,
If the remaining polygons need to be divided, the above-described series of processing is repeated, so the number of convex polygons obtained by dividing the concave polygons is, at worst, the number of triangles obtained by the triangulation method. In many cases, the number of triangles can be made smaller than the number of triangles obtained by the triangulation method, and since new vertices are not created unlike the trapezoidal division method, errors are prevented from occurring. be able to.

According to the method for geometrically dividing a concave polygon according to claim 2, a concave vertex list is created by detecting concave vertices from geometric data representing a concave polygon to be divided, and one concave vertex list is formed from the concave vertex list. Is selected, two vertices in a predetermined direction are selected on the basis of the selected concave vertex, and it is determined whether or not a triangle formed by the selected three vertices can be cut, and the triangle can be cut. When it is determined that, the number of vertices selected on the basis of the concave vertex is increased, it is determined whether the convex polygon formed by all the selected vertices including the concave vertex can be cut, As long as the number of selected vertices does not exceed the preset number, repeat the process of increasing the number of selected vertices and determine whether the convex polygon formed by all the selected vertices can be cut or not And the most top The convex polygon having a large number is cut out from the concave polygon to be divided, and the above-mentioned series is performed under the condition that the remaining polygon is a concave polygon or a convex polygon having a larger number of vertices than a preset number of vertices. Is repeated, and if it is determined that the triangle is not cuttable, another concave vertex is selected from the concave vertex list, and the series of processes is repeated. Therefore, the concave polygon is divided. The number of convex polygons obtained by this can be made at worst equal to the number of triangles obtained by the triangulation method, and in many cases, can be smaller than the number of triangles obtained by the triangulation method, and Since a new vertex is not created unlike the trapezoidal division method, it is possible to prevent an error from occurring.

[0013]

Embodiments of the present invention will be described below in detail with reference to the accompanying drawings. FIGS. 1 and 2 are flow charts for explaining one embodiment of the method for geometrically dividing concave polygons according to the present invention. In step SP1, it is determined whether each vertex of the concave polygon is a concave vertex or a convex vertex, and all the concave vertices are stored in a concave vertex list.
In P2, it is determined whether or not a concave vertex exists in the concave vertex list. If there is a concave vertex in the concave vertex list, in step SP3, a concave vertex is extracted from the top of the concave vertex list. In step SP4, the concave vertex is extracted in a predetermined direction (for example, counterclockwise). Focusing on three vertices continuous in the direction
At 5, it is determined whether the triangle formed by these three vertices can be cut. If it is determined that the triangle can be cut, it is determined in step SP6 whether or not the N polygon can be cut. Here, N is an integer of 4 or more, and in step SP6 and step SP12 to be described later, the value of N is sequentially increased until it is determined that cutout is impossible.

Then, if the N polygon can be cut off,
In step SP7, the N polygon is cut out. Conversely, if the N polygon cannot be cut out, in step SP8, the triangle is cut out. After the processing of step SP7 or step SP8 is performed, in step SP9, it is determined whether or not the extracted concave vertex has become a convex vertex by cutting a triangle or N-gon. And
If a convex vertex is reached, the determination in step SP2 is performed again. Conversely, if it is not a convex vertex, the process of step SP4 is performed again.

If it is determined in step SP5 that the triangle cannot be cut, step SP1
At 0, focusing on three vertices that are continuous in the opposite direction (for example, clockwise) including the extracted concave vertex, at step SP11, it is determined whether a triangle formed by these three vertices can be cut out. Is determined.
If it is determined that the triangle can be cut, it is determined in step SP12 whether the N-gon can be cut.

Then, if the N polygon can be cut out,
In step SP13, the N polygon is cut out,
If the polygon cannot be cut, a triangle is cut in step SP14. After the processing of step SP13 or step SP14 is performed, step SP15
In, it is determined whether or not the extracted concave vertex has become a convex vertex by cutting a triangle or N-gon. And when it becomes a convex vertex, the determination of step SP2 is performed again. Conversely, if it is not a convex vertex, the process of step SP10 is performed again.

