WO2014181598A1 - Dispositif de trace de surface courbe et programme de trace de surface courbe - Google Patents

Dispositif de trace de surface courbe et programme de trace de surface courbe Download PDF

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Publication number
WO2014181598A1
WO2014181598A1 PCT/JP2014/058770 JP2014058770W WO2014181598A1 WO 2014181598 A1 WO2014181598 A1 WO 2014181598A1 JP 2014058770 W JP2014058770 W JP 2014058770W WO 2014181598 A1 WO2014181598 A1 WO 2014181598A1
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Prior art keywords
curved surface
curve
line segment
line
plane
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PCT/JP2014/058770
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English (en)
Japanese (ja)
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保弘 清水
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日本ユニシス株式会社
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T17/00—Three dimensional [3D] modelling, e.g. data description of 3D objects
    • G06T17/30—Polynomial surface description
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]

Definitions

  • the present invention relates to a curved surface drawing apparatus and a curved surface drawing program, and more particularly to a curved surface drawing apparatus that draws a curved line or curved surface obtained by smoothly interpolating the sides and the inside of a polygon mesh from each vertex and intermediate control point of the polygon mesh. It is suitable for use.
  • the calculation is performed by approximating the target region with discrete lattice points. Accordingly, the object surface shape included in the region is represented by a polyhedron model (triangle mesh, quadrilateral mesh) composed of triangles and quadrangles.
  • a polyhedron model triangle mesh, quadrilateral mesh
  • Nagata patch is an effective technique for simply interpolating the roundness of a curved surface from normal vectors (directions orthogonal to the shape surface) assigned to the vertices of a triangular mesh and a quadrilateral mesh.
  • normal vectors directions orthogonal to the shape surface assigned to the vertices of a triangular mesh and a quadrilateral mesh.
  • the Nagata patch defines a curved surface equation that interpolates with roundness in the polygonal area by a quadratic polynomial from each vertex of the polygon (triangle or square) and the unit normal vector given to each vertex. ing. In the following, for simplification of description, the description will be limited to interpolation of a triangular region.
  • the basis of the Nagata patch is a curve formula that interpolates each side (line segment) of a polygon from two end points and a unit normal vector at each end point using a quadratic polynomial.
  • the curve obtained by this curve formula is referred to as “Nagata segment”.
  • the start point of the line segment is P 0
  • the end point is P 1
  • the respective position vectors are x 0 , x 1
  • the unit normal vectors are n 0 , n 1 .
  • the Nagata patch interpolates x 0 , x 1 , n 0 , and n 1 with a quadratic polynomial (1) of the following form using 0 ⁇ ⁇ ⁇ 1 as a parameter.
  • x ( ⁇ ) x 0 + (dc) ⁇ + c ⁇ 2
  • x is a position vector representing a point on the curve
  • c is an unknown constant vector called “curvature parameter” calculated from x 0 , x 1 , n 0 , and n 1 , and defines the bulge of the curve.
  • the point P m must be on the start point side tangent plane ⁇ x 0 , n 0 ⁇ and on the end point side tangent plane ⁇ x 1 , n 1 ⁇ . That is, when the both tangent planes ⁇ x 0 , n 0 ⁇ , ⁇ x 1 , n 1 ⁇ have an intersection line, the point P m must be on the intersection line.
  • the both tangent planes ⁇ x 0 , n 0 ⁇ , ⁇ x 1 , n 1 ⁇ have intersecting lines
  • n 0 ⁇ n 1 is parallel to the direction vector of the intersecting lines, so the vector c is Must be orthogonal.
  • the point P m is a leg of a perpendicular drawn from the midpoint x m of the chord to the intersecting line.
  • a line segment connecting each end point P 0 , P 1 from the position vector x 0 , x 1 of each end point P 0 , P 1 of the line segment and the position vector of the point P m is curved with a quadratic polynomial. Interpolate.
  • the second-order polynomial (2) for this curve interpolation it is possible to use a Bezier curve as shown in FIG. 11 that is often used in CAD.
  • P 20 x 0 (position vector of start point P 0 ),
  • P 02 x 1 (position vector of end point P 1 ),
  • P 11 (x 0 + x 1 ) / 2 ⁇ c / 2 (intermediate control point P m position vector).
  • the Nagata segment is a quadratic polynomial curve, and is a plane curve because it has a shape obtained by cutting a part of a parabola.
  • the plane on which the curve is placed is a plane including a string connecting the start and end points x 0 and x 1 , and this is called a “curve plane”.
  • the curve plane is determined.
  • the intermediate control point P m that hits the foot of the perpendicular line drawn from the midpoint x m of the chord to the tangential plane intersects largely laterally with respect to the chord, and as a result, the secondary Bezier curve is curved.
  • the Nagata segment is bent because the curved plane falls to a state that is nearly parallel to the tangential plane.
