JP3643237B2 - Approximate polygon surface smoothing method - Google Patents

Description

[0001]
BACKGROUND OF THE INVENTION
The present invention relates to a technique for smoothing an approximate polygon surface formed by a set of a plurality of original triangles by subdividing a plurality of original triangles that are approximately expressed by polygonizing a given curved surface. Is.
[0002]
[Prior art]
In Japanese Patent Laid-Open No. 8-263692, prior to subdivision, vertices of a plurality of triangles to be generated by the subdivision and not existing in the original triangle are displayed at both ends of each side of the original triangle. Describes a technique based on two vertices and two vertex normal vectors that are normal vectors at each of the two vertices, but not on the sides but on the curved elements that the sides should approximate. ing.
This publication also relates to a method for determining a vertex normal vector: “For example, an image acquisition measuring apparatus 1 such as a CT X-ray tomography apparatus or an MR apparatus is capable of producing a number of tomographic objects to be inspected. An image or a three-dimensional image is formed. Based on the image, the image processing apparatus 2 forms an object model that approximately represents the surface of the object by a large number of triangles. The position information of the vertices of the triangle and the normal to the surface of the object at the vertices are stored in the image memory 3. Is described.
[0003]
[Problems to be Solved, Problem Solving Means, Actions and Effects of Invention]
However, this publication does not specifically teach “a normal to the surface of the object at the vertex”, that is, a method for determining the vertex normal vector.
On the other hand, in the approximate polygon surface smoothing method using the vertex normal vector, the vertex normal vector can be easily determined, and is true, that is, a normal vector for a given curved surface and passes through the vertex. It is desirable to be able to determine with high accuracy so as to match as much as possible.
[0004]
In view of the above circumstances, the present invention provides an approximate polygon surface smoothing method that can easily and accurately reproduce a given curved surface by determining vertex normal vectors easily and accurately. It was made as an issue.
[0005]
  The problem is solved by the following aspect. In the following description, each aspect of the present invention is described in the same format as the claims, with each item numbered.
[0006]
  (1) A method of smoothing an approximate polygon surface formed by a set of a plurality of original triangles by subdividing a plurality of original triangles that are approximated by polygonizing a given curved surface. Prior to subdivision, a plurality of vertices of a plurality of triangles that should be generated by the subdivision and not new vertices, two vertices at both ends of each side of the original triangle, and An approximate polygon surface smoothing method based on two vertex normal vectors, which are normal vectors at each of two vertices, and generated on a curved element whose side should be approximated instead of on that side,
  A memory stores data of the plurality of original triangles;
  A processor, for each of the two vertices based on data stored in its memory, extracting the plurality of original triangles sharing each of the vertices;
  A processor, with respect to the extracted plurality of original triangles, obtaining a normal vector for the surface of the original triangle based on the three vertices of the original triangle as a plurality of surface normal vectors;
  The processor determines the vertex normal vector for each of the two vertices based on the obtained plurality of surface normal vectors, and determines the influence of each surface normal vector to the vertex normal vector in each determination. Exerting more strongly as the angle at the shared vertex of the original triangle corresponding to the surface normal vector increases;
  A processor obtaining the curve element based on the determined two vertex normal vectors and the two vertices;
  Generating a new vertex on the obtained curve element;
An approximate polygon surface smoothing method characterized by comprising:
  The present inventors can obtain a surface normal vector that is a normal vector with respect to the surface of the original triangle if the positions of the three vertices of the original triangle are known, and also about a plurality of original triangles sharing the same vertex. I noticed that if the surface normal vector is known, the vertex normal vector can be determined. In addition, the surface normal vector is considered to be sufficiently coincident with the typical normal vector for the curved surface element to be approximated by the original triangle of the given curved surface. It has also been found that the determination of 頂点 results in the determination of the vertex normal vector in consideration of the normal vector for the curved surface element that passes through the vertex.
  Based on the above knowledge, in the method described in this section, for a plurality of original triangles sharing the same vertex, the surface normal vector of the original triangle is acquired based on the three vertices of the original triangle, A vertex normal vector at the vertex is determined based on the plurality of surface normal vectors acquired for the plurality of original triangles.
  Therefore, according to this method, the vertex normal vector can be easily and accurately determined, and as a result, a given curved surface can be easily and accurately reproduced.
  Moreover, the influence of each surface normal vector is not always exerted on the vertex normal vector to be determined with the same strength between the original triangles sharing the same vertex, but the angle of the shared vertex. The larger the triangle is, the stronger the influence of the surface normal vector corresponding to the triangle is on the vertex normal vector to be determined. Therefore, the vertex normal vector can be determined to be true, that is, closer to the normal vector for a given curved surface that passes through the vertex.
  (2) The processor performs the plurality of original triangles based on the plurality of surface normal vectors.The two verticesAfter determining the normal vector,The two peaksPoint normal vectorAt least one of a plurality of other vertex normal vectors determined in the same manner as each vertex normal vector and adjacent to each vertex normal vector.The method for smoothing an approximate polygon surface according to item (1), including a step of correcting based on the method (Claim 2).