If it is determined in step SP11 that the triangle cannot be cut, the process proceeds to step SP11.
In step 16, the extracted concave vertex is stored at the end of the concave vertex list, and the process of step SP3 is performed again. If it is determined in step SP2 that there are no concave vertices in the concave vertex list, a polygon exceeding the N angle is divided into polygons having the N angle or less in step SP17, and the series of processing ends as it is.

FIGS. 3 and 4 are flow charts for explaining in more detail the main parts of the flow charts of FIGS. 1 and 2. In step SP1, a concave vertex is extracted from the head of the concave vertex list, and in step SP2, it is determined whether three consecutive vertices including the extracted concave vertex in a predetermined direction form a convex polygon. When forming a convex polygon, in step SP3,
It is determined whether there is no other vertex in the convex polygon and whether there is an intersection between another edge (including the edge of the already divided polygon) and the edge of the convex polygon. If it is determined that there is no other vertex in the convex polygon and there is no intersection between the other edge and the edge of the convex polygon, in step SP4, the vertex from the extracted concave vertex to the current vertex is determined. It is determined whether or not the polygon constituted by is a convex polygon having N angles. If it is determined that it is an N-angle convex polygon, it is determined in step SP5 whether the extracted concave vertex becomes a convex vertex by cutting out the N-angle convex polygon. If it is determined that the vertices will be convex vertices, step SP
At 6, the convex polygon composed of the extracted concave vertex to the current vertex is cut out. Conversely, if it is determined that the vertices will not be convex vertices, in step SP7,
The current vertex is shifted by one in a predetermined direction, and step SP is performed again.
Perform the determination of 2.

If it is determined in step SP4 that the polygon is not an N-angle convex polygon, then in step SP8, the number of polygons formed by the vertices from the extracted concave vertex to the current vertex is 4 or more. It is determined whether or not the number of corners is four or more. In step SP9, a polygon having the current number of corners minus one is cut out.

If it is determined in step SP3 that there is another vertex in the convex polygon or that there is an intersection between another edge and the edge of the convex polygon, in step SP10, the extracted concave vertex is extracted. It is determined whether or not the number of angles of the polygon formed by the vertices from to the current vertex is 4 or more. If it is determined that the number of angles is 4 or more, in step SP11, the current angle is -1
Cut out polygons with the number of corners.

If it is determined in step SP2 that the polygon is not a convex polygon, step SP8, step SP10
In, when it is determined that the number of corners is not 4 or more,
In step SP12, it is determined whether three vertices, including the extracted concave vertices, which are continuous in the direction opposite to the predetermined direction, form a convex polygon. When forming a convex polygon, in step SP13, there is no other vertex in the convex polygon, and another edge (including the edge of the already divided polygon) and the edge of the convex polygon are combined. It is determined whether there is no intersection. If it is determined that there is no other vertex in the convex polygon and there is no intersection between another edge and the edge of the convex polygon,
In step SP14, it is determined whether or not the polygon composed of the vertices from the extracted concave vertex to the current vertex is an N-angle convex polygon. If it is determined that the polygon is a convex polygon having N angles, in step SP15,
It is determined whether or not the extracted concave vertex becomes a convex vertex by cutting out a convex polygon having N angles. And
If it is determined to be a convex vertex, step SP16
In, a convex polygon formed from the extracted concave vertex to the current vertex is cut out. Conversely, if it is determined that the vertices will not be convex vertices, in step SP17,
The current vertex is shifted by one in a predetermined direction, and step SP is performed again.
A determination of 12 is made.

If it is determined in step SP14 that the polygon is not an N-angle convex polygon, then in step SP18, the number of polygons consisting of the vertices from the extracted concave vertex to the current vertex is 4 or more. It is determined whether or not the number of corners is four or more. In step SP19, a polygon having the current number of corners minus one is cut out.

