JP6026949B2 - Curved surface drawing apparatus and curved surface drawing program - Google Patents
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Description
本発明は、曲面描画装置および曲面描画用プログラムに関し、特に、多角形メッシュの各頂点と中間制御点とから多角形メッシュの辺や内部を滑らかに補間した曲線または曲面を描画する曲面描画装置に用いて好適なものである。 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.
通常、有限要素法（ＦＥＭ）による数値シミュレーションでは、対象領域を離散的な格子点群で近似して計算を行う。これに伴い、領域内に含まれる対象物表面形状は、三角形や四角形から構成される多面体モデル（三角形メッシュ、四角形メッシュ）で表される。 Usually, in the numerical simulation by the finite element method (FEM), the calculation is performed by approximating the target region with a discrete lattice point group. Accordingly, the object surface shape included in the region is represented by a polyhedron model (triangle mesh, quadrilateral mesh) composed of triangles and quadrangles.
一方、ＣＡＤ（Computer Aided Design：コンピュータ支援設計）分野では、伝統的に滑らかな曲面モデルが使われてきた。近年では、計測情報処理の発達により、多面体モデルと曲面モデルとを混在させた形状処理が行われるようになってきている。このような混在モデルでは、多面体表面を構成する各三角形や四角形の辺および内部を簡便な曲面式により滑らかに補間したい場面が生ずる。 On the other hand, in the CAD (Computer Aided Design) field, traditionally smooth curved surface models have been used. In recent years, with the development of measurement information processing, shape processing in which a polyhedral model and a curved surface model are mixed has come to be performed. In such a mixed model, there arises a scene where it is desired to smoothly interpolate the sides and the inside of each triangle or quadrangle constituting the polyhedral surface by a simple curved surface formula.
例えば、曲面を三角形メッシュに多面体近似し、その三角形メッシュを数値シミュレーションで変形した後、変形されたメッシュに合わせて曲面を滑らかに変形するという処理を考えると、多面体の頂点位置で算出される変形量を頂点間の中間位置にも滑らかに補間する計算が必要になる。従来、補間によって曲線や曲面を描画する技術が数多く提供されている（例えば、特許文献１，２参照）。 For example, considering the process of approximating a polyhedron to a triangular mesh and deforming the triangular mesh by numerical simulation, and then smoothly deforming the curved surface according to the deformed mesh, the deformation calculated at the vertex position of the polyhedron A calculation that smoothly interpolates the quantity also in the middle position between the vertices is required. Conventionally, many techniques for drawing a curve or a curved surface by interpolation have been provided (see, for example, Patent Documents 1 and 2).
ところで、理化学研究所のＶＣＡＤプロジェクトの中で長田隆氏により創案された曲面式として、「長田パッチ」と呼ばれるものがある。長田パッチは、三角形メッシュ、四角形メッシュの各頂点に付与された法線ベクトル（形状表面に直交する方向）から、曲面の丸みを簡便に補間するのに有効な技術である。以下に、この長田パッチの概要を説明する。 By the way, there is a so-called “Nagata patch” as a curved surface formula created by Takashi Nagata in the VCAD project of RIKEN. The 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. The outline of the Nagata patch will be described below.
長田パッチでは、多角形（三角形または四角形）の各頂点と、その各頂点に与えられた単位法線ベクトルとから、２次多項式により多角形領域内に丸みを持って補間する曲面式を定義している。なお、以下では説明の簡略化のため、三角形領域の補間に限定して説明する。長田パッチで基本となるのは、多角形の各辺（線分）を、２つの端点と各端点での単位法線ベクトルとから２次多項式で曲線補間する曲線式である。以下、この曲線式で得られる曲線を「長田セグメント」と呼ぶ。 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 quadrangle) 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. Hereinafter, the curve obtained by this curve formula is referred to as “Nagata segment”.