  According to this method,Multiple other adjacentCompared to the case where the vertex normal vector is determined without considering the vertex normal vector at all, the plurality of vertex normal vectors adjacent to each other can be determined such that the directions are more continuous with each other.
  (3) The processor includes a step of acquiring at least one of a plurality of constituent lines configured by connecting sides of the plurality of original triangles as a feature line, and the processor The step of determining a vertex normal vector for each of two vertices is such that if the vertex for which the vertex normal vector is to be determined is not on the acquired feature line, all the original triangular face methods that share the vertex Determining the vertex normal vector based on a line vector, and, if on the acquired feature line, based on the surface normal vector of at least one original triangle located on one side of the feature line Determining a second vertex normal vector based on the first vertex normal vector and the surface normal vector of at least one original triangle located on the other side; The approximate polygon surface smoothing method according to item (1) or (2), including: (Claim 3).
  When the orientations of two parts of a given curved surface with a line as a boundary are greatly different from each other, the vertex normal vectors at the vertices on the boundary are based on a plurality of surface normal vectors located on both sides of the boundary. When the decision is made, the surface of the approximate polygon is smoothed so that the angle between the surfaces of the two portions is finally reduced. Accepting such a situation would reduce the reproducibility of the approximate polygon surface.
  On the other hand, in the method described in this section, for the vertices on the boundary, vertex normal vectors are determined independently from each other on one side and the other side of the boundary.
  Therefore, according to this method, the vertex normal vector can be appropriately determined even for the portion of the curved surface where the directions of the two portions are greatly different from each other.
  (4) When two original triangles sharing the same side among the plurality of original triangles are extracted, and the angle at which the extracted two original triangles are connected at the shared side is smaller than a reference value The approximate polygon surface smoothing method according to (3), wherein the shared side is extracted as the feature line, and if it is equal to or larger than a reference value, the step of not extracting is included.
  In this method, “feature line extraction” may be performed by an operator or a computer.
  (5) The processor obtains a curve element based on two vertex normal vectors and two vertices;
  (a) For the side corresponding to the curve element to be acquired, a first function having a first end point which is one of both end points thereof and a second end point which is the other end point, and a second end point Defining a second function having a start point and a first end point as an end point;
  (b) A value obtained by integrating the product of the first function and the first vertex normal vector, which is the vertex normal vector at the first end point, from the first end point to an arbitrary point on the side is expressed as A first movement vector representing a direction and an amount for moving an arbitrary point on the side in order to obtain a first curve element corresponding to the side below;
  (c) a value obtained by integrating the product of the second function and the second vertex normal vector, which is the vertex normal vector at the second end point, from the second end point to an arbitrary point on the side, A second movement vector representing a direction and an amount for moving an arbitrary point on the side in order to obtain a second curve element corresponding to the side below;
  (d) obtaining a final curve element corresponding to the side as the sum of the first and second movement vectors, wherein each of the first function and the second function is:
    (i) having a variable t that varies from 0 to the length L of the side as it moves from the start point to the end point on the side;
    (ii) The value integrated from 0 to the side length L is 0 for the variable t,
    (iii) a value h when the variable t is 0, and a value whose magnitude is half the value h when the variable t is a half value L / 2 of the side length and whose sign is opposite to the value h − h / 2, 0 when the variable t is the side length L,
    (iv) the value h is set such that the final curve element is orthogonal to the first and second vertex normal vectors at first and second endpoints, respectively.
The approximate polygon surface smoothing method according to any one of items (1) to (4), including: [Claim 5].
  According to this method, a curved element can be expressed by the same type of function regardless of the type of shape, and thus an approximate polygon surface smoothing method having high versatility with respect to the type of shape of the curved element can be obtained. .
  (6) In each of the first function and the second function, in the coordinate plane in which the horizontal axis represents the variable t and the vertical axis represents the value of each function, the variable t is 0, and the function value is A point where the value is h, a variable t is the half value L / 2, and a function value is the value −h / 2, a variable t is the length L, and a function value is The approximate polygon surface smoothing method according to item (5), which defines a relationship represented by a graph in which points that are 0 are connected to each other by a single straight line.
  According to this method, a user-friendly function can be used as the function described in the previous section.
  (7) Prior to the subdivision, when the distance between each side of the original triangle and the corresponding curve element is larger than a reference value, the processor generates the new vertex for the side, The approximate polygon surface smoothing method according to any one of items (1) to (6), which is not generated when the value is equal to or less than a reference value.
  In this method, among the plurality of sides of the plurality of original triangles, the side segmentation is omitted for those that do not require side segmentation (generation of new vertices). Therefore, according to this method, useless segmentation of sides can be avoided.
  In this method, “original triangle” means a triangle to be subdivided. Therefore, the “original triangle” means the first triangle at the time of the first subdivision, and means the triangle that exists immediately before the subdivision at the time of the second subdivision.