  • the present invention has been made to solve such a problem, and can be obtained by performing curve interpolation on a line segment connecting two end points of the polygon mesh without finely dividing the polygon mesh. It is an object of the present invention to prevent the occurrence of a bending phenomenon such as a large curve.
  • the end points of the line segment are P 30 and P 03
  • the intermediate control points are P 21 and P 12
  • the position vectors of the end points are x 0 and x 1
  • the curvature of the curve is interpolated with the following cubic polynomial, with the parameter c and 0 ⁇ ⁇ ⁇ 1.
  • the line segment is interpolated by a cubic polynomial in which the degree of the Bezier curve is expanded to the third order, so that there are two intermediate control points P 21 and P 12. It is possible to prevent the intermediate control points P 21 and P 12 from being largely displaced laterally with respect to the line segment. Accordingly, it is possible to prevent the occurrence of a bending phenomenon in which a curve obtained by performing curve interpolation on a line segment connecting two end points of the polygon mesh is greatly curved without finely dividing the polygon mesh.
  • FIG. 1 is a functional block diagram illustrating a configuration example of the curved surface drawing apparatus according to the present embodiment.
  • the curved surface drawing apparatus of the present embodiment draws a curved line obtained by smoothly interpolating the sides of the polygon mesh from each vertex of the polygon mesh and intermediate control points, or a curved surface obtained by smoothly interpolating the inside.
  • a curve calculation unit 1 a curved surface calculation unit 2, a curve drawing unit 3, a curved surface drawing unit 4, and a display control unit 5 are provided.
  • the above functional blocks 1 to 5 can be configured by any of hardware, DSP (Digital Signal Processor), and software.
  • each of the functional blocks 1 to 5 is actually configured to include a CPU, RAM, ROM, etc. of a computer, and is a curved surface stored in a recording medium such as RAM, ROM, hard disk, or semiconductor memory. This is realized by operating a drawing program.
  • the functions of the functional blocks 1 to 5 can be realized by recording a curved surface drawing program on a recording medium such as a CD-ROM and reading it into a computer.
  • a recording medium for recording the curved surface drawing program a flexible disk, a hard disk, a magnetic tape, an optical disk, a magneto-optical disk, a DVD, a nonvolatile memory card, and the like can be used in addition to the CD-ROM. It can also be realized by downloading a curved surface drawing program to a computer via the Internet.
  • FIG. 2 is a diagram for explaining the processing contents by the curve calculation unit 1.
  • the curve calculation unit 1 uses P 30 and P 03 for the end points of the line segment, P 21 and P 12 for the intermediate control points, and x 0 and x for the position vectors of the end points P 30 and P 03. 1
  • the unit normal vector at both end points P 30 and P 03 is n 0 , n 1
  • the curvature parameter of the curve to be obtained is c
  • 0 ⁇ ⁇ ⁇ 1 is the parameter
  • the line segment is expressed by the following cubic polynomial (3) Interpolate curves.
  • the curve drawing unit 3 draws the curve obtained by the curve calculation unit 1.
  • the display control unit 5 displays the curve drawn by the curve drawing unit 3 on a display (not shown). By performing the processing of the curve calculation unit 1, the curve drawing unit 3 and the display control unit 5 for each line segment of the polygon mesh, it is possible to display a curve obtained by smoothly interpolating the sides of each polygon mesh on the display. It is.
  • the processing content of the curve calculating part 1 is demonstrated in detail.
  • the bending phenomenon of the Nagata segment occurs because the curved plane on which the curve of the Nagata segment rides falls to a state close to being parallel to the tangential plane at both end points of the line segment. Therefore, as a starting point for measures to prevent curving of the curve, the curve plane is made as close to the orthogonal plane as possible with respect to the tangent planes ⁇ x 0 , n 0 ⁇ , ⁇ x 1 , n 1 ⁇ .
  • the plane includes a line segment connecting the start and end points x 0 and x 1 and passes through the start point x 0 of the line segment and is orthogonal to the start point side tangent plane ⁇ x 0 , n 0 ⁇ .
  • a plane that includes the line segment, passes through the end point x 1 of the line segment, and is orthogonal to the end point tangent plane ⁇ x 1 , n 1 ⁇ is defined as a second plane ⁇ 1 .
  • the intermediate control point P 11 In order to obtain a curve, the intermediate control point P 11 should be on the both-side plane, so the position of the intersection between the curve plane ⁇ and the intersection line of the both-side planes ⁇ x 0 , n 0 ⁇ , ⁇ x 1 , n 1 ⁇
  • the intermediate control point P 11 must be arranged in Therefore, when the position of the tangent plane intersection line is in the wrong position relative to the strings, as shown in FIG. 4, the intermediate control points P 11 is the range between the orthogonal plane at both end points of the string It happens to go out.
