KR20030070426A - Exact geometry based triangle mesh generating method - Google Patents

Abstract

PURPOSE: A shape-based triangle mesh generating method is provided to generate a triangle mesh rapidly and simply, reduce system resources and enable fast display. CONSTITUTION: Boundary curves represented in three dimension are converted into trimming curves on a two-dimensional parametric domain(S210). For the trimming curves, cells are divided in such a manner that each cell has only one external loop, and a cell loop is constructed(S220). Triangulation is carried out in the parametric domain on the basis of only the cell boundary for each cell, to generate a triangle(S230). A triangle on the three-dimensional space is generated from the triangle on the parametric domain(S240).

Description

Form-based triangular net generation method {EXACT GEOMETRY BASED TRIANGLE MESH GENERATING METHOD}

The present invention relates to a triangular net generation method used in an engine for displaying a shape in 3D graphics, and in particular, considering the boundary and curvature from a bounded surface modeled in 3D. To triangulate all the surfaces on the original surface without converting them to non-uniform Rational B-spline surfaces (NURBS).

In 3D graphics, for the display of 3D shapes, the inside of the shape is sculpted into fine unit elements, which are represented by triangles. The set of unit elements is usually a triangle, and a set of these triangles is called a triangle mesh. A device for generating a triangular network is called a triangulator.

Various conventional triangular net generation methods and conventional triangular net generators for triangular net generation have already been provided.

1 is a flowchart illustrating an example of a conventional triangular network generating method.

First, the base surface is converted to a non-uniform Rational B-spline surface (NURBS) (S 110), and then the boundary curve is converted to a trimming curve (S 120). In other words, regardless of the shape of the underlying surface, the curved surface is uniformly converted to NURBS, and then the boundary curve is converted to a trimming curve by converting the boundary curve on the converted NURBS parameters.

Construct the next trimming loop (S 130), create each triangle to form a triangle using the Delaunay triangulation scheme (S 140), and finally, each of the parametric triangles thus created Is converted to the actual triangle (S 150).

That is, the conventional triangular network generation method converts a boundary surface into NURBS, converts boundary curves into parametric curves on NURBS, and generates a seam line to process internal boundaries. After that, we use the Delaunai triangulation method in the NURBS parametric domain to generate nearly uniform triangles.

The triangles generated by the conventional Delaunai triangulation are suitable for the purpose of analysis, such as finite element analysis (FEA), in which all triangles should be close to equilateral triangles, but there are many excessive triangles separate from shapes or boundaries. The problem of wasting system resources and slowing down data transmission and display speeds has been pointed out.

In addition, due to the characteristics of the triangular network generation algorithm, since the parameter space for generating triangles is made in the parameter space of NURBS, iterative numerical calculation (Iteration) is used, so the time required for triangular network generation is long. It has also been pointed out that it takes a long time to find a point in space, so the overall triangle generation time is long.

In addition, the conventional triangular network generation method is a three-dimensional graphics field that requires rapid triangle generation optimized for shape due to such data capacity and speed problems, that is, 3D CAD, virtual reality, speed and data size is important authority The problem is pointed out that the efficiency of the virtual reality on the Internet, the architectural design that must show many elements at once, interior design simulation, etc. has a certain limit and the efficiency decreases.

The present invention has been made to solve the problems of the prior art, and an object of the present invention is to provide a triangular net generation method that can improve both of the triangle network generation speed and the characteristics of the generated triangle simultaneously.

In addition, the present invention improves preprocessing speed and triangle generation speed for triangle network generation by configuring NURBS transformation for the first and second curved surfaces, and optimizes the generated triangles to shape and boundary to improve the amount of data. An object of the present invention is to provide a triangular net generation method that minimizes system resources, enables fast display, and facilitates the movement of triangular net data over the Internet.

1 is a flowchart illustrating an example of a conventional triangular network generating method.

2 is a flowchart schematically showing each step of the triangular network generating method according to the present invention.

3 is a schematic view showing a parameterization method of a plane by a triangular network generation method of the present invention.

4 is a schematic diagram showing a parameterization method of a part of a cylinder by the triangular net generating method of the present invention.

5 and 6 are schematic diagrams showing a method of parameterizing a sphere and a toric in 3D by the triangular network generating method of the present invention.

7 is an exemplary diagram illustrating an example of forming a cell loop by dividing cells with respect to trimming curves in a generated parameter space.

8 is a schematic diagram illustrating cell division and results for a cylinder.

9 is a flowchart showing in detail the steps of the triangle generation method by the triangular network generation method of the present invention considering only the nodes of the boundary loop.