  (8) Prior to the subdivision, the processor determines that a distance between each side of the original triangle and the corresponding curve element is a first reference value in an area of the curved surface designated by an operator. If it is larger, the new vertex is generated with respect to the side, while if it is less than or equal to the first reference value, it is not generated, and if the gap is larger than the second reference value outside the specified area. Whether or notAllThe approximate polygon surface smoothing method according to any one of (1) to (6), wherein the new vertex is generated with respect to the side.
  According to this method, for example, when the first reference value is set to be smaller than the second reference value, and when it is attempted to subdivide the finer sides outside on the inside of the designated area, Thus, it is possible to avoid that fine side segmentation is performed wastefully.
  (9) The approximate polygon surface smoothing method according to any one of (1) to (6), wherein the processor performs the subdivision only in an area of the curved surface designated by an operator. Item 9].
  (10) A computer-readable recording medium on which a program executed by a computer for performing the approximate polygon surface smoothing method according to any one of (1) to (9) is recorded [Claim 10].
  Examples of the “recording medium” herein include a flexible disk, a magnetic tape, a magnetic disk, a magnetic drum, a magnetic card, an optical disk, a magneto-optical disk, a ROM, a CD-ROM, an IC card, and a punched tape.
[0007]
DETAILED DESCRIPTION OF THE INVENTION
Hereinafter, an approximate polygon surface smoothing method according to some embodiments of the present invention will be described in detail with reference to the drawings.
[0008]
First, the first embodiment will be described.
FIG. 1 shows a data processing system used to implement this embodiment. In this system, an input device 12, an output device 14, a display device 16 and an external storage device 18 are connected to a computer 10. The computer 10 includes a processor 20 and a memory 22. The input device 12 includes a keyboard, a mouse, a digitizer, and the like. The output device 14 is configured by a printer, a plotter, or the like. The display device 16 is configured by a CRT, a liquid crystal display, or the like. The external storage device 18 is configured to include a portion that reads and writes data from and to the portable recording medium 24. The recording medium 24 records an approximate polygon surface smoothing program (hereinafter simply referred to as “smoothing program”) as an application program executed by the computer 10, and the smoothing program is executed as an object to be executed. STL (stereolithography) data is recorded.
[0009]
STL data is known as standard data for expressing the shape of a three-dimensional model to be manufactured by an optical modeling method. The STL data is data that approximately represents a given curved surface by a set of a plurality of triangles, and includes a plurality of triangle data each representing information about the triangles. Each triangle data is attributed as
(1) Data representing a triangle number that is a triangle number;
(2) Data representing edge numbers that are numbers of the three sides of the triangle;
(3) Data representing end point coordinate values that are the coordinate values of both end points of each side;
(4) Data indicating the relationship between the triangle number, side number and end point coordinate value
have. However, the STL data does not have adjacent relationship defining data that directly defines an adjacent relationship that is a relationship between each triangle and another triangle that is adjacent to the triangle.
[0010]
After turning on the computer 10, the operator inputs an execution command for smoothing the approximate polygon surface and specifies STL data that is the execution target. In response, the external storage device 18 reads the smoothing program and the STL data from the recording medium 24 and stores them in the memory 22. Thereafter, the processor 20 reads the smoothed program from the memory 22 and executes it on the STL data.
[0011]
When the smoothing program is executed, first, in step S1 in FIG. 2 (hereinafter, simply expressed as “S1”, the same applies to other steps), the adjacency defining data is added to the STL data. Specifically, first, each of a plurality of original triangles is set as a target triangle in order according to its triangle number, and next, an adjacent triangle whose target triangle is another original triangle adjacent to each side of the target triangle is displayed. Next, the data obtained by associating the side number of each side of the target triangle, the triangle number of the adjacent triangle, and the side number of the shared side shared with the target triangle among the three sides of the adjacent triangle are adjacent to each other. It is stored in the memory 22 as the relationship defining data. If the state in which the structure of the STL data changes due to the addition of the adjacency relationship definition data is described conceptually, before the addition of the adjacency relationship definition data, as shown in FIG. Are not directly related to each other but exist separately, but after the adjacency definition data is added, multiple triangle data are directly related to each other as shown in (b) of the figure .
[0012]
Next, in S2, what characterizes the shape of the curved surface to be expressed by STL data is extracted as a feature line from a plurality of sides of a plurality of original triangles. Specifically, when an angle formed by two original triangular surfaces connected to each other on the same side is smaller than a reference value, the side is extracted as a feature line. In the example of FIG. 4, since the connection angle of the two original triangles 102 and 104 connected to each other at the side 100 is smaller than the reference value, the side 100 is extracted as a feature line. Similarly, since the connection angle of the two original triangles 108 and 110 connected to each other at the side 106 is smaller than the reference value, the side 106 is extracted as a feature line. After all, in this example, a plurality of sides indicated by two closed thick solid lines are extracted as feature lines. The connection angle between the two original triangles can be defined as an angle inside the two original triangles. In this case, the reference value can be set to 100 to 120 degrees.
[0013]
Subsequently, in S3, a vertex normal vector is determined for each of a plurality of vertices of the plurality of original triangles. Details of this step are shown in a flowchart in FIG. 5 as a vertex normal vector determination routine.
