JP2828759B2 - Free-form surface division method - Google Patents

Description

Description: TECHNICAL FIELD The present invention relates to graphics and three-dimensional CAD (Computed Ai
ded Design) For more information on how to divide a free-form surface in a system, see shading.
The present invention relates to a method of dividing a curved surface into polygons at a hidden line or a hidden line.

Conventional technology Conventional technology renders (depicts) a trimmed surface
There were few ways to do this. The trimmed surface refers to a surface formed by changing the boundary of the surface by calculation of interference between two surfaces, cutting by a plane, or making a hole in the surface at the design stage. The advantage of the trimmed surface is that the original surface data is retained even if the boundary of the surface is changed.

The present invention is designed so that such a curved surface can be easily divided into triangles. In a conventional method of dividing a trimmed surface into polygons, the surface itself is divided, the surface is re-patched, and the divided surface is further divided into triangles.
The problem with this method is that the re-patched surface data is not accurate, which causes a problem of roughness. Also, G ¹ connections between patches generated by Ripatchingu is (continuity of the surface and the adjacent surface - degree - smoothness) on the problem.

Rockwood and colleagues view trimmed patches in real time
It describes a modular approach to segmenting and rendering in a coordinate system. This method divides the trimming range into a single closed loop of curves containing each range in parameter space. To divide into these ranges, we must find the local maxima and minima of the trimming curve using the Newton method. There is overhead due to the repetition of the Newton method, and cracking occurs in this method because the trimming range is divided. The method of the present invention does not divide the trimming area into patches, but only creates one base polygon first.

Many CAD systems use NURBS as their surface representation
(Non-uniform rational B-spline).
Therefore, many applications dealing with NURBS have been proposed. Rendering of trimmed NURBS using modified forward difference has also been proposed, but it is very complex.

The present invention has been made in view of the above situation, and has been made to provide a method of dividing a free-form surface, which is intended to divide a polygon of a curved surface into triangles by a simple method. is there.

In order to achieve the above object, the present invention provides a three-dimensional CAD system in which a boundary curve of a curved surface and a boundary curve of a hole are divided into polylines, all the curved surfaces of the boundary curves are poly-lined, and the poly-line data is converted to a curved surface. The point that becomes the shortest distance between the base polygon and the polygon that forms the hole in the curved surface in the two-dimensional parameter space, using the outer surface boundary mapped to the two-dimensional parameter space as the base polygon. , Connect the points that form the shortest distance to each other, connect the polygons that make up the hole to the base polygon, convert the boundary of the curved surface in three-dimensional space into one polygon in two-dimensional parameter space, Except for the points included in the polygon in the parameter space, add the internal points in the two-dimensional parameter space corresponding to the points of the curved surface in the three-dimensional space, When necessary, the main polygon is divided into triangular polygons using the internal points, and the divided polygons are returned from the two-dimensional parameter space to the original three-dimensional space. It is characterized by being converted into. Hereinafter, a description will be given based on examples of the present invention.

 The present invention performs the following processes (A) and (B).

(A) Create a polygon from the boundary of a curved surface.

Generate polygons, following the surface boundaries and all holes.

(B) Triangulate the main polygon.

The division is repeated until the original polygon becomes a triangular polygon.

Here, the above two processes (A) and (B) are considered in the depth direction. Hereinafter, each of the processes (A) and (B) will be described in order.

FIG. 1 is a flowchart for explaining the process (A) of the method for dividing a free-form surface according to the present invention. Less than,
The processing (A) will be described.

Process (A): Creating a polygon from a boundary of a curved surface The purpose of this process (A) is to create a polygon by following a curve representing an original curved surface boundary and a hole of the curved surface. All edges of the polygon are straight segments, and all curved ridges are divided into straight segments by a fixed number of divisions. The structure for storing polygons is as shown in Table 1 below.

In this structure when this processing (A) is completed, points in the three-dimensional space are stored in the array ptabR, and points in the two-dimensional parameter space are stored in ptabP. Points in the parameter space are represented by parameters u and v at points on the screen, and the points are represented in a two-dimensional space. With this concept, the speed can be greatly increased when the polygon is divided into triangles in the second stage.