図１０に示すように、線分の始点をＰ0、終点をＰ1、各々の位置ベクトルをｘ0，ｘ1、単位法線ベクトルをｎ0，ｎ1とする。この場合、長田パッチでは、０≦ζ≦１をパラメータとして、次の形の２次多項式(1)でｘ0，ｘ1，ｎ0，ｎ1を補間する。
ｘ(ζ)＝ｘ0＋(ｄ−ｃ)ζ＋ｃζ2・・・(1)
ここに、ｘは曲線の線上点を表す位置ベクトル、ｄは始点から終点に向かう差ベクトル（ｄ＝ｘ1−ｘ0）である。また、ｃはｘ0，ｘ1，ｎ0，ｎ1から算出される「曲率パラメータ」と呼ばれる未知定数ベクトルであり、曲線の膨らみを規定する。
As shown in FIG. 10, 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 , and the unit normal vectors are n 0 , n 1 . In this case, the Nagata patch interpolates x 0 , x 1 , n 0 , and n 1 with a quadratic polynomial (1) of the following form using 0 ≦ ζ ≦ 1 as a parameter.
x (ζ) = x 0 + (dc) ζ + cζ 2 (1)
Here, x is a position vector representing a point on the curve, and d is a difference vector (d = x 1 −x 0 ) from the start point to the end point. Further, 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.
さて、位置ベクトルｘ0，ｘ1を結ぶ弦（線分）の中点の位置ベクトルをｘm（＝ｘ0＋ｄ／２）とし、中点ｘmからベクトル（−ｃ／２）だけ移動した位置をＰm（＝ｘm−ｃ／２）とする。また、ｘj（ｊ＝０，１）を通り法線ベクトルｎjに直交する平面を｛ｘj，ｎj｝で表す。この平面は曲線が接するべき平面なので、「接平面」とよぶ。 Now, the position vector of the middle point of the string (line segment) connecting the position vectors x 0 and x 1 is x m (= x 0 + d / 2), and the vector (−c / 2) is moved from the middle point x m . The position is P m (= x m −c / 2). A plane that passes through x j (j = 0, 1) and is orthogonal to the normal vector n j is represented by {x j , n j }. This plane is called the “tangential plane” because it is the plane that the curve should touch.
ここで、点Ｐmは、始点側接平面｛ｘ0，ｎ0｝上にも、終点側接平面｛ｘ1，ｎ1｝上にもなければならない。すなわち、両接平面｛ｘ0，ｎ0｝，｛ｘ1，ｎ1｝が交線を持つ場合には、点Ｐmは交線上になければならない。また、両接平面｛ｘ0，ｎ0｝，｛ｘ1，ｎ1｝が交線を持つ場合には、ｎ0×ｎ1は交線の方向ベクトルと平行なので、ベクトルｃは交線と直交しなければならない。したがって、両接平面｛ｘ0，ｎ0｝，｛ｘ1，ｎ1｝が交線を持つ場合、点Ｐmは弦の中点ｘmから交線に下した垂線の足ということになる。 Here, 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. In addition, when 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. Accordingly, if the tangent planes {x 0 , n 0 }, {x 1 , n 1 } have intersecting lines, the point P m is a leg of a perpendicular drawn from the midpoint x m of the chord to the intersecting line. .
長田パッチでは、線分の各端点Ｐ0，Ｐ1の位置ベクトルｘ0，ｘ1と、点Ｐmの位置ベクトルとから、各端点Ｐ0，Ｐ1を結ぶ線分を２次多項式で曲線補間する。この曲線補間の２次多項式(2)として、ＣＡＤでよく使われる図１１のようなベジエ曲線を用いることが可能である。すなわち、２次ベジエ曲線を
ｘ(ζ)＝Ｐ20(１−ζ)2＋Ｐ11・２(１−ζ)ζ＋Ｐ02ζ2・・・(2)
として、Ｐ20＝ｘ0（始点Ｐ0の位置ベクトル）、Ｐ02＝ｘ1（終点Ｐ1の位置ベクトル）、Ｐ11＝（ｘ0＋ｘ1）／２−ｃ／２（中間制御点Ｐmの位置ベクトル）のように当て嵌めることができる。
In the Nagata patch, 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. As 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. That is, a quadratic Bezier curve is expressed as x (ζ) = P 20 (1−ζ) 2 + P 11 · 2 (1−ζ) ζ + P 02 ζ 2 (2)
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).
長田セグメントは２次多項式曲線であり、放物線の一部を切り取った形状であるため、平面曲線である。曲線が乗っている平面は始終点ｘ0，ｘ1を結ぶ弦を含む平面であり、これを「曲線平面」と呼ぶ。長田セグメントの２次多項式曲線を２次ベジエ曲線として表現する場合には、中間制御点Ｐmの位置を選ぶことにより、曲線平面が定まる。 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”. In case of expressing a quadratic polynomial curves Nagata segment as a quadratic Bezier curve, by selecting the position of the intermediate control points P m, the curve plane is determined.