[0014]
In this routine, first, in S11, each of a plurality of vertices of a plurality of original triangles is sequentially determined as a target vertex, and then in S12, the determined target vertices are selected from among the plurality of original triangles. A plurality of original triangles that share are extracted. Subsequently, in S13, the position data of the three vertices of each of the extracted plurality of original triangles is read from the memory 22, and based on the read position data, the surface of each of the plurality of original triangles is read. A surface normal vector that is a normal vector is acquired. Specifically, a plane including three vertices of the original triangle is acquired, and a surface normal vector is acquired as a unit vector orthogonal to the acquired one plane.
[0015]
Thereafter, in S14, it is determined whether or not the current target vertex is not on the extracted feature line. If it is assumed that this is not the case, the determination is yes, and one vertex normal vector is determined for the target vertex in S15. Specifically, a weighted average of a plurality of surface normal vectors corresponding to the extracted plurality of original triangles is determined as one vertex normal vector. The weight in the weighted average is set to be larger as the angle of the three vertices of the original triangle to which each surface normal vector corresponds matches the target vertex.
[0016]
FIG. 6A shows an example where the target vertex is not on the feature line. In this case, one vertex normal vector 142 is used as a weighted average of all surface normal vectors 132, 134, 136, 138, and 140 of the plurality of original triangles 122, 124, 126, 128, and 130 that share the target vertex 120. Is determined.
[0017]
On the other hand, when the current target vertex is on the extracted feature line, the determination in S14 is NO, and in S16, two vertex normal vectors for the target vertex, that is, the first and first vectors are used. A two-vertex normal vector is determined. Specifically, a plurality of original triangles sharing the target vertex are divided into a group on one side and a group on the other side with respect to the feature line, and the vertex normal vector for each group is It is determined in the same way. The first vertex normal vector is determined to correspond to the original triangle group on one side of the feature line, and the second vertex normal vector is determined to correspond to the original triangle group on the other side of the feature line.
[0018]
FIG. 6B shows an example in which the target vertex is on the feature line. In this case, with respect to the target vertex 152 on the feature line 150, the original triangles 154, 156, and 158 constitute the first group, and the original triangles 160 and 162 constitute the second group. Then, the first vertex normal vector 164 is determined as a weighted average of the three surface normal vectors of the three original triangles 154, 156, 158 belonging to the first group, while the two original elements belonging to the second group A second vertex normal vector 166 is determined as a weighted average of the two surface normal vectors of the triangles 160 and 162.
[0019]
In any case, thereafter, in S17, it is determined whether or not the vertex normal vectors have been determined for all the vertices. If it is assumed that this is not the case, the determination is no, the process returns to S11, a new vertex is determined as the target vertex, and the following steps are executed as in the above case. On the other hand, if it is assumed that the vertex normal vectors have been determined for all the vertices, the determination in S17 is YES, and the process proceeds to S18.
[0020]
In S18, the determined plurality of vertex normal vectors are sequentially determined as the target vertex normal vector, and the direction of the target vertex normal vector considers the direction of at least one other adjacent vertex normal vector To be corrected. At least one other vertex normal vector adjacent to one another is a sphere whose radius is a set value centered on the vertex through which the target vertex normal vector passes among other vertex normal vectors. Extracted as existing. In addition, among the at least one other vertex normal vectors adjacent to each other, the shorter the distance between the vertex that passes through and the vertex through which the target vertex normal vector passes, the stronger the influence is exerted on the target vertex normal vector. To be corrected. By doing so, the continuity of the direction between the plurality of vertex normal vectors is improved. This completes one execution of this routine.
[0021]
If S3 is completed as described above, then the triangle is subdivided in S4 of FIG. Details of this step are shown in a flowchart in FIG. 7 as a triangle segmentation routine.
[0022]
First, in S31, a plurality of sides of a plurality of original triangles are set as target sides in order, and a Yamatani function f (t) is defined for the target side. As shown in FIG. 8, the Yamatani function f (t) is represented by a downwardly convex line graph. In the figure, “L” represents the length of the target side, and “t” represents the start point P on the target side.₀To end point P_LIs a variable that changes from 0 to L as it moves to. This Yamatani function f (t) is uniquely specified if the length L and the value h are set. The length L is set based on two vertices at both ends of the target side. The setting of the value h will be described later.
[0023]
This Yamatani function f (t) is defined to be 0 when integration is performed using t = 0 to L as an integration interval. Further, the product of the Yamatani function f (t) and the vertex normal vector N is integrated with t = 0 to t (0 ≦ t ≦ L) as an integration interval, that is,
∫N ・ f (t) dt
Is an arbitrary point P on the target side S as shown in FIG._tIs a movement vector X indicating the direction and amount of movement from the target side S to the curved element C_tIt is. The direction is of course parallel to the vertex normal vector N. In the example shown, X_t1Is an arbitrary point P_t1Vector of movement, X_t2Is an arbitrary point P_t2The movement vector of is shown.