If it is determined in step SP13 that there is another vertex in the convex polygon or that there is an intersection between another edge and the edge of the convex polygon, in step SP20, the extracted concave vertex is extracted. It is determined whether or not the number of angles of the polygon constituted by the vertices from to the current vertex is 4 or more. If it is determined that the number of angles is 4 or more, in step SP21, the current angle −
Cut out the polygon with 1 corner.

If it is determined in step SP12 that the polygon is not a convex polygon, step SP18, step SP18
If it is determined in step 20 that the number of corners is not four or more, the series of processing ends. Therefore, by performing the processing of the above-described flowchart, it is possible to divide the concave polygon into polygons having the largest number of corners within a range not exceeding a preset number of corners. As a result, the amount of data handled in the computer graphics application can be reduced, and real-time processing can be easily handled. In the case of a radiosity application, the calculation load is 2 polygons.
Since the number increases in proportion to the power, reducing the number of polygons after the division by performing the processing of the above-described flowchart will greatly reduce the calculation load.

This will be described in more detail by taking the concave hexagonal division shown in FIG. 5 as an example. Note that, in FIG.
-5. (1) Set the maximum value N of the number of corners of the cut polygon. Specifically, N = 5 is set. (2) Check which vertex of the original hexagon is a concave vertex. Specifically, the cross product of vectors representing two sides sandwiching the vertex is calculated, and whether or not the cross product is a concave vertex is determined based on whether or not the value of the cross product is 0 or more. In the case of the concave hexagon in FIG. 5, the concave vertex is zero.

(3) Starting from the concave vertex 0, it is determined whether three points (0-1-2) can be cut counterclockwise. In the case of the concave hexagon shown in FIG. 5, cutting can be performed. (4) It is determined whether the cut polygon is an N-convex polygon or less. In this case, since it is 0-1-2, it is a tri-convex polygon. (5) It is determined whether the concave vertex 0 has become a convex vertex.
In the case of the concave hexagon shown in FIG.

(6) Move the current center vertex counterclockwise by 1
Stagger. In this case, vertex 1 becomes vertex 2. (7) Starting from the concave vertex 0, 3 points (0-2
It is determined whether or not -3) can be cut. In the case of the concave hexagon shown in FIG. 5, cutting can be performed. (8) It is determined whether or not the cut polygon is an N convex polygon or less. In this case, since it is 0-1-2-3, it is a quadrilateral convex.

(9) It is determined whether the concave vertex 0 has become a convex vertex. In the case of the concave hexagon shown in FIG. 5, it is a convex vertex (it is already a convex vertex). (10) Shift the current center vertex one counterclockwise.
In this case, vertex 2 becomes vertex 3. (11) Starting from the concave vertex 0, three points (0-
It is determined whether 3-4) can be cut. In the case of the concave hexagon shown in FIG. 5, cutting can be performed.

(12) It is determined whether the cut polygon is an N-convex polygon or less. In this case, since it is 0-1-2-3-4, it is a 5-convex polygon (N = 5). (13) It is determined whether the concave vertex 0 has become a convex vertex. In the case of the concave hexagon shown in FIG. 5, it is a convex vertex (it is already a convex vertex). (14) Cut out the 5-convex polygon (0-1-2-3-4).

(15) It is determined whether the remaining polygons are N-gons or less. In this case, it is equal to or smaller than the N-sided polygon. (16) The division processing is terminated as it is. Therefore, when the triangulation is performed, four triangles are obtained, whereas when the division is performed based on this embodiment, the number of the obtained polygons is two.

This will be described in more detail with reference to the division of a concave pentagon shown in FIG. In FIG. 6, each vertex is set to 0.
-8. (1) Set the maximum value N of the number of corners of the cut polygon. Specifically, N = 5 is set. (2) Check which vertices among the vertices of the original concave octagon are concave vertices, and register them in the concave vertex list. Recess 9 in FIG.
In the case of a square, the concave vertices are 3 and 4.