しかしながら、始終点ｘ0，ｘ1に与えられる単位法線ベクトルｎ0，ｎ1の方向が平行に近いとき、図１２に示すように、２次ベジエ曲線で表されるパッチ形状が乱れる（曲線が大きく湾曲する）場合があるという問題があった。曲線の湾曲現象は、単位法線ベクトルｎ0，ｎ1が平行に近く、始点側接平面｛ｘ0，ｎ0｝と終点側接平面｛ｘ1，ｎ1｝とが始終点ｘ0，ｘ1を結ぶ弦から大きく隔たった位置に交線を持つ場合に発生する。このとき、弦の中点ｘmから接平面交線に下した垂線の足に当たる中間制御点Ｐmは、弦に対して大きく横にずれ、結果として２次ベジエ曲線は湾曲するに至る。すなわち、曲線平面が接平面と平行に近い状態まで倒れるため、長田セグメントの湾曲が起きる。 However, when the directions of the unit normal vectors n 0 and n 1 given to the start and end points x 0 and x 1 are close to parallel, the patch shape represented by a quadratic Bezier curve is disturbed as shown in FIG. There is a problem that there is a case where the curve is greatly curved). The curve bending phenomenon is such that the unit normal vectors n 0 and n 1 are nearly parallel, and the start point side tangent plane {x 0 , n 0 } and the end point side tangent plane {x 1 , n 1 } are the start and end points x 0 , It occurs when having a line of intersection largely spaced apart position from the chord connecting the x 1. At this time, 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. In other words, the Nagata segment is bent because the curved plane falls to a state that is nearly parallel to the tangential plane.
長田パッチでは、このような問題が生じるのは始終点ｘ0，ｘ1の間隔が大き過ぎるためであるとして、精度を満たすまで三角形メッシュの分割を細かくすることが推奨されている。確かにメッシュを十分に細分すると、法線ベクトルｎ0 ，ｎ1と弦は直交に近づくので、曲率パラメータｃ＝０で長田セグメントが弦と一致しても、法線直交条件は破綻しない。しかし、現実の応用では三角形メッシュを随意に細かくすることができず、与えられた粗さのメッシュで処理を進めなければならない場合も多い。 In the Nagata patch, such a problem occurs because the interval between the start and end points x 0 and x 1 is too large, and it is recommended that the triangular mesh be divided finely until the accuracy is satisfied. Certainly, if the mesh is sufficiently subdivided, the normal vectors n 0 , n 1 and the chord approach orthogonality, so even if the curvature parameter c = 0 and the Nagata segment matches the chord, the normal orthogonality condition does not break down. However, in actual applications, the triangular mesh cannot be made arbitrarily small, and in many cases, the processing must proceed with a mesh having a given roughness.
本発明は、このような問題を解決するために成されたものであり、多角形メッシュの分割を細かくすることなく、当該多角形メッシュの２端点を結ぶ線分を曲線補間することによって得られる曲線が大きく湾曲するといった湾曲現象の発生を防止できるようにすることを目的とする。 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.
上記した課題を解決するために、本発明では、線分の両端点をＰ30，Ｐ03、中間制御点をＰ21，Ｐ12、両端点の位置ベクトルをｘ0，ｘ1、曲線の曲率パラメータをｃ、０≦ζ≦１をパラメータとして、次の３次多項式によって線分を曲線補間するようにしている。
ｘ(ζ)＝Ｐ30(１−ζ)3＋Ｐ21・３(１−ζ) 2ζ＋Ｐ12・３(１−ζ)ζ2＋Ｐ03ζ3
ここで、
Ｐ30＝ｘ0 ，Ｐ03＝ｘ1
Ｐ21＝(ｘ0＋２Ｐ11)／３，Ｐ21＝(２Ｐ11＋ｘ1)／３
Ｐ11＝(ｘ0＋ｘ1)／２−ｃ／２
In order to solve the above-described problem, in the present invention, 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 , and the curvature of the curve. The line is interpolated with the following cubic polynomial, with the parameter c and 0 ≦ ζ ≦ 1.
x (ζ) = P 30 (1−ζ) 3 + P 21 · 3 (1−ζ) 2 ζ + P 12 · 3 (1−ζ) ζ 2 + P 03 ζ 3
here,
P 30 = x 0 , P 03 = x 1
P 21 = (x 0 + 2P 11 ) / 3, P 21 = (2P 11 + x 1 ) / 3
P 11 = (x 0 + x 1 ) / 2−c / 2
上記のように構成した本発明によれば、ベジエ曲線の次数を３次に拡張した３次多項式により線分が曲線補間されるので、中間制御点はＰ21，Ｐ12の２つとなり、当該中間制御点Ｐ21，Ｐ12が線分に対して大きく横にずれてしまうことを防ぐことができる。これにより、多角形メッシュの分割を細かくすることなく、当該多角形メッシュの２端点を結ぶ線分を曲線補間することによって得られる曲線が大きく湾曲するといった湾曲現象の発生を防止することができる。 According to the present invention configured as described above, 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.