[0024]
The Yamatani function f (t) is the first end point Q of the target side S₁Starting point P₀, Second end point Q₂To end point P_LThe first Yamatani function f₁(t) and the second end point Q₂Starting point P₀, First end point Q₁To end point P_LThe second Yamatani function f₂(t) and FIG. 10 shows the first and second Yamatani functions f.₁(t), f₂first and second curve elements C represented by (t)₁, C₂Are shown respectively.
[0025]
And the first Yamatani function f₁First movement vector X by (t)_tAnd the second Yamatani function f₂Second movement vector Y by (t)_tAre obtained for the entire target side S, the first and second curve elements C are obtained.₁, C₂Is obtained, and this is the final curve element C corresponding to the target side S_FNLIt becomes. This final curve element C_FNLAre the vertex normal vectors N at both end points of the object side S₁, N₂Orthogonal to FIG. 11 shows the first movement vector X for the same value of the variable t._tAnd the second movement vector Y_tIs the arbitrary point P on the target side S corresponding to the variable t._tIs the last curve element C_FNLMovement vector Z with end point on it_tIt means that. In the figure, the variable t is t₁, T₂, T_ThreeThree movement vectors Z when_t1, Z_t2, Z_t3It is shown.
[0026]
Here, a method for setting the value h will be described.
The value h is the final curve element C corresponding to the target side S_FNLIs the end point Q of the target side S₁, Q₂Each vertex normal vector N₁, N₂Is set to be orthogonal to each other.
[0027]
Arbitrary point P on target side S_tLet P (t) be the position vector of the first curve element C₁Arbitrary point R above_1tAnd an arbitrary point P on the target side S_tA position vector R corresponding to₁(t) is
R₁(t) = P (t) + ∫N₁・ F₁(t) dt
It is represented by On the other hand, the second curve element C₂Arbitrary point R above_2tAnd an arbitrary point P on the target side S_tA position vector R corresponding to₂(L-t) is
R₂(L-t) = P (L-t) + ∫N₂・ F₂(L-t) dt
It is represented by Therefore, the final curve element C_FNLFirst end point Q₁The position vector R (0) at
R₁(0) + R₂(L)
= {P (0) + ∫N₁・ F₁(t) dt} + {P (L) + ∫N₂・ F₂(L-t) dt}
= P (0) + N₁・ H₁+ P (L) +0
It becomes. Here, the first end point Q₁Starting point, second end point Q₂Is defined as U, the position vector R (0) is
U + N₁・ H₁
It becomes. The first valley function f is set so that the position vector R (0) becomes zero.₁the value h of (t)₁Is set.
Similarly, the final curve element C_FNL2nd end point Q₂The position vector R (L) at
R₁(L) + R₂(0)
= {P (L) + ∫N₁・ F₁(t) dt} + {P (0) + ∫N₂・ F₂(L-t) dt}
= P (L) + 0 + P (0) + N₂・ H₂
It becomes. This position vector R (L) is
U + N₂・ H₂
It becomes. The second valley function f is set so that the position vector R (L) becomes zero.₂the value h of (t)₂Is set.
[0028]
Next, in S32 of FIG._FNL(Hereinafter simply referred to as “curve element C”)_MPoint R corresponding to (t = L / 2)_MIs acquired. The corresponding point R_MAnd midpoint P_MVector Z between_MAs is clear from the above description,
Z_M= ∫N₁・ F₁(L / 2) dt + ∫N₂・ F₂(L / 2) dt
It is represented by In FIG. 12, the movement vector Z_MAn example is shown.
[0029]
Subsequently, in S33 of FIG. 7, a plurality of original triangles are sequentially set as target triangles, and the target triangles are subdivided into four triangles. Three vertices of the target triangle and the three corresponding points R acquired in S32 above_MAre divided into a plurality of triangles by connecting the six points consisting of FIG. 13 shows three vertices Q₁, Q₂, Q_ThreeIs shown, and the original triangle 170 has three vertices Q₁, Q₂, Q_ThreeAnd three corresponding points R_M1, R_M2, R_M3A state is shown in which six new dots 172 are connected to a plurality of linear elements and divided into four new triangles 172, 174, 176, and 178. In each figure after FIG. 13, “C₁"," C₂"," C_Three"Is the side S₁, S₂, S_ThreeThe final curve element corresponding to is shown.
[0030]
Thereafter, in S34, a plurality of corresponding points acquired in S32, that is, vertex normal vectors at new vertices generated for subdivision are determined. The vertex normal vector is normalized by averaging two known vertex normal vectors at both end points of the side where the new vertex is located among the three sides of the original triangle (the length of the vector is 1). Determined). As illustrated in FIG. 14, the new vertex R_M1Z₁For, its corresponding edge S₁Both end points Q₁, Q₂Vertex normal vector N₁, N₂Is a normalized vector, i.e.,
(N₁+ N₂/ N₁+ N₂｜
Is acquired. Other new vertex R_M2, R_M3The same applies to.