(3) Starting from the first concave vertex 3 in the concave vertex list, it is determined whether three points (3-4-5) can be cut counterclockwise. In the case of the concave octagon in FIG. 6, cutting is impossible. (4) Starting from concave vertex 3 and clockwise 3 points (3-2-
It is determined whether or not 1) can be cut. In the case of the concave octagon in FIG. 6, cutout is possible.

(5) It is determined whether or not the cut polygon is an N convex polygon or less. In this case, since it is 3-2-1, 3
It is a convex polygon. (6) It is determined whether the concave vertex 3 has become a convex vertex.
In the case of the concave octagon in FIG. 6, it is a concave vertex. (7) Shift the current center vertex one clockwise. In this case, vertex 2 becomes vertex 1.

(8) Starting from concave vertex 3 and clockwise 3
It is determined whether or not the point (3-1-0) can be cut. FIG.
Can be cut off in the case of a concave octagon. (9) It is determined whether the cut polygon is an N-convex polygon or less. In this case, since it is 3-2-1-0, it is a quadrangle convex. (10) It is determined whether the concave vertex 0 has become a convex vertex. In the case of the concave octagon in FIG. 6, it is a concave vertex.

(11) Move the current center vertex clockwise by 1
Stagger. In this case, vertex 1 becomes vertex 0. (12) Starting from the concave vertex 0, three points clockwise (3-0
It is determined whether or not -8) can be cut. In the case of the concave octagon in FIG. 6, cutout is possible. (13) It is determined whether or not the cut polygon is N convex or less. In this case, since it is 3-2-1-0-8, it is a 5-convex polygon (N = 5).

(14) It is determined whether the concave vertex 0 has become a convex vertex. In the case of the concave octagon in FIG. 6, it becomes a convex vertex. (15) Cut out the 5-convex polygon (3-2-1-0-8) and delete vertex 3 from the concave vertex list. (16) Extract the processing target concave vertex from the concave vertex list. In this case, the number of concave vertices is four.

(17) Starting from the concave vertex 4, it is determined whether three points (4-5-6) can be cut counterclockwise.
In the case of the concave octagon in FIG. 6, cutout is possible. (18) It is determined whether or not the cut polygon is an N convex polygon or less. In this case, since it is 4-5-6, it is a tri-convex polygon. (19) It is determined whether the concave vertex 3 has become a convex vertex. In the case of the concave octagon in FIG. 6, it becomes a convex vertex.

(20) Shift the current center vertex by one counterclockwise. In this case, vertex 5 becomes vertex 6. (21) Starting from concave vertex 4 and turning 3 points counterclockwise (4-
It is determined whether 6-7) can be cut. In the case of the concave octagon in FIG. 6, cutout is possible. (22) It is determined whether or not the cut polygon is an N convex polygon or less. In this case, since it is 4-5-6-7, it is a quadrangle.

(23) It is determined whether the concave vertex 4 has become a convex vertex. In the case of the concave octagon in FIG. 6, it is a convex vertex (already a convex vertex). (24) Shift the current center vertex one counterclockwise.
In this case, vertex 6 becomes vertex 7. (25) Starting from concave vertex 4 and turning 3 points counterclockwise (4-
It is determined whether or not 7-8) can be cut. In the case of the concave octagon in FIG. 6, cutout is possible.

(26) It is determined whether or not the cut polygon is N convex or less. In this case, since it is 4-5-6-7-8, it is a 5-concave polygon (N = 5). (27) It is determined whether the concave vertex 4 has become a convex vertex. In the case of the concave octagon in FIG. 6, the vertexes are convex (they are already convex vertices). (28) Decrease the number of vertices of the cut polygon by one.

(29) The four convex polygons (4-5-6-7) are cut out, and vertex 4 is deleted from the concave vertex list. (30) Since there are no more concave vertices in the concave vertex list, a series of division processing is terminated on condition that the number of vertices of the remaining polygons is N or less. Therefore, if it is not possible to cut a convex polygon even if the vertices are selected counterclockwise starting from the concave vertex, N is set to the upper limit of the number of vertices by selecting the vertex clockwise starting from the concave vertex. Convex polygons can be cut out.