以下、本発明の一実施形態を図面に基づいて説明する。図１は、本実施形態による曲面描画装置の構成例を示す機能ブロック図である。図１に示すように、本実施形態の曲面描画装置は、多角形メッシュの各頂点と中間制御点とから多角形メッシュの辺を滑らかに補間した曲線または内部を滑らかに補間した曲面を描画するものであり、その機能構成として、曲線演算部１、曲面演算部２、曲線描画部３、曲面描画部４および表示制御部５を備えて構成されている。 Hereinafter, an embodiment of the present invention will be described with reference to the drawings. FIG. 1 is a functional block diagram illustrating a configuration example of the curved surface drawing apparatus according to the present embodiment. As shown in FIG. 1, 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. As a functional configuration thereof, 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.
上記各機能ブロック１〜５は、ハードウェア、ＤＳＰ（Digital Signal Processor）、ソフトウェアの何れによっても構成することが可能である。例えばソフトウェアによって構成する場合、上記各機能ブロック１〜５は、実際にはコンピュータのＣＰＵ、ＲＡＭ、ＲＯＭなどを備えて構成され、ＲＡＭやＲＯＭ、ハードディスクまたは半導体メモリ等の記録媒体に記憶された曲面描画用プログラムが動作することによって実現される。 Each of the functional blocks 1 to 5 can be configured by any of hardware, DSP (Digital Signal Processor), and software. For example, when configured by software, each of the functional blocks 1 to 5 actually includes 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.
したがって、上記各機能ブロック１〜５の機能は、曲面描画用プログラムをＣＤ−ＲＯＭのような記録媒体に記録し、コンピュータに読み込ませることによって実現できるものである。曲面描画用プログラムを記録する記録媒体としては、ＣＤ−ＲＯＭ以外に、フレキシブルディスク、ハードディスク、磁気テープ、光ディスク、光磁気ディスク、ＤＶＤ、不揮発性メモリカード等を用いることができる。また、曲面描画用プログラムをインターネットを介してコンピュータにダウンロードすることによっても実現できる。 Therefore, 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 the program into a computer. As 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.
図２は、曲線演算部１による処理内容を説明するための図である。図２に示すように、曲線演算部１は、線分の両端点をＰ30，Ｐ03、中間制御点をＰ21，Ｐ12、両端点Ｐ30，Ｐ03の位置ベクトルをｘ0 ，ｘ1、両端点Ｐ30，Ｐ03における単位法線ベクトルをｎ0，ｎ1、求める曲線の曲率パラメータをｃ、０≦ζ≦１をパラメータとして、次の３次多項式(3)によって線分を曲線補間する。
ｘ(ζ)＝Ｐ30(１−ζ)3＋Ｐ21・３(１−ζ) 2ζ＋Ｐ12・３(１−ζ)ζ2＋Ｐ03ζ3・・・(3）
ここで、
Ｐ30＝ｘ0 ，Ｐ03＝ｘ1
Ｐ21＝(ｘ0＋２Ｐ11)／３，Ｐ21＝(２Ｐ11＋ｘ1)／３
Ｐ11＝(ｘ0＋ｘ1)／２−ｃ／２：２次多項式で曲線補間する場合の中間制御点
FIG. 2 is a diagram for explaining the processing contents by the curve calculation unit 1. As shown in FIG. 2, 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, and 0 ≦ ζ ≦ 1 is the parameter, and the line segment is expressed by the following cubic polynomial (3) Interpolate curves.
x (ζ) = P 30 (1−ζ) 3 + P 21 · 3 (1−ζ) 2 ζ + P 12 · 3 (1−ζ) ζ 2 + P 03 ζ 3 (3)
here,
P 30 = x 0 , P 03 = x 1
P 21 = (x 0 + 2P 11 ) / 3, P 21 = (2P 11 + x 1 ) / 3
P 11 = (x 0 + x 1 ) / 2−c / 2: Intermediate control point when curve interpolation is performed with a quadratic polynomial
曲線描画部３は、曲線演算部１により求められた曲線を描画する。表示制御部５は、曲線描画部３により描画された曲線をディスプレイ（図示せず）に表示させる。多角形メッシュの各線分に対して曲線演算部１、曲線描画部３および表示制御部５の処理を行うことにより、各多角形メッシュの辺を滑らかに補間した曲線をディスプレイに表示させることが可能である。 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.