Fig. 10 is a schematic diagram showing a preferred example of generating triangles by the triangle network generating method of the present invention.

Hereinafter, the configuration and operation of the present invention will be described in detail with reference to the accompanying drawings.

2 is a flowchart schematically showing each step of the triangular network generating method according to the present invention. Compared with the conventional triangular network generation method shown in FIG. 1, the input and output are the same in each of the two flowcharts, but each step undergoes a different process.

First, the boundary curves expressed in three dimensions are converted to trimming curves in a two-dimensional parameter space (S 210). This step is performed in common in the conventional triangular network generation method and the method of the present invention, but there is a difference in the content. In other words, in order to map a curve in three-dimensional space in two dimensions, the base surface should be represented by a two-dimensional parameter. In the conventional method, a planar surface and a cylinder are used for this purpose. The boundary curves are also transformed on the NURBS parameters by converting the underlying surface of the cylindrical surface, the conical surface, the spherical surface, the toroidal surface, etc. into NURBS (S 110, see FIG. 1). S120, see FIG. 2), this triangular network generation method is constructed to introduce the concept of the object to these base surface to enable the parameter expression without conversion to NURBS. In order to enable such a representation, the present invention devised a conversion method between 3D-2D or 2D-3D of each of a plane, a cylinder, a cone, a sphere, and a torus.

3 is a schematic view showing a parameterization method of a plane by a triangular net generation method of the present invention, when a hypothetical plane 302 that can sufficiently cover an original planar figure 301 to be handled is simulated, All 3D points can be represented by first-order interpolation of virtual rectangular edges P ₀ , P ₁ , P ₂ , and P ₃ ; 303. In other words, if the interpolation parameter between P ₀ and P ₁ is u and the interpolation parameter between P ₀ and P ₂ is v, both the two-dimensional to three-dimensional transformation or the three-dimensional to two-dimensional transformation Obtained by vector calculations. If the three-dimensional point to be transformed is P (x, y, z), the corresponding parameters u and v can be obtained as follows.

u = (PP ₀ ) · (P ₁ -P ₀ ) / (| PP ₀ || P ₁ -P ₀ |)

v = (PP ₀ ) · (P ₂ -P ₀ ) / (| PP ₀ || P ₂ -P ₀ |)

(0≤u≤1, 0≤v≤1)

Conversely, the three-dimensional point P from the parameters u and v can be obtained from the following equation.

P = u (P ₁ -P ₀ ) + v (P ₂ -P ₀ ) + P ₀

4 is a schematic view showing a method of parameterizing a portion of a cylinder by the triangular network generating method of the present invention, in which the original portion of the cylinder portion 401 to be handled is represented by a loop of the base surface 402 and a boundary curve. The base surface 402 is represented by an axis 404 and a radius representing the local coordinate system of the cylinder in order to represent an infinite cylinder. The x, y, z of the axis 404 is not an absolute coordinate system but is a vector representing the axis. A built or rotated cylinder can also be represented. The method of representing a 3D point in a 2D parametric coordinate system is based on a parameter u, the angle of the projection of the 3D point on the plane formed by x and y of the axis 404 with the x vector, and the 3D point as the axis 404. The displacement of the point projected on the z vector of) with the origin of the axis 404 is taken as the parameter v.

That is, if the center point of the local coordinate axes x, y, z is C, and the height of the virtual cylinder is h, the parameters u and v can be obtained from the three-dimensional point P using the following equation.

u = ∠ [ X , ((PC) · X / ｜ PC ｜) X + (PC) · Y / ｜ PC ｜) Y ]

v = (PC) Z / | PC |

(0≤u≤2π, 0≤v≤1, X , Y , Z are direction vectors of each coordinate system)

Conversely, the three-dimensional point P can be obtained from the parameters u and v as follows.

P = r (cosu) X + r (sinu) Y + vh Z + C

On the right side of FIG. 4, a parameterized cylindrical region 406 and the boundary 407 of the original curved surface represented in this region are represented. In this case, since the original surface was along the x-axis of the cylindrical local coordinate system, it is divided into two loops by parameterization. In the case of the conventional method disclosed in FIG. 1, when the boundary is cut off, the process returns to the step of converting the base surface into NURBS (S 110), and the boundary is not cut off. In order to express the cut in the step S 130 of constructing the use of a method of inserting a seam (seam line) to avoid this case, but the method according to the present invention knows that such a case is not a problem at all. Can be. The cone is also parameterized in a very similar way because the phase is not different from the cylinder.