  • the intermediate control point P 11 is located on this tangent line, but if the intersection point of the opposite end point with the tangent plane goes backward to the chord direction, the intermediate control point is placed on the opposite end side on the tangent line to avoid the reverse direction. Must. However, this is not possible with a quadratic Bezier curve. Therefore, in this embodiment, the degree of the Bezier curve is expanded to the third order, and the line segment is subjected to curve interpolation by the above-described third order polynomial (3).
  • the correction control points P 11 (0) and P 11 (1) are determined as in the following equations (9) and (10).
  • P 21 (P 20 + 2P 11 (0)) / 3 (9)
  • P 21 (2P 11 (1) + P 02 ) / 3 (10)
  • the correction control points P 11 (0) and P 11 (1) coincide with the control point P 11 of the quadratic Bezier curve when no retrograde occurs.
  • the tangent of the other side (the end point side at the start point side tangent and the start point side at the end point side tangent) is symmetric about the chord on the curve plane on the tangent line of each end point of the quadratic Bézier curve The intersection obtained by placing the
  • the occurrence of a bending phenomenon such that the curve obtained by interpolating the line connecting the two end points of the polygon mesh is greatly curved without making the polygon mesh divided finely. Can be prevented. That is, when the original Nagata patch has good surface quality, the same surface quality can be provided, and even when the original Nagata patch has a disturbed surface shape, the surface shape can be prevented from being disturbed.
  • a well-known interpolation method is the Coons curved surface.
  • the Coons curved surface is a combination of one frame line and a frame line opposite to it, and a ruled surface that is linearly interpolated from all combinations is added for all combinations, and extra boundary conditions are added.
  • the curved surface calculation unit 2 in FIG. 1 includes a plurality of frame lines calculated by the curve calculation unit 1 for each line segment of the polygon mesh, each vertex of the polygon mesh, and an intermediate control point for each line segment between the respective vertexes.
  • the Bezier curved surface that is linearly interpolated from the combination of one frame line and the opposite frame line is added for all combinations, and the Bezier curved surface is defined by linear interpolation from each vertex of the polygon mesh. Is subtracted to calculate the Coons surface.
  • the curved surface drawing unit 4 draws the Coons curved surface obtained by the curved surface calculation unit 2.
  • the display control unit 5 displays the Coons curved surface drawn by the curved surface drawing unit 4 on a display (not shown). For example, a curved surface obtained by smoothly interpolating the inside of each polygonal mesh by performing processing of the curved surface calculation unit 2, the curved surface drawing unit 4 and the display control unit 5 on a plurality of polygonal meshes constituting a three-dimensional shape. Can be displayed on the display.
  • a triangular mesh is taken as an example of a polygonal mesh, and a triangular Coons curved surface that interpolates a curved triangular area surrounded by three frame lines (curves) generated by the curved line calculation unit 1 will be described.
  • the three frame lines of the Nagata patch are changed to a cubic Bezier curve, and therefore, a cubic triangular Bezier curved surface must be used for the curved surface expression in the Bezier format.
  • x ( ⁇ , ⁇ , ⁇ ) P 003 ⁇ ⁇ 3 + P 102 ⁇ 3 ⁇ 2 + P 201 ⁇ 3 ⁇ 2 ⁇ + P 300 ⁇ ⁇ 3 + P 012 ⁇ 3 ⁇ 2 + P 111 ⁇ 6 ⁇ + P 210 ⁇ 3 ⁇ 2 ⁇ + P 021 ⁇ 3 ⁇ 2 ⁇ + P 120 ⁇ 3 ⁇ 2 + P 030 ⁇ ⁇ 3 (11)
  • the total number of control points required to draw the cubic triangular Bezier curved surface is 10 including the three vertices P 003 , P 300 , and P 030 .
  • x ( ⁇ , 0,1- ⁇ ) x 00 (1- ⁇ ) 3 + P 102 ⁇ 3 (1- ⁇ ) 2 ⁇ + P 201 ⁇ 3 (1- ⁇ ) ⁇ 2 + x 10 ⁇ 3 (12)
  • x (1- ⁇ , ⁇ , 0) x 10 (1- ⁇ ) 3 + P 210 ⁇ 3 (1- ⁇ ) 2 ⁇ + P 120 ⁇ 3 (1- ⁇ ) ⁇ 2 + x 11 ⁇ 3 (13)
  • x (0, ⁇ , 1- ⁇ ) x 00 (1- ⁇ ) 3 + P 012 ⁇ 3 (1- ⁇ ) 2 ⁇ + P 021 ⁇ 3 (1- ⁇ ) ⁇ 2 + x 11 ⁇ 3 (14)
  • control points of the cubic triangular Bezier curved surface 9 except for the control point P 111 located at the center of the curved triangular area are determined from the three frame lines by the above equations (12) to (14).
  • the central control point P 111 a method for determining this is separately required. This method will be described below.
  • the three frame lines can be expressed as follows from the equations (12) to (14).
  • x ( ⁇ , 0,1- ⁇ ) P 003 (1- ⁇ ) 3 + P 102 ⁇ 3 (1- ⁇ ) 2 ⁇ + P 201 ⁇ 3 (1- ⁇ ) ⁇ 2 + P 300 ⁇ 3
  • x (1- ⁇ , ⁇ , 0) P 300 (1- ⁇ ) 3 + P 210 ⁇ 3 (1- ⁇ ) 2 ⁇ + P 120 ⁇ 3 (1- ⁇ ) ⁇ 2 + P 030 ⁇ 3
  • x (0, ⁇ , 1- ⁇ ) P 003 (1- ⁇ ) 3 + P 012 ⁇ 3 (1- ⁇ ) 2 ⁇ + P 021 ⁇ 3 (1- ⁇ ) ⁇ 2 + P 030 ⁇ 3 (17)
  • the second ruled surface is defined as S 2
  • a triangular Bezier curved surface with a minute point as a control point and is expressed as follows.
  • a curved plane and a cubic Bézier curve are fitted by a cubic polynomial (3) as a curved surface patch for interpolating the given three vertices P 003 , P 300 , P 030.
  • a triangular mesh has been described as an example of a polygonal mesh, but the present invention is not limited to this.
  • the present invention can be applied to a quadrilateral mesh.