以下に、曲線演算部１の処理内容を詳しく説明する。従来技術について説明したように、長田セグメントの湾曲現象は、当該長田セグメントの曲線が乗っている曲線平面が線分の両端点における接平面と平行に近い状態まで倒れるために起きる。そこで、曲線の湾曲を防止する対策の出発点として、曲線平面を両接平面｛ｘ0，ｎ0｝，｛ｘ1，ｎ1｝に対して可能な限り直交に近づける。 Below, the processing content of the curve calculating part 1 is demonstrated in detail. As described with respect to the prior art, 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 }.
具体的には、図３に示すように、始終点ｘ0，ｘ1を結ぶ線分を含み、当該線分の始点ｘ0を通り始点側接平面｛ｘ0，ｎ0｝に直交する平面を第１平面Π0とする。また、上記線分を含み、当該線分の終点ｘ1を通り終点側接平面｛ｘ1，ｎ1｝に直交する平面を第２平面Π1とする。そして、第１平面Π0と第２平面Π1とを等角で２等分する２等分平面のうち、第１平面Π0と第２平面Π1となす角が小さい方を曲線平面Πとして選択する。 Specifically, as shown in FIG. 3, 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 }. the a first plane [pi 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 Π 1 . Of the bisector plane and the first plane [pi 0 bisects the second plane [pi 1 and an isometric, who curve plane [pi the angle formed between the first plane [pi 0 [pi 1 and the second plane is less Choose as.
なお、図１２に示したように 両接平面｛ｘ0，ｎ0｝，｛ｘ1，ｎ1｝の交線が線分（弦）から大きく隔たっている場合には、上記のように曲線平面Πを選択するだけでは問題は解決しない。式(2)のような２次多項式で線分を曲線補間する場合、始終点ｘ0，ｘ1で各接平面｛ｘ0，ｎ0｝，｛ｘ1，ｎ1｝に接する２次ベジエ曲線を得るには、中間制御点Ｐ11は両接平面上にあるべきなので、曲線平面Πと両接平面｛ｘ0，ｎ0｝，｛ｘ1，ｎ1｝の交線との交点位置に中間制御点Ｐ11を配置しなければならない。このため、接平面交線の位置が弦に対して不適切な位置にあると、図４に示すように、中間制御点Ｐ11が、弦の両端点での直交平面で挟まれた範囲の外に出てしまうことが起きる。 In addition, as shown in FIG. 12, when the intersection line of both tangent planes {x 0 , n 0 }, {x 1 , n 1 } is largely separated from the line segment (string), the curve is as described above. Simply selecting the plane will not solve the problem. When a line segment is interpolated with a quadratic polynomial such as equation (2), a quadratic bezier that touches each tangent plane {x 0 , n 0 }, {x 1 , n 1 } at the start and end points x 0 and x 1. 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 Π 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 For this reason, when the position of the tangent plane intersection line is in an inappropriate position with respect to the string, as shown in FIG. 4, the intermediate control point P 11 is in a range sandwiched by orthogonal planes at both end points of the string. It happens to go out.
図４の状況を２次ベジエ曲線の数式として表現すると、
(Ｐ11−Ｐ20，Ｐ02−Ｐ20)＜０または(Ｐ02−Ｐ11，Ｐ02−Ｐ20)＜０・・・(4)
である。中間制御点Ｐ11と弦の端点Ｐ20，Ｐ02との差ベクトルは、端点Ｐ20，Ｐ02での曲線の接線の方向に対応しているので、式(4)の前半の不等式は、始点Ｐ20での接線方向が弦の方向（始点Ｐ20から終点Ｐ02に向かう方向）と逆行していることを表している（図５（ａ）参照）。また、式(4)の後半の不等式は、終点Ｐ02での接線方向が弦の方向と逆行していることを表している（図５（ｂ）参照）。
If the situation of FIG. 4 is expressed as a mathematical expression of a quadratic Bezier curve,
(P 11 -P 20, P 02 -P 20) <0 or (P 02 -P 11, P 02 -P 20) <0 ··· (4)
It is. Since the difference vector between the intermediate control point P 11 and the chord end points P 20 and P 02 corresponds to the direction of the tangent of the curve at the end points P 20 and P 02 , the inequality in the first half of equation (4) is This indicates that the tangent direction at the start point P 20 is opposite to the chord direction (the direction from the start point P 20 toward the end point P 02 ) (see FIG. 5A). Further, the inequality in the latter half of the equation (4) indicates that the tangent direction at the end point P 02 is opposite to the chord direction (see FIG. 5B).