5 and 6 are schematic diagrams showing a method of parameterizing a sphere and a toric in 3D by the triangular network generating method of the present invention. In the present invention, in order to parameterize them, the sphere uses angles corresponding to the longitude and latitude as u and v parameters, respectively, and the torus has u as a cylinder or cone with respect to the coordinate system x, y, z. Then, the v parameter is determined by the angle formed with the x 'axis in the coordinate system x' and y '. Their equations are also the same as the cylindrical parameterization equations, but they apply differently to the reference coordinate system.

As described above, the method of parameterizing the first and second curved surfaces based on the original shape is performed by simple vector operation and trigonometric functions, and thus, the existing NURBS which requires numerical iteration. Compared to the parametric method, the performance is excellent in terms of computation speed and computation stability.

Returning to the flowchart of FIG. 2, the process of converting the boundary curve to the trimming curve (S 210) is completed. The cell is divided and a cell loop is formed (S 220).

The cell is a divided region in the parameter space, which is a concept used in the triangulation method according to the triangulation network generation module. In the present invention, a cell is mainly used for two reasons. The first is to eliminate an inner / hole loop and the second is to generate an optimized triangle according to the curvature of a curved surface.

7 is an exemplary diagram illustrating an example of forming a cell loop by dividing cells with respect to a trimming curve in a parameter space generated by the process of S 210. In order to represent the plane 701 shaped like a central hole, there is an outer boundary 703 and an inner boundary 704 on the virtual parameter area 702. In order for the triangle to be created only inside the geometry, there must be a distinction between the inside and outside of the boundary, which can be determined in the direction of the boundary loop. If the direction of the boundary loop is counterclockwise (703) the interior of the loop should be drawn and if the direction of the loop is clockwise (704) the outside of the loop should be drawn.

Conventional triangular network generators take care of the direction of these loops based on both boundary loops 703 and 704 when generating a triangle for a shape where these outer and inner boundaries 703 and 704 exist simultaneously. A triangle was created. On the contrary, the present invention provides a new method for shortening the time by generating only the inner loop when generating the triangle. That is, when there is a clockwise loop in the dividing cell (S 220), the parameter space is divided in the direction passing therethrough so that only the counterclockwise loop exists. 706 of FIG. 7 shows a cell boundary created to eliminate an inner loop, and is divided into two cells by this boundary. When the cell is divided, the boundary element crossing the cell is also cut at the cell boundary and the loop is reconstructed in the cell (707). This results in two cells, each with only one outer loop.

FIG. 8 is a schematic diagram showing cell division for a cylinder and its result, illustrating a second reason for dividing a cell, for example, of the cylinder of this drawing. In the case of a cylinder there is a constant curvature in the direction of the parameter u and no curvature in the direction of the parameter v. If this is expressed in three dimensions, it is appropriate not to triangulate in the direction without curvature, so by dividing the cells as shown in 801 and forming a cell loop for each cell, a triangle having a shape optimized for a cylinder such as 802 can be obtained. In the example of the cylinder shown in FIG. 8, in the free curved surface where the curvature is uniform but the curvature is uneven for each part, the optimized triangle can be generated by dividing the cell size by the curvature. This, of course, applies to both the u and v directions depending on the curvature.

When each cell is divided and has only one outer loop per cell, triangulation is performed in the parameter area based on the cell boundary only (S 230). Since the generation of the boundary loop considering the curvature and the directionality is completed in the previous step (S 220), the triangulation step (S 230) generates a triangle considering only the shape of one boundary loop.

9 is a flowchart showing in detail the steps of the triangle generation method by the triangular network generation method of the present invention considering only the nodes of the boundary loop.

First, a start index for starting triangle network generation is set (S 910). When setting the starting index, you usually use the first point of the loop, but you can set it differently if the first point is a concave point or unusual.

After that, we create a triangle by looping through the program loop and delete the nodes on the boundary loop. That is, a triangle consisting of an index point and three points immediately following the next point and the next point is set (S 920), and it is determined whether the set triangle is a clockwise triangle or another point (S930), and clockwise. If it is or includes another point, increase the index to move to the next point (S 940), otherwise generates a set triangle (S 950).

When the triangle is generated, the next point after the reference starting index point is removed from the loop (S960).

It is determined whether the size of the next remaining loop is greater than 3, that is, the number of points is greater than three (S970). If there are more than three points, the process returns to step 920, sets the index point, the next point after the point removed from the loop, and the triangle formed by the next point, and repeats the following steps (S 920 to S 970).