[0031]
Subsequently, in S35 of FIG. 7, it is determined whether or not the triangle subdivision should be terminated. Specifically, as the distance between each side of the original triangle and the corresponding curve element, the midpoint P of each side of the original triangle_MThe movement vector Z corresponding to (t = L / 2)_MIs used (hereinafter referred to as “movement amount M”), and it is determined whether or not the movement amount M is less than or equal to the error designated by the operator for all three sides of the original triangle. . In FIG. 15, side S₁And curve element C₁Travel distance M₁, Side S₂And curve element C₂Travel distance M₂, Side S_ThreeAnd curve element C_ThreeTravel distance M_ThreeIs shown.
[0032]
If the movement amount M is less than the specified error for all three sides of the original triangle, it is determined that the subdivision should be terminated. On the other hand, if the movement amount M for at least one side is larger than the specified error, it is determined that the subdivision should not be terminated. In this case, thereafter, in S36, each subdivided triangle is selected as an original triangle to be subdivided next time, and then S31 to S35 are executed in the same manner as described above.
[0033]
When one execution of the smoothing program is completed as described above, the original triangle 170 shown in FIGS. 13 and 14 is subdivided into a plurality of triangles as shown in FIG.
[0034]
Next, a second embodiment of the present invention will be described. However, the present embodiment is different from the previous first embodiment in that the target triangle is divided, and the other points are the same. Therefore, only the different points will be described in detail, and the common points are denoted by the same reference numerals. Detailed description will be omitted by using.
[0035]
In the first embodiment, regardless of whether the distance between each side of the original triangle and the corresponding curve element is large, all the curve elements corresponding to all the sides are divided into two, and the division points are It can be generated as a new vertex. Note that dividing a curve element into two parts and dividing each side into two do not exactly match, but note that the curve element is generated in association with each side, and the original triangle In order to emphasize the relationship with each side, the two divisions of the curve element are also referred to as two divisions of each side. In the present embodiment, only two sides with a large distance are divided into two parts.
[0036]
FIG. 17 is a flowchart showing the triangle subdivision routine in the present embodiment. Hereinafter, this routine will be described, but steps common to the routine of FIG. 7 will be briefly described.
[0037]
First, in S51, the mountain-valley function f (t) is defined in the same manner as in the first embodiment. Next, in S52, the midpoint P of the target side S_MIs obtained. This acquisition is performed in the same manner as in the first embodiment. Thereafter, in S53, it is determined whether or not the acquired movement amount M is larger than the specified error (reference value) for all three sides of the target triangle.
[0038]
If the movement amount M is greater than the specified error for all three sides, the determination is yes, and in S54, each of these three sides is divided into two, thereby generating three new vertices for all three sides. Thereafter, in S55, the target triangle is subdivided by connecting six points including the generated three new vertices and the three vertices of the target triangle with a plurality of linear elements. An example of this is shown in FIG.
[0039]
On the other hand, when the movement amount M is larger than the specified error for only two sides, the determination in S53 is NO and the determination in S56 is YES. In S57, the two sides are each divided into two parts. Two new vertices are generated for two sides. Thereafter, in S56, the target triangle is subdivided by connecting five points including the generated two new vertices and the three vertices of the target triangle with a plurality of linear elements.
[0040]
There are two methods for connecting five points with a plurality of linear elements. When five points are connected by one closed broken line, one pentagon is generated, and when the pentagon is divided by one linear element, one triangle and one quadrangle are generated. When dividing the quadrilateral into two triangles by one linear element, there are two methods depending on which of the two diagonal lines of the quadrangle is selected as the dividing line. In the present embodiment, the diagonal line that passes through the four vertices of the quadrilateral having the maximum angle among the two diagonal lines is selected as the dividing line. It is designed not to occur as much as possible. An example is shown in FIG. 18 (b).
[0041]
If the movement amount M is larger than the specified error for only one side of the target triangle, the determinations in S53 and S56 are both NO, and the determination in S59 is YES. Thereafter, in S60, the one side is divided into two, thereby generating one new vertex. Subsequently, in S61, the target triangle is divided by connecting four points formed by the generated new vertex and the three vertices of the target triangle by a plurality of linear elements. An example is shown in FIG. 18 (c).
[0042]
In any case, after that, in S62, each subdivided triangle (a new triangle generated by subdivision) is selected as a triangle to be subdivided next time. The same processing as described above is performed for the triangle.
[0043]
In the above, the case where the movement amount M of any one of the three sides of the target triangle is larger than the specified error has been described. Both of these are NO, and one execution of this routine ends without generating new vertices or subdividing the target triangle.
[0044]
Next, a third embodiment of the present invention will be described. However, since this embodiment has many points in common with the previous first and second embodiments, only different points will be described in detail.
[0045]
As for the second embodiment, if the operator issues an execution command for smoothing the approximate polygon surface to the computer 10, the entire curved surface automatically becomes a subdivided region, and a plurality of execution targets are set. All of the original triangles are subject to subdivision. On the other hand, in the present embodiment, it is possible to specify an area of the curved surface that the operator wants to subdivide.