このように、両接平面｛ｘ0，ｎ0｝，｛ｘ1，ｎ1｝の交線が弦から大きく隔たっていると、接線方向と弦の方向が逆行するという問題は、２次ベジエ曲線である以上、避けられない。すなわち、各端点Ｐ20，Ｐ02での曲線への接線は、曲線平面Πと各接平面｛ｘ0，ｎ0｝，｛ｘ1，ｎ1｝との交線である。中間制御点Ｐ11はこの接線上に位置するが、相手端点の接平面との交点位置が弦の方向と逆行する場合、逆行を回避するには中間制御点を接線上、相手端点側に配置しなければならない。しかしながら、これは２次ベジエ曲線では不可能である。そこで、本実施形態では、ベジエ曲線の次数を３次に拡張し、上述した３次多項式(3)により線分を曲線補間することとする。 Thus, when the intersecting line of both tangent planes {x 0 , n 0 }, {x 1 , n 1 } is greatly separated from the string, the problem that the direction of the tangent and the direction of the string are reversed is a second order Bezier. As long as it is a curve, it is inevitable. That is, the tangent to the curve at each end point P 20 , P 02 is the intersection of the curve plane Π and each tangent plane {x 0 , n 0 }, {x 1 , n 1 }. 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).
なお、４個の制御点Ｐ30，Ｐ21，Ｐ12，Ｐ03は同一平面上にあるとは限らないので、一般に３次ベジエ曲線は平面曲線とは限らない。そこで、本実施形態では、図２に示すように、線分の両端点ｘ0＝Ｐ30，ｘ1＝Ｐ03および２つの中間制御点Ｐ21，Ｐ12を曲線平面Π上に配置することで、平面曲線として考える。ただし、３次ベジエ曲線では中間制御点がＰ21，Ｐ12の２個になり、その決め方を別途案出する必要が生じる。本実施形態では、２次ベジエ曲線の形状を可能な限り継承する手法を採用する。以下にこの手法を説明する。 Since the four control points P 30 , P 21 , P 12 , and P 03 are not necessarily on the same plane, the cubic Bezier curve is generally not necessarily a plane curve. Therefore, in this embodiment, as shown in FIG. 2, both end points x 0 = P 30 and x 1 = P 03 of the line segment and the two intermediate control points P 21 and P 12 are arranged on the curve plane Π. Then, think as a plane curve. However, in the cubic Bezier curve, there are two intermediate control points P 21 and P 12 , and it is necessary to devise how to determine them. In the present embodiment, a method of inheriting the shape of the quadratic Bezier curve as much as possible is adopted. This method will be described below.
２次多項式は、３次の項＝０の場合の特別な３次多項式と言えるので、２次ベジエ曲線を３次ベジエ曲線の特別形と考えたときの制御点を計算する。これは、「ベジエ曲線の次数上げ」として知られた手法を用いて、２次ベジエ曲線の２次多項式(2)に定数１を表す恒等式１＝(１−ζ)＋ζを掛けて、形式的に３次式にすればよい。なお、次数上げしても、曲線の形状は変化しない。 Since the quadratic polynomial can be said to be a special cubic polynomial when the third-order term = 0, the control point when the quadratic Bezier curve is considered as a special form of the cubic Bezier curve is calculated. This is obtained by multiplying the quadratic polynomial (2) of the quadratic Bezier curve by the identity 1 = (1-ζ) + ζ representing the constant 1, using a method known as “order increase of the Bezier curve”. The cubic equation may be used. Even if the order is increased, the shape of the curve does not change.
