JPH01239676A - Polyhedron inside/outside setting method - Google Patents

Abstract

PURPOSE:To eliminate defective setting work and to realize labor saving by setting the direction of the surface of a plane including an intersection most separated from a straight line extending from an arbitrary point on a polyhedron in its inside as reference, and setting the directions of the surfaces of all of the planes other than that. CONSTITUTION:The straight line in parallel with an axis (y) passing a point P on the polyhedron is extended. A constitution plane crossing at the intersection Q where the maximum (y) coordinate value is obtained out of the intersections of the straight line and the polyhedron is set as R. When the intersection Q exists on a plane in parallel with a side or the axis (y) and does not cross with the polyhedron, fluctuation is attached on the (x) or the (z) coordinate value of the point P, then it is crossed. Next, the (y) component of a normal vector on the plane R is defined so as to be set a positive value, and its direction is designated as the surface side of the plane R. The normal vectors of all of the planes of the polyhedron are decided by utilizing the fact that the order of the apex shared by the adjacent planes is set reversely with each other setting the plane R as reference, and by dissolving the direction of the surface automatically, the inside and the outside of the polyhedron can be set.

Description

[Detailed Description of the Invention] [Field of Industrial Application] The present invention is applicable to computer graphics, CAD/CA
This paper relates to a method for setting the inside and outside of a polyhedron in M, etc.

[Conventional technology]

There are various ways to represent a polyhedron, but basically you just need to set vertices, edges, and faces. At this time, by setting the front and back sides of the faces, the inside and outside of the polyhedron can be identified, which is useful for body size and mass calculations, cross-sectional diagram creation, mesh division using finite element methods, etc. Furthermore, in computer graphics, for efficiency, only the face facing forward is often displayed.

The front and back sides of a surface are determined by the ordering of the vertices or edges that make up the surface. For example, as shown in Figure 2, the vertices of the surface are arranged in order in a counterclockwise direction when viewed from the outside of the object, and each vertex is traced to ensure that the direction in which the right-handed screw advances is the front side of the surface. There is.

Conventionally, an operator manually orders the vertices or edges to set the front and back sides of a surface when creating a polyhedron.

[Problem to be solved by the invention]

The ordering of vertices or edges as described above involves the setting of a vertex row (or edge row) with adjacency to form a surface, and the direction of the vertex row, i.e., clockwise or counterclockwise. The settings can be divided into two tasks. A vertex sequence with an adjacency relationship is, for example, for the surface 1 in Figure 2, v
l, v2. v3゜v4 or its directions v4. v3. v
2. It is vl. To set the front and back sides of a face, you must choose one of these.

Such work requires the operator to imagine a three-dimensional space. Although it is relatively easy to set the adjacency relationship between vertex rows, there are many errors in setting the orientation of vertex rows, which requires caution.

The present invention aims to automatically set the orientation of a vertex sequence in order to solve the above problems.

[Means to solve the problem]

The method of the present invention starts from an arbitrary point existing on the polyhedron and extends a straight line in a direction that intersects the polyhedron, and the distance from the point of intersection of the polyhedron and the straight line that is the farthest from the starting point is Find the constituent faces of the polyhedron containing points inside, set the direction of the face of the constituent faces based on the direction vector of the straight line, and calculate the faces of all other faces of the polyhedron based on the constituent faces. It is characterized by setting the direction.

[Effect]

First, prepare a straight line that intersects the polyhedron. To do this, as shown in FIG. 4, a point P existing on the polyhedron is set, and a straight line passing through the point P and parallel to the y-axis is extended. Such a straight line has a high probability of intersecting a polyhedron.

Next, from among the intersections of the above straight line and the polyhedron, the intersection Q where the y-located ring value is maximum is selected, and the constituent face of the polyhedron that intersects the above straight line at the intersection Q is set as R. If the intersection point Q is on a side of the polyhedron, or if the point P is on a side parallel to the y-axis of the polyhedron and does not intersect the polyhedron well, the intersect with the faces of the polyhedron.

Next, so that the y component of the normal vector of surface R is positive,
Determine the normal vector of surface R, and set its direction as the front side of R. Determine the normal vectors of all the faces of the polyhedron based on the surface R. If you know the direction of the front of all the faces, you can set the inside and outside of the polyhedron.

〔Example〕

FIG. 3 is an explanatory diagram showing an example of configuration information of a polyhedron. x, y, z coordinate values of each vertex of the polyhedron (Figure 3 (a))
, the shape of a polyhedron is expressed by a set of vertices forming each side (FIG. 3(b)) and a row of vertices forming each face (FIG. 3(C)).

Instead of the vertex sequence, a sequence of edges constituting the surface may be used as information about the surface. In addition, the winged edge data structure (PIXEL Nα13 rCAD/C
There are various data structures such as "Introduction to Shape Modeling for AM Users"), but no matter which data structure is used, what determines the front and back of a surface is the order in which the vertices or edges that make up the surface are arranged. . Therefore, although explanation will be given here based on the data structure shown in FIG. 3, this does not mean that it loses its membrane properties.

FIG. 2 is a conceptual diagram showing how to decide on the front and back sides of a surface. To determine the front and back sides of the face f1, arrange the vertices counterclockwise when viewed from the outside of the polyhedron (vl, v2, v
3. v4), and when tracing each vertex, the direction in which the right-hand screw advances is guaranteed to be the front side of the surface f1. Of course, the convention can be set in any way as long as the front and back sides of a face can be uniquely determined for a certain order of vertices (or edges).