By repeating these steps, if finally three points remain, a triangle is generated from the remaining points (S980).

FIG. 10 illustrates a triangulation method for a loop composed of eight points as a preferred example of generating triangles by the triangular net generation method of the present invention. Referring to FIG. 10, a triangle is generated by the method of the flowchart of FIG. 9.

In step 1, point 0 is set as a starting point, that is, a starting index (S910; see i).

Step 2, the triangle formed by the point 012 is set (S 920; ii). At this time, since the other point 6 is included in the triangle (S 930), the start point is moved to the next point without actually generating the triangle (S 940).

Step 3, the triangle formed by the point 123 is set (S920; see ⅲ). Since the triangle is clockwise (S 930), the triangle is abandoned and moved to the next point (S 940).

Step 4, a triangle formed by the point 234 is set (S920; see ⅳ). Since it is counterclockwise and does not include another point or line segment, a triangle is generated (S 950), and the second point of the triangle (point 3) is removed from the loop (S 960; ⅴ). The starting point stays at point 2 as it is. Even if point 3 is removed from the loop, the number of remaining points, that is, the loop size, is 7, so it exceeds 3. Therefore, the step S920 is repeated after the loop size determination process (S970).

In step 5, a triangle formed by the point 245 is set (S920; see ⅴ) to generate a triangle (S950). Remove point 4, the second point of the next triangle (S 960; see ⅵ).

In step 6, a triangle formed by the point 256 is set (S920) to generate a triangle (S950). Remove point 5, the second point of the next triangle (S 960; see ⅶ).

In step 7, a triangle formed by the point 267 is set (S920; see ⅶ). Since the triangle is clockwise (S 930), the starting point is moved to the next point 6 (S 940; ⅷ).

In step 8, a triangle formed by the point 670 is set (S920; see ⅷ), and is generated (S950).

In step 9, a triangle formed by the point 601 is set (S950) and is generated (S950). If the center point 0 is removed, there are three remaining points (S970), and 612 is counterclockwise to generate the last triangle of 612 (S980).

After generating the triangle in the parameter space as described above, a triangle in the real world, that is, three-dimensional space is generated (S 240). Since the triangle in the parameter space is represented by the value (u, v), the inverse transformation of the surface parameterization method according to the present invention described above, that is, the equation for obtaining a three-dimensional point from the two-dimensional parameter (for example, Equation 2 above) And by using the equation (4) to obtain a three-dimensional point and to send these triangles to the graphics library of the application program can be drawn on the screen. In this case, smooth rendering (Phong shading or Goroud shading) is required when there is a curved normal vector for each point. In the present invention, a method of converting a surface class from a three-dimensional point to a u and v parameter (for example, Equation 1 and Equation 3) or a method for obtaining a three-dimensional normal vector from u and v parameters in addition to a method of converting from u and v to a three-dimensional point (for example, Equations 2 and 4 above). To provide.

That is, when the basis surface is a cylinder, a method of deriving a normal vector will be described. The above equation 4, which is a equation for deriving P, may be partially divided in the u and v directions, respectively, and two resultant equations that are partial in each direction may be obtained. Cross product can yield the desired normal vector. The same applies to equation (2) for parameterization of planes. That is, according to the present invention, it is possible to easily derive a normal vector by using the original parameterization equation as it is, so that it is easy to render.

While the present invention has been described with reference to preferred embodiments, it will be understood that the invention is for the purpose of illustration and not of limitation. It will also be understood that various modifications and variations can be made by those skilled in the art without departing from the spirit and scope of the invention.

According to the configuration of the present invention, by designing and selecting algorithms fast and concise, compared to the conventional NURBS and Delaunay triangulation based triangulation method, triangulation that can expect stability and high performance There is an effect that it can provide a network generation method.

In addition, according to the present invention, the generated triangles are well represented in the curvature of the curved surface while being optimized for shape and boundary, so in terms of the application program using the same, light and improved performance can be expected, and the amount of data consumes less system resources and provides a faster display. It has the effect of enabling it.

In addition, according to the present invention, it can be used in any 3D CAD or 3D graphics system, especially a virtual reality system over the Internet, a system that requires real-time rendering, a viewer that must be able to show light and fast, many elements on one screen It can be effectively applied to fields such as architectural design or interior simulation that should be given.