３次ベジエ曲線の多項式(3)と、２次ベジエ曲線の多項式(2)を３次に次数上げした曲線の多項式との関係式から、以下の関係式が得られる。
Ｐ30＝Ｐ20（＝ｘ0）・・・(5)
Ｐ21＝(Ｐ20＋２Ｐ11)／３・・・(6)
Ｐ21＝(２Ｐ11＋Ｐ02)／３・・・(7)
Ｐ03＝Ｐ02（＝ｘ1）・・・(8)
The following relational expression is obtained from the relational expression between the polynomial (3) of the cubic Bezier curve and the polynomial of the curve obtained by increasing the degree of the cubic (2) of the quadratic Bezier curve.
P 30 = P 20 (= x 0 ) (5)
P 21 = (P 20 + 2P 11 ) / 3 (6)
P 21 = (2P 11 + P 02 ) / 3 (7)
P 03 = P 02 (= x 1 ) (8)
ただし、線分の両端点での座標値と法線とから曲線を作るので、接線方向の正負は必要に応じて反転しても構わない。そこで、図５に示したように２次ベジエ曲線の接線方向が弦の方向と逆行している場合には、逆行しない側に補正した制御点から３次ベジエ曲線の制御点を決めることとする。 However, since the curve is formed from the coordinate values and normal lines at the two end points of the line segment, the positive / negative of the tangential direction may be reversed as necessary. Therefore, as shown in FIG. 5, when the tangential direction of the secondary Bezier curve is reverse to the direction of the chord, the control point of the cubic Bezier curve is determined from the control point corrected to the non-reverse side. .
すなわち、次の式(9),(10)のように補正制御点Ｐ11(0)，Ｐ11(1)を定める。
Ｐ21＝(Ｐ20＋２Ｐ11(0))／３・・・(9)
Ｐ21＝(２Ｐ11(1)＋Ｐ02)／３・・・(10)
この補正制御点Ｐ11(0)，Ｐ11(1)は、逆行が起きていない場合には２次ベジエ曲線の制御点Ｐ11に一致する。一方、逆行が起きている場合には、２次ベジエ曲線の各端点の接線上で相手側（始点側接線では終点側、終点側接線では始点側）の接線を曲線平面上で弦について線対称に配置して得られる交点とする。
That is, 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. On the other hand, when a 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
このことを、図６を用いて始点側を例に説明する。図６において、始点Ｐ20から両接線の交点Ｐ11に向かう方向は弦の方向と逆行しているので、交点Ｐ11自体を不採用とする。拘束条件は制御点が接線上にあることであり、接線と弦とのなす角度を維持するため、相手＝終点側接線を弦について線対称（上向きの白抜き矢印）に配置して始点側接線と交点Ｐ11(0)を求める。逆に、始点側接線を弦について線対称（下向きの白抜き矢印）に配置して終点側接線と交点Ｐ11(1)を求めてから、式(9),(10)で３次ベジエ曲線の制御点Ｐ21，Ｐ12を決めると、逆行が回避される。 This will be described using the starting point side as an example with reference to FIG. In FIG. 6, the direction from the starting point P 20 toward the intersection P 11 of both tangents is opposite to the direction of the chord, so the intersection P 11 itself is not adopted. The constraint condition is that the control point is on the tangent line, and in order to maintain the angle between the tangent line and the chord, the other end = tangent line is symmetrical with respect to the chord (upward white arrow) and the start point side tangent line And the intersection P 11 (0) is obtained. Conversely, the starting point side tangent line is placed symmetrically with respect to the chord (downward white arrow), the end point side tangent line and the intersection point P 11 (1) are obtained, and then a cubic Bezier curve is obtained using equations (9) and (10). When the control points P 21 and P 12 are determined, retrograde is avoided.
以上のような演算を行うことにより、多角形メッシュの分割を細かくすることなく、当該多角形メッシュの２端点を結ぶ線分を曲線補間することによって得られる曲線が大きく湾曲するといった湾曲現象の発生を防止することができる。すなわち、オリジナルの長田パッチで面質良好な場合には同等の面質を与え、オリジナルの長田パッチでは面形状が乱れる場合でも面形状の乱れを抑止することができる。 By performing the above calculation, 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.