In this process (A), the number of internal points corresponding to the number of screen divisions is generated. The purpose of adding the internal points is to make the shape of the generated polygon as close as possible to an equilateral triangle.

 This processing (A) includes the following steps.

step1; Create a base polygon from the boundary of the curved surface.

step2-4; Each hole is made into an individual polygon.

step5; Join all other generated polygons to the base polygon.

step6; add internal points Hereinafter, each step will be described in order.

step1; Make a base polygon.

For the ridge line at each boundary of the curve, the terminal point and the parameter values on the curved surface at the terminal point are obtained, and all data are stored in the polygon structure. The polygon variable tabsize1 stores the number of stored points. All curved edges are further divided into straight line segments. The polygon variable bond is set to 1 for all straight line segments in this process. Every stored point is given a unique number id (vertex) corresponding to the address of that point's coordinates in the array structures ptabP and ptabR.
This polygon is called a base polygon. In this processing, the variable bond is 1 at all points. In the example of FIG. 3A, the ids of the base polygons are 1, 2, 3, and 4.

step2,3,4; Generate polygons from each hole.

For each hole on the curved surface, create polygons individually in the same way as the above process, and
Link to and join. In fact, the structure One Polygon
Join the polygon to the pointer next. In this step, the bond flag is set to 1 for all points. Third
In the example of FIG. 7A, the ids of the holes are created in the polygon as 5, 6, and 7.

step5; Join all other generated polygons to the base polygon.

List onepoly to which the base polygon is linked
To other generated polygons.

 The joining procedure is as follows.

Find the polygon that is the closest distance from any point of the base polygon to all polygons other than the base polygon. The state of both end points to be connected is checked, and if any point is in the state of 2 (when already connected to another polygon), the point is discarded. It also checks whether the new edge between the two points does not intersect with either the base polygon or the candidate polygon, and if so, checks the other points. Of newly created edges
The bond flag is set to a value of 0 because it is not the original boundary. When the two selected points satisfy all the conditions, a point on the base polygon is connected to a point on the candidate polygon, and the two polygons are linked. Note that when linking two polygons, the base polygon can be traced counterclockwise and the candidate polygon can be traced clockwise. The two polygons taken together become the new base polygon, and the base polygon is always generated in a counterclockwise fashion.

This stpe5 is repeated for all other candidates,
Eventually, only one base polygon will be generated.

In the example of FIG. 3B, since the distance between the point 2 of the base polygon and the point 7 of the hole is the shortest, 2 and 7 are combined to generate a polygon. Array id of polygon id
s is “1,2,7,5,6,7,2,3,4.” Table 2 shows the meanings of the status flags.

step6; Add interior points.

Add the interior points of the surface to the polygon data structure except when the surface type is flat. These internal points are added as auxiliary data in order to make the polygon a shape close to a curved surface when forming the polygon and to make the shape of the polygon as close to an equilateral triangle as possible. Also, each interior point is ptabP and p
Given an id corresponding to the address of the coordinate of the point in tabR. The maximum number of points that can be added is (divnum-
1) * (divnum-1). For example, if the number of divisions is four, the maximum number of points is nine. These interior points are generated using a simple procedure based on the increment of the surface parameters u and v so as to have a certain distance in the parameter space of the surface. Check if any point is included in the hole on the curved surface, and if so, do not use that point.

Whether or not an internal point is included in a polygon is checked as follows. That is, to check if the point is inside the surface polygon and outside any holes on the surface. This procedure is based on the intersection of two-dimensional line segments, and the procedure itself is very simple and fast.

Create an arbitrary two-dimensional point in the parameter space.

Make a straight line by connecting that point to the point to be checked. Count the intersections with all edges of the curved polygon.

If the count ends with an even number, it is a point outside the surface boundary.

For each hole, count the intersections with all edges of the hole polygon.

If the count ends in an odd number, it is in the hole.

FIG. 3 (c) shows a polygon with point states. Interior points are marked with x. Since the remaining points are not included in the polygon, the number of divisions is four, but seven points are valid.

The table storing the ids of the internal points is defined as shown in Table 3.