The present invention automatically sets the orientation of vertices forming a surface, given the adjacency relationship between them. For example, the arrangement of the constituent vertices that sets the front direction of the face f1 is vl
, v2. v3. v4 or its reverse order v4. v3.
v2. This is what determines vl.

FIG. 1 is a flowchart showing the steps of the method for setting the front and back sides of a polyhedron according to the present invention.

First, in step S1, an arbitrary point on a polyhedron is found. To do this, select any one of the faces that make up the polyhedron (referred to as face F), and as shown in Figure 5,
The center of gravity G of a triangle (vlv2v3 in FIG. 5) consisting of three consecutive points among the vertices constituting the surface F can be found.

However, if the surface F is a concave shape, as shown in FIG. 5(b), the center of gravity G exists outside the surface F, and there is no guarantee that it will exist on the polyhedron. Therefore, the obtained center of gravity G is the surface F
It is necessary to check whether it is inside the .

To determine whether a point is inside a polygon, for example, when looking around each vertex of the polygon from that point once, calculate the sum of the swing angles of the line segments connecting that point and the vertex. (Refer to PIXEL N11.30 "Geometric figure processing by computer"). Shift three successive vertices one by one until the center of gravity is inside the surface F (in Fig. 5(b), v2v3v4, v3
v4v5, etc.) Perform the above operations. The point on the polyhedron obtained in this way will be called P.

Next, in step S2, the rotation of the polyhedron with a straight line passing through point P and parallel to the y-axis is determined, and the intersection point with the maximum y-coordinate value is selected (FIG. 4). If this intersection exists on the boundary line of the face of the polyhedron, give a slight fluctuation to the point PLy') is included inside the surface excluding the boundary.

Let us call the face of the polyhedron containing this intersection point R. The y component of the normal vector indicating the front side of such surface R is always a positive value. If this is not the case, the polyhedron will not be closed unless there is another surface above the surface R (in the y direction), which contradicts the above process.

Note that the use of straight lines parallel to the y-axis here is for convenience, and any straight line may be used. In that case,
Let R be a surface that includes not the intersection with the maximum y-coordinate value but the intersection with the maximum distance from the point P. However, for processing efficiency, it is preferable to use a straight line parallel to any one of the x, y, and z axes.

Next, in step S3, the order of the vertices of the surface R is determined. To do this, first find the normal vector N of the surface R from the given order of the constituent vertices (basically,
All you have to do is consider each side of the surface as a vector and find the sum of its cross products. For details, see "Hidden Surface Elimination of Polyhedra Using PIXELNQ, 25r Depth Buffer").

From step S2, since the value of the y component of the surface normal vector of the surface R is positive, the order of the constituent vertices remains as given if the value of the y component of the normal vector N is positive; If the value of the y component of is negative, the given order can be reversed. If the value of the y component of the normal vector N is 0, the surface R is parallel to the y-axis and is included when the intersection point is on the boundary line of the surface in step S2, so it has already been excluded.

Finally, in step S4, the vertex order of all the faces of the polyhedron is determined based on the vertex order of the face R. To do this, use the fact that the order of the vertices shared by adjacent faces is opposite to each other.

For example, in Figure 2, the shared vertex of surface 1 and surface f2 is v
3. v4, and its direction is v3-> when viewed from plane f1.
v4, and when viewed from the plane f2, v4->v3. In this way, starting from surface R, for all surfaces of the polyhedron,
The order of the constituent vertices can be set so that vertices shared with adjacent faces are in opposite directions.

In this way, if the order of the constituent vertex rows of all the faces is determined and the front and back sides of the faces are set, the inside and outside of the polyhedron have been set.

〔Effect of the invention〕

According to the present invention, since constituent vertices that correctly set the front and back sides of each face of a polyhedron are automatically determined, the labor of the operator is reduced and work errors can be eliminated.

[Brief explanation of the drawing]

Fig. 1 is a flowchart showing the procedure of the present invention, Fig. 2 is an explanatory drawing showing how to set the front and back sides of the faces of a polyhedron, Fig. 3 is an explanatory drawing showing configuration information of the polyhedron, and Fig. 4 is an explanatory drawing showing the configuration information of the polyhedron. point P
FIG. 5 is an explanatory diagram showing how a polyhedron intersects with a straight line passing through and parallel to the y-axis, and FIG. 5 is an explanatory diagram showing how to find points within the polygon. In Fig. 2, 1 is a polyhedron, fl, and f2 is a surface, vl. v2. v3. v4. v5. v6 represents the vertex. In Figure 4, 2 represents a polyhedron, P represents a point on the polyhedron, Q represents the point with the largest y-coordinate value among the intersections of polyhedron 2 with a straight line passing through P and parallel to the y-axis, and R represents the surface containing Q. . 3 in Figure 5.
4 is a plane (polygon), vl. V2. v3. v4. v5 is the vertex, G is vl, v2゜V3
represents the center of gravity of a triangle consisting of

Claims (1)

[Claims] In the method of setting the inside and outside of each face of a polyhedron used in computer graphics, CAD/CAM, etc.
Starting from an arbitrary point existing on the polyhedron, extend a straight line in a direction that intersects the polyhedron, and forming a straight line that includes the point furthest from the starting point among the intersections of the polyhedron and the straight line. Find the constituent faces of the polyhedron, set the front direction of the constituent faces based on the direction vector of the straight line, and set the front directions of all other faces of the polyhedron based on the constituent faces. Characteristic polyhedral interior and exterior setting method.