Claims (9)

Converting the boundary curves represented in three dimensions into a trimming curve in two-dimensional parameter space; Dividing cells and configuring cell loops so that each cell has only one outer loop for the converted trimming curve; Generating a triangle by performing triangulation in the parameter region based on only a cell unit boundary for each cell; And Generating a triangle in the three-dimensional space from the triangle in the parameter space; Shape-based triangle network generation method comprising a. 2. The shape-based method of claim 1, wherein the first step is to parameterize planar surfaces, cylinders, cones, spheres, and torics in three dimensions in a manner specific to each shape without conversion to NURBS. How to create a triangle network. The reference point P ₀ according to claim 2, wherein when the three-dimensional boundary curve of the first step is a plane, the reference point P ₀ is based on an edge P ₀ , P ₁ , P ₂ of an imaginary rectangle surrounding the plane figure. The interpolation parameter between one edge P ₁ is u, and the interpolation parameter between reference point P ₀ and the other edge P ₂ is v, and each coordinate of the three-dimensional planar figure is determined by a predetermined equation. A method for generating a shape-based triangle network comprising converting (P) into parameters u and v to convert a three-dimensional boundary curve into a trimming curve in two-dimensional parameter space. The method of claim 3, wherein the predetermined formula is, u = (PP ₀ ) · (P ₁ -P ₀ ) / (| PP ₀ || P ₁ -P ₀ |) v = (PP ₀ ) · (P ₂ -P ₀ ) / (| PP ₀ || P ₂ -P ₀ |) (0≤u≤1, 0≤v≤1) P = u (P ₁ -P ₀ ) + v (P ₂ -P ₀ ) + P ₀ Shape-based triangle network generation method characterized in that. 3. The method of claim 2, wherein when the three-dimensional boundary curve of the first step is a portion of the cylinder or cone, the predetermined local coordinate axes x, y, and z of the cylinder or cone are set, and the point on the three-dimensional boundary curve is set to the axis. Using the angle u of the point projected on the plane formed by x and y as the x vector, and the displacement of the point projected on the axis z by the point projected on the axis z as the parameter v Shape-based coordinates are characterized by converting the three-dimensional boundary curve into a trimming curve in two-dimensional parameter space by displaying each coordinate P of the three-dimensional boundary curve by parameters u and v by a predetermined equation. How to create a triangle network. The method according to claim 5, wherein the predetermined formula is u = ∠ [ X , ((PC) · X / ｜ PC ｜) X + (PC) · Y / ｜ PC ｜) Y ] v = (PC) Z / | PC | (0≤u≤2π, 0≤v≤1, X , Y , Z are direction vectors of each coordinate system) P = r (cosu) X + r (sinu) Y + vh Z + C Shape-based triangle network generation method characterized in that. The method according to claim 1, wherein when the three-dimensional boundary curve of the first step is a sphere, the angles corresponding to longitude and latitude are respectively defined as parameters u and v for the local coordinate axes x, y, and z of the sphere. Shape-based triangle characterized by converting the three-dimensional boundary curve into a trimming curve in two-dimensional parameter space by displaying the coordinates (P) of the three-dimensional boundary curve with the parameters u and v by the equation. How to create a network. 2. The plane of claim 1, wherein when the three-dimensional boundary curve of the first step is a torus, the plane of the axis x, y forms a point on the three-dimensional boundary curve with respect to x, y, and z, which is the coordinate system of the torus. Using the parameter u as the angle formed by the point projected at x as the parameter u, and the angle at which the point on the three-dimensional boundary curve forms the axis x 'as the parameter v with respect to the minor coordinate system x', y 'of the torus, Shape-based, characterized by converting the three-dimensional boundary curve into a trimming curve in two-dimensional parameter space by displaying the coordinates (P) of the three-dimensional boundary curve by the parameters u, v by a predetermined equation How to create a triangle network. The method of claim 1, wherein the third step, Step 3-1 of setting an index point starting in the cell loop; Setting a triangle consisting of the index point, the next point, and three points next to the third point; Determining whether the set triangle is a clockwise triangle or includes another point; Step 3-4 of increasing the index to move to the next point when the set triangle is clockwise or includes another point; Generating a set triangle when the set triangle is not clockwise and does not include another viscosity at the same time; Step 3-6 of removing the next point of the reference starting index point from the loop when the triangle is generated; Steps 3-7 to determine whether the size of the remaining loop is greater than 3, that is, the number of points is greater than three; If the loop size exceeds 3, the process returns to step 3-2 to set the triangle formed by the index point, the next point after the point removed from the loop, and the next point. Steps 3-8 to repeat step 7; And Generating a triangle from the remaining points when the loop size is 3; Shape-based triangle network generation method comprising a.