Also, the variable tabsize in the structure of the main polygon
Stores the number of internal points.

FIG. 2 is a flowchart for explaining the process (B) of the method for dividing a free-form surface according to the present invention. Less than,
The processing (B) will be described.

Process (B): The main polygon is triangulated.

In the second stage of the processing (B), the base polygon is divided into triangular polygons. Interior points are used to generate triangular polygons. At the end of this step, the polygon table is linked to all triangulated polygons. This is where speed is evaluated.

The procedure at this stage includes the following steps. Hereinafter, the steps will be sequentially described.

step1; Allocate the base edge of the base polygon.

The base edge may be a polygon. Here, the first and second points are set as starting points. The base edge forms one edge of the triangular polygon. To do so, find a third point that connects to the base edge forming the triangle.

step2; If there are no polygons to be processed, the process ends.

 Every polygon has three edges.

step3,4; If the polygon has three edges, get a new base edge for the next polygon.

First, start with the base polygon. As you continue to process the base polygons, many polygons (some of which have more than two edges) are linked to the list.

step5; Get all possible candidate points from the base edge.

There are many points that can be joined to the base point, but some are inappropriate because they intersect the base polygon. Find all candidate points related to edges that are close to the base edge.

step6; Select one point.

One point is selected from candidate points having the maximum angle with respect to the base edge. That is, it gives the maximum angle when connecting to both end points of the base edge.

step7: Form a triangulated polygon and link to the list.

Connect the selected point to the base edge to create a triangular polygon. Link this polygon to the list.

step8; Change the base polygon and base edge and return to step2.

The base polygon is updated to generate a triangular polygon. The triangular polygon may be divided into two polygons by dividing the base polygon. One of these will be the new base polygon and the other will link to the list for later processing. Update a new base edge.

 Next, the steps 5, 6, and 8 will be described in more detail.

step5; Selection of appropriate candidate point An appropriate candidate point is selected depending on the edge close to the base edge. The structure shown in Table 4 is used to hold the current base edge.

There are two cases, convex or concave, at the angle with each adjacent edge. All selected points must be inside the boundary defined for the base edge. If the angle to a certain edge is concave, use 90 degrees,
If the angle is convex, use your own angle. If the point is not inside the defined boundary, the concave corner is not 90 degrees
Use 180 degrees. If no suitable point is found, select a new base edge and repeat this process. Care should be taken not to select a point on the same line as the base edge because the area will be a zero triangle.

FIG. 4 (a) shows the search space when there are two convex corners with arrows.

FIG. 4 (b) shows the search space when there are convex corners and concave corners with arrows.

FIG. 4 (c) shows the search space when there are two concave corners with arrows. The angle is 90 degrees for a concave corner, but if the search space has no points, the angle is increased to 180 degrees. When selecting a candidate point, the only point in the structure of the base edge used is its corner. This makes the selection process independent and therefore very compact.

step6; Select one point in the selection process Select the point that gives the largest angle among the angles made between the points and both ends of the base edge. This situation is shown in FIG.
Shown in There are two candidate points 3 and 4. The angle formed by the base edge and point 3 is greater than the angle formed by the base edge and point 4. Therefore, choose point 3 and point 3
Is formed from points 3,10,11. This is shown in FIG. 5 (b).

step8; Updating the base polygon There are two types of points that can be selected when combining with the base edge. That is, the points on the boundary of the current base polygon and the interior points. When a point on the boundary is picked up, two parts, the left part and the right part of the point that picked up the base polygon,
Divided into two parts. In this algorithm, when dividing into two parts, the part with many points is
The other parts are kept in a polygon table for later processing and become new polygons.
This retained polygon point must be deleted from the new base polygon and the point state updated. If the selected point has two states (ie, overlaps), a special check is made to determine the portion of the base polygon.

Update of the base polygon will be described in detail below.

Updating points and states There are three cases when updating a base polygon.

(Case 1): When picking up a point close to the base edge One of the points of the base edge is deleted from the base polygon list. In the example of FIGS. 6 (a) and 6 (b),
14 is picked up because it is close to the base edge. The updated base polygon does not include the remaining points of the base edge.