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Abstract

La présente invention, quand P30, P03 représentent des extrémités opposées d'un segment de droite, P21, P12 représentent des points de contrôle intermédiaires, c représente un paramètre de courbure d'une ligne courbe, et un paramètre représente 0 ≤ ζ ≤ 1, permet d'interpoler le segment de droite pour la courbe à l'aide d'un polynôme cubique, dans lequel l'ordre d'une courbe de Bezier est étendu jusqu'au troisième ordre, par interpolation du segment de droite pour la courbe à l'aide d'un polynôme cubique donné ci-dessous, ce qui rend possible de prévenir un déplacement latéral sensible des points de contrôle intermédiaires relativement au segment de droite. x(ζ) = P30(1 - ζ)3 + P21 × 3(1 - ζ)2 ζ + P12 × 3(1 - ζ) ζ2 + P03 ζ3 (où P21 = (x0 + 2P11)/3, P21 = (2P11 + x1)/3, et P11 = (x0 + x1)/2 - c/2).
PCT/JP2014/058770 2013-05-07 2014-03-27 Dispositif de trace de surface courbe et programme de trace de surface courbe WO2014181598A1 (fr)

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JP2013097657A JP6026949B2 (ja) 2013-05-07 2013-05-07 曲面描画装置および曲面描画用プログラム
JP2013-097657 2013-05-07

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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN111127590A (zh) * 2019-12-26 2020-05-08 新奥数能科技有限公司 一种二阶贝塞尔曲线绘制方法及装置

Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
JPH01120675A (ja) * 1987-11-04 1989-05-12 Sony Corp 物体の表面形状データ作成方法
JPH07311858A (ja) * 1994-05-18 1995-11-28 Sony Corp 自由曲面作成方法及び自由曲面作成装置
JP2001331812A (ja) * 2000-05-18 2001-11-30 Sony Corp 図形情報生成装置、画像処理装置、情報処理装置、図形情報生成方法及び記録媒体

Family Cites Families (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
GB9223315D0 (en) * 1992-11-06 1992-12-23 Canon Res Ct Europe Ltd Processing image data

Patent Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
JPH01120675A (ja) * 1987-11-04 1989-05-12 Sony Corp 物体の表面形状データ作成方法
JPH07311858A (ja) * 1994-05-18 1995-11-28 Sony Corp 自由曲面作成方法及び自由曲面作成装置
JP2001331812A (ja) * 2000-05-18 2001-11-30 Sony Corp 図形情報生成装置、画像処理装置、情報処理装置、図形情報生成方法及び記録媒体

Non-Patent Citations (1)

* Cited by examiner, † Cited by third party
Title
YASUHIRO SHIMIZU: "Extension of Nagata Patch to Cubic Degree", UNISYS TECHNOLOGY REVIEW, vol. 32, no. 3, 30 November 2012 (2012-11-30), pages 141 - 162 *

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN111127590A (zh) * 2019-12-26 2020-05-08 新奥数能科技有限公司 一种二阶贝塞尔曲线绘制方法及装置
CN111127590B (zh) * 2019-12-26 2023-06-20 新奥数能科技有限公司 一种二阶贝塞尔曲线绘制方法及装置

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