[0046]
FIG. 19 is a flowchart showing the approximate polygon surface smoothing program according to this embodiment. Hereinafter, this program will be described, but there are steps common to the approximate polygon surface smoothing program in the first embodiment and the steps common to the triangle segmentation routine in the immediately preceding second embodiment. These common steps will be briefly described.
[0047]
First, in S71, it is determined whether or not an area to be subdivided has been specified for a plurality of original triangles by the operator operating the input device 12. If it is assumed that it has not been performed this time, the determination is NO and one execution of this program ends. If it is assumed that the determination has been made, the determination is YES and the process proceeds to S72.
[0048]
FIG. 20A shows a state in which a subdivided area is designated. The operator designates a subdivided region by surrounding a part of the graphic displayed on the screen of the display device 16 with a closed line 200 (a broken line or a curved line).
[0049]
In S72, a plurality of original triangles that are present in the subdivision area are extracted. FIG. 20B shows a plurality of original triangles extracted as existing in the subdivision area. Thereafter, in S73 to S75, adjacency defining data is added, feature lines are extracted, and vertex normal vectors are determined in the same manner as in the first embodiment. Subsequently, in S76, triangular segmentation is performed in the same manner as in the second embodiment. FIG. 20 (c) shows an enlarged view of how a plurality of original triangles in the subdivision area are subdivided. This completes one execution of this program.
[0050]
In addition, as in the example illustrated in FIG. 20, when the feature line 202 is present within the region surrounded by the closed line 200, the entire region surrounded by the closed line 200 is defined as a subdivided region. Or a region surrounded by the closed line 200 and the feature line 202 can be defined as a subdivided region.
[0051]
Although some embodiments of the present invention have been described in detail with reference to the drawings, various modifications and improvements can be made based on the knowledge of those skilled in the art without departing from the scope of the claims. The present invention can be carried out in the applied form.
[Brief description of the drawings]
FIG. 1 is a block diagram showing a data processing system for implementing an approximate polygon curved surface smoothing method according to a first embodiment of the present invention.
FIG. 2 is a flowchart showing an approximate polygon curved surface smoothing program recorded on the recording medium of FIG. 1;
FIG. 3 is a perspective view for explaining S1 of FIG. 2;
FIG. 4 is a perspective view for explaining S2 of FIG.
FIG. 5 is a flowchart showing details of S3 of FIG. 2 as a vertex normal vector determination routine;
FIG. 6 is a perspective view for explaining the vertex normal vector determination routine;
FIG. 7 is a flowchart showing details of S4 of FIG. 2 as a triangle segmentation routine.
FIG. 8 is a graph showing a valley function in S31 of FIG.
FIG. 9 is a perspective view for explaining a movement vector in S31.
FIG. 10 is a perspective view illustrating a state in which a curve element is acquired by superimposing two of the above-mentioned Yamatani functions.
FIG. 11 is a perspective view for explaining a movement vector in S31.
12 is a perspective view for explaining S32 in FIG. 7; FIG.
FIG. 13 is a perspective view for explaining S33 of FIG.
14 is a perspective view for explaining S34 in FIG. 7; FIG.
FIG. 15 is a perspective view for explaining S35 in FIG. 7;
FIG. 16 is a perspective view showing an execution result of the approximate polygon surface smoothing program.
FIG. 17 is a flowchart showing a triangle segmentation routine used in the approximate polygon surface smoothing method according to the second embodiment of the present invention;
FIG. 18 is a perspective view for explaining the triangular subdivision routine.
FIG. 19 is a flowchart showing an approximate polygon surface smoothing routine used in the approximate polygon surface smoothing method according to the third embodiment of the present invention;
FIG. 20 is a perspective view for explaining the approximate polygon surface smoothing routine;
[Explanation of symbols]
10: Computer
24: Recording medium
100, 106: side
102, 104, 108, 110, 122, 124, 126, 128, 130, 154, 156, 158, 160, 162, 170: original triangle
120, 152: Target vertex
132, 134, 136, 138, 140: surface normal vector
142: Vertex normal vector
150: Feature line
164: first vertex normal vector
166: second vertex normal vector
172, 174, 176, 178: New triangle
N₁, N₂, N_Three: Vertex normal vector
C₁, C₂, C_Three: Curve element
S₁, S₂, S_Three: Side
Q₁, Q₂, Q_Three: Endpoint
R_M1, R_M2, R_M3: Corresponding point
P_M: Midpoint
M₁, M₂, M_Three:Amount of movement
Z_t1, Z_t2, Z_t3: Movement vector
P_t: Arbitrary point on side S
R_t: Arbitrary point on curve element C

Claims (10)

A method of smoothing an approximate polygon surface formed by a set of a plurality of original triangles by subdividing a plurality of original triangles that are approximated by polygonizing a given curved surface. Prior to conversion, a plurality of vertices of a plurality of triangles to be generated by subdividing, and new vertices that are not in the original triangle, two vertices at both ends of each side of the original triangle, and the two vertices An approximate polygon surface smoothing method based on two vertex normal vectors that are normal vectors in each of the above, and that is generated on a curved element whose side should be approximated instead of on that side,
A memory stores data of the plurality of original triangles;
A processor, for each of the two vertices based on data stored in its memory, extracting the plurality of original triangles sharing each of the vertices;
A processor, with respect to the extracted plurality of original triangles, obtaining a normal vector for the surface of the original triangle based on the three vertices of the original triangle as a plurality of surface normal vectors;
The processor determines the vertex normal vector for each of the two vertices based on the obtained plurality of surface normal vectors, and determines the influence of each surface normal vector to the vertex normal vector in each determination. Exerting more strongly as the angle at the shared vertex of the original triangle corresponding to the surface normal vector increases;