(Case 2): When picking up internal points The selected points are added to the base polygon list,
The state of the internal point is 1. FIG. 6 (c) and FIG. 6 (d)
In the example, the updated base polygon contains interior points inside the two newly created base edges.

(Case 3): When picking up points on the polygon that separates the base polygon The base polygon is divided into two parts: a part that becomes a new base polygon and another part that is added to the polygon list for later processing. Divided into two. FIG. 6 (e)
In the example of FIG. 6 (f), the original polygon is divided into a new triangular polygon and two polygons (still to be processed). The id of the polygon on the left is 13,1
4,15,16,9, and the id of the polygon on the right is 10,11,12,
It is 13. The polygon on the left is a new base polygon because it contains many points. Since the polygon on the right has three or more points,
Linked to the list.

You have to be careful when picking up duplicate points.
When a polygon is generated from points having a state of 2 (overlapping points), a special reverse check is required so that the generated polygon is counterclockwise.

In the example of FIGS. 7A and 7B, the point 13 overlaps and is selected as a part of the generated triangular polygon. In this case, since the point 13 appears twice in the polygon id list, care must be taken in generating the polygon. In order to solve this problem, a fast concave /
The problem is solved by employing the inverse algorithm of the convex angle algorithm.

In this example, edges composed of points 12, 13, and 14 are used collectively. Check whether the base edge points 9 and 10 are suitable candidate points by the concave / convex angle algorithm. If they are not appropriate, the edge consisting of the other points 12, 13, 15 must be correct.

The algorithm also makes the conical surface type a special case for the problem of the point at the tip of the cone head.

Due to the conical surface, the edges of the cone are subdivided by increasing the number of divisions. This is the case when dividing by straight edges, which produces good quality polygons for conical surfaces. In addition, special processing at ordinary points also requires the point at the tip of the head of the cone.

As examples, some examples of curved surfaces divided into polygons are shown in FIGS. 8 (a) to 8 (c).

Effect As is apparent from the above description, according to the present invention, 3
The method of polygonal division is very simple and uses internal points to make a dense triangular polygon. It is very simple and straightforward and applicable to all surface types.

[Brief description of the drawings]

FIG. 1 is a flowchart for explaining a process of forming a polygon from a curved surface boundary which is an embodiment of the free-form surface dividing method according to the present invention, and FIG. 2 is an embodiment of the free-form surface dividing method according to the present invention. FIG. 3 is a flowchart for explaining the process of dividing the main polygon into triangles.
FIGS. 7A and 7B are diagrams showing a process of generating a polygon from a hole, FIG. 7A is a diagram showing a surface including a hole, FIG. 8B is a diagram showing generation of a single polygon, and FIG. FIG. 4 is a diagram showing a search space when there are irregular corners, and FIG. 5 is a diagram showing a candidate point selection process based on the maximum angle and generation of a triangulated polygon. , Sixth
FIG. 7 is a diagram for explaining updating of a base polygon, FIG. 7 is a diagram showing a case where a polygon is generated from overlapping points,
FIG. 8 is a diagram showing some of the curved surfaces divided into polygons.

Claims (1)

(57) [Claims] In a three-dimensional CAD system, a boundary curve of a curved surface and a boundary curve of a hole are divided into polylines, all the curved surfaces of the boundary curves are polylined, and the polyline data is mapped to a two-dimensional parameter space of the curved surface. Using the outer curved surface boundary mapped to the two-dimensional parameter space as a base polygon, find a point at the shortest distance between the base polygon and the polygon constituting the hole of the curved surface in the two-dimensional parameter space, and form the shortest distance The polygons that connect the points to each other and form a hole are connected to the base polygon, and the boundary of the curved surface in the three-dimensional space is made into one polygon in the two-dimensional parameter space. , Except for the internal points in the two-dimensional parameter space corresponding to the points of the curved surface in the three-dimensional space, and use the internal points when necessary. A free-form surface characterized by dividing a main polygon into triangular polygons, returning the divided polygons from a two-dimensional parameter space to an original three-dimensional space, and converting the trimmed free-form surface into a polygon. Split method.