Further, while the number of obtained triangles is four when triangulation is performed, the number of obtained polygons is two when division is performed according to this embodiment. In the case of dividing a concave polygon as shown in FIG. 7 and when processing a concave vertex sandwiched between concave vertices first, whether the vertex is selected in any of the clockwise direction and the counterclockwise direction is selected. Then, it becomes impossible to cut out the convex polygon.
However, in such a case, the corresponding concave vertex may be added to the end of the concave vertex list, and by performing the above-described series of processing with the concave vertex 2 or the concave vertex 4 as a reference,
The division of concave polygons can be achieved.

8 to 10 show the result of dividing a concave polygon by the triangulation method {see each figure (A)} and the result of dividing a concave polygon by the method of the present invention {see each figure (B)}.
In each case, the number of polygons obtained by the method of the present invention is small. However,
In the case of FIG. 8, N = 5 or more, and in the case of FIG.
As described above, in the case of FIG. 10, N is set to 8 or more.

[0044]

According to the first aspect of the present invention, the number of convex polygons obtained by dividing a concave polygon is made equal to the number of triangles obtained by a triangulation method even in the worst case. Since the number of triangles can be made smaller than the number of triangles obtained by the triangulation method, and a new vertex is not created unlike the trapezoidal division method, there is a specific effect that an error can be prevented from occurring.

[Brief description of the drawings]

FIG. 1 is a flowchart illustrating a part of an embodiment of a method of geometrically dividing a concave polygon according to the present invention.

FIG. 2 is a flowchart illustrating the rest of the embodiment of the method for geometrically dividing a concave polygon according to the present invention;

FIG. 3 is a flowchart for explaining a part of a main part of the flowcharts of FIGS. 1 and 2 in more detail;

FIG. 4 is a flowchart describing in more detail the remaining part of the flowcharts of FIGS. 1 and 2;

FIG. 5 is a diagram showing an example of a concave polygon divided by the method of the present invention.

FIG. 6 is a diagram showing another example of a concave polygon divided by the method of the present invention.

FIG. 7 is a diagram showing still another example of a concave polygon divided by the method of the present invention.

FIG. 8 is a diagram showing a result of triangulation of an example of a concave polygon and a result of division by the method of the present invention.

FIG. 9 is a diagram showing a result obtained by dividing another example of a concave polygon into triangles and a result obtained by dividing the triangle by the method of the present invention.

FIG. 10 is a diagram showing a result obtained by dividing still another example of a concave polygon into triangles and a result obtained by dividing according to the method of the present invention.

Claims (2)

[Claims] 1. A concave vertex list is created by detecting concave vertices from geometric data representing a concave polygon to be divided, one concave vertex is selected from the concave vertex list, and a convex vertex is selected based on the selected concave vertex. Select the maximum number of vertices that can form a polygon, cut out the convex polygon consisting of the selected concave vertex and the selected maximum number of vertices from the concave polygon, and split the remaining polygons. If
A method of geometrically dividing a concave polygon, comprising repeating the series of processes. 2. A concave vertex list is created by detecting concave vertices from geometric data representing a concave polygon to be divided, one concave vertex is selected from the concave vertex list, and a predetermined direction is determined based on the selected concave vertex. Is selected, and it is determined whether or not a triangle formed by the selected three vertices can be cut. When it is determined that the triangle can be cut, the triangle is selected based on the concave vertex. Increase the number of vertices, and determine whether the convex polygon formed by all the selected vertices including the concave vertex can be cut out, and the number of selected vertices does not exceed the preset number In a range,
It repeats the process of increasing the number of selected vertices and determining whether or not the convex polygon formed by all the selected vertices can be cut out. The above-described series of processing is repeated on condition that the remaining polygon is a concave polygon or a polygon having a larger number of vertices than a preset number of vertices, and it is determined that the triangle cannot be cut. If so, select another concave vertex from the concave vertex list,
A method of geometrically dividing a concave polygon, comprising repeating the series of processes.