A processor obtaining the curve element based on the determined two vertex normal vectors and the two vertices;
Generating a new vertex on the obtained curve element; and a method for smoothing an approximate polygon surface. Wherein the processor, after determining the two vertices normal vectors for the plurality of original triangles based on the plurality of surface normal vector, each of the two vertex normals vectors that they determine their respective 2. The approximate polygon surface smooth of claim 1 including the step of modifying based on at least one of a plurality of other vertex normal vectors determined in the same manner as the vertex normal vectors and adjacent to each of the vertex normal vectors. Method. The processor includes a step of acquiring at least one of a plurality of constituent lines configured by connecting sides of the plurality of original triangles as a feature line, and the processor includes the vertex normal The step of determining a vector for each of the two vertices is such that if the vertex for which the vertex normal vector is to be determined is not on the acquired feature line, all the original triangle surface normal vectors sharing the vertex The vertex normal vector based on the first feature based on the surface normal vector of at least one original triangle located on one side of the feature line if the vertex normal vector is on the acquired feature line. Respectively determining a second vertex normal vector based on the vertex normal vector and the surface normal vector of at least one original triangle located on the other side. Approximation polygon surface Smoothing method according to no claim 1 or 2. If two original triangles that share the same side among the plurality of original triangles are extracted and the angle at which the extracted two original triangles are connected at the shared side is smaller than a reference value, The approximate polygon surface smoothing method according to claim 3, further comprising a step of extracting an edge as the feature line and not extracting the edge when the edge is equal to or greater than a reference value. The processor obtaining a curve element based on two vertex normal vectors and two vertices;
(a) For the side corresponding to the curve element to be acquired, a first function having a first end point which is one of both end points thereof and a second end point which is the other end point, and a second end point Defining a second function having a start point and a first end point as an end point;
(b) A value obtained by integrating the product of the first function and the first vertex normal vector, which is the vertex normal vector at the first end point, from the first end point to an arbitrary point on the side is expressed as A first movement vector representing a direction and an amount for moving an arbitrary point on the side in order to obtain a first curve element corresponding to the side below;
(c) a value obtained by integrating the product of the second function and the second vertex normal vector, which is the vertex normal vector at the second end point, from the second end point to an arbitrary point on the side, A second movement vector representing a direction and an amount for moving an arbitrary point on the side in order to obtain a second curve element corresponding to the side below;
(d) obtaining a final curve element corresponding to the side as the sum of the first and second movement vectors, wherein each of the first function and the second function is:
(i) having a variable t that varies from 0 to the length L of the side as it moves from the start point to the end point on the side;
(ii) The value integrated from 0 to the side length L is 0 for the variable t,
(iii) a value h when the variable t is 0, and a value whose magnitude is half the value h when the variable t is a half value L / 2 of the side length and whose sign is opposite to the value h − h / 2, 0 when the variable t is the side length L,
(iv) The value h includes a step in which the final curve element is set to be orthogonal to the first and second vertex normal vectors at first and second endpoints, respectively. The approximate polygon surface smoothing method according to any one of the above. Each of the first function and the second function has a variable t of 0 on the coordinate plane in which the horizontal axis represents the variable t and the vertical axis represents the value of each function, and the function value is the value h. A point where the variable t is the half-value L / 2, a function value is the value −h / 2, a variable t is the length L, and the function value is 0. 6. The approximate polygon surface smoothing method according to claim 5, wherein a relationship represented by a graph in which points are connected to each other by a single straight line is defined. Prior to the subdivision, when the distance between each side of the original triangle and the curve element corresponding thereto is larger than a reference value, the processor generates the new vertex for the side, while not exceeding the reference value. 7. The method for smoothing an approximate polygon surface according to any one of claims 1 to 6, wherein the smoothing is not generated. Prior to the subdivision, the processor has a gap between each side of the original triangle and the corresponding curve element larger than a first reference value in an area of the curved surface designated by an operator. In the case of generating the new vertex with respect to the side, if it is less than or equal to the first reference value, it is not generated, and whether or not the gap is larger than the second reference value outside the specified region. 7. The approximate polygon surface smoothing method according to claim 1, wherein the new vertex is generated for all the sides. The approximate polygon surface smoothing method according to claim 1, wherein the processor performs the subdivision only in an area of the curved surface designated by an operator. 10. A computer-readable recording medium on which a program executed by a computer for executing the approximate polygon surface smoothing method according to claim 1 is recorded.

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