次に、与えられた枠線だけから曲面内部を内挿する方法について述べる。当該内挿の手法としてよく知られているものとして、クーンズ曲面がある。クーンズ曲面は、図７に示すように、１つの枠線とそれに対向する枠線との組合せから１次式補間される線織面（ruled surface）を全ての組合せについて足し合わせ、境界条件の余分を調整して定義される曲面である。枠線間の１次補間から“ピンと張った”曲面が得られる特性があるので、長田パッチ定義式を拡張する曲面式に適している。 Next, a method for interpolating the inside of a curved surface from only a given frame line will be described. A well-known interpolation method is the Coons curved surface. As shown in FIG. 7, the Coons curved surface is obtained by adding a ruled surface that is linearly interpolated from a combination of one frame line and a frame line opposite to the frame line for all the combinations. Is a curved surface defined by adjusting. Since it has a characteristic that a “pinched” curved surface can be obtained from linear interpolation between frame lines, it is suitable for a curved surface formula that extends the Nagata patch definition formula.
図１の曲面演算部２は、多角形メッシュの各線分について曲線演算部１により算出された複数の枠線と、多角形メッシュの各頂点と、当該各頂点間の各線分に関する中間制御点とを用い、１つの枠線とそれに対向する枠線との組合せから１次式補間されるベジエ曲面を全ての組合せについて足し合わせるとともに、多角形メッシュの各頂点から１次補間で定義されるベジエ曲面を差し引いてクーンズ曲面を算出する。 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.
曲面描画部４は、曲面演算部２により求められたクーンズ曲面を描画する。表示制御部５は、曲面描画部４により描画されたクーンズ曲面をディスプレイ（図示せず）に表示させる。例えば、３次元形状を構成する複数の多角形メッシュに対して曲面演算部２、曲面描画部４および表示制御部５の処理を行うことにより、各多角形メッシュの内部を滑らかに内挿した曲面をディスプレイに表示させることが可能である。 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 the 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.
なお、以下では、多角形メッシュの一例として三角形メッシュをとりあげ、曲線演算部１により生成した３本の枠線（曲線）で囲まれた曲線三角形領域を内挿する三角クーンズ曲面について説明する。本実施形態では、上述したように、長田パッチの３本の枠線を３次ベジエ曲線に変更したので、ベジエ形式での曲面式は３次の三角ベジエ曲面を用いなければならない。 In the following, a triangular mesh is taken as an example of a polygon 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. In the present embodiment, as described above, 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.
例えば、図８のように、３つの頂点Ｐ003，Ｐ300，Ｐ030とそれらを結ぶ３本の枠線（曲線）とがある場合、３次三角ベジエ曲面は、μ，ν，ωを三角形の重心座標と考えたとき、０≦μ，ν，ω≦１（μ＋ν＋ω＝１）をパラメータとして、次の式(11)により表される。
ｘ(μ，ν，ω)＝Ｐ003・ω3＋Ｐ102・３μω2＋Ｐ201・３μ2ω＋Ｐ300・μ3
＋Ｐ012・３νω2＋Ｐ111・６μνω＋Ｐ210・３μ2ν
＋Ｐ021・３ν2ω＋Ｐ120・３μν2＋Ｐ030・ν3・・・(11)
この式(11)から明らかなように、３次三角ベジエ曲面を描画するために必要な制御点は、３つの頂点Ｐ003，Ｐ300，Ｐ030を含めて全部で１０個である。
For example, as shown in FIG. 8, when there are three vertices P 003 , P 300 , P 030 and three frame lines (curves) connecting them, the cubic triangular Bezier curved surface is a triangle of μ, ν, ω. Is expressed by the following equation (11) using 0 ≦ μ, ν, ω ≦ 1 (μ + ν + ω = 1) as parameters.
x (μ, ν, ω) = P 003 · ω 3 + P 102 · 3 µω 2 + P 201 · 3 µ 2 ω + P 300 · µ 3
+ P 012 · 3νω 2 + P 111 · 6μνω + P 210 · 3μ 2 ν
+ P 021・ 3ν 2 ω + P 120・ 3μν 2 + P 030・ ν 3 (11)
As is clear from this equation (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 .
ここで、長田パッチの３頂点との対応は、Ｐ003＝ｘ00，Ｐ300＝ｘ10，Ｐ030＝ｘ11である。また、パラメータとの対応は、μ＝η−ζ，ν＝ζ，ω＝１−ηであり（０≦ζ≦η≦１）、各枠線には上述の方法で決められた３次ベジエ曲線を次のように当て嵌める。
ｘ(η,0,１−η)＝ｘ00(１−η)3＋Ｐ102・３(１−η)2η＋Ｐ201・３(１−η)η2＋ｘ10η3・・・(12）
ｘ(１−ζ,ζ,0)＝ｘ10(１−ζ)3＋Ｐ210・３(１−ζ)2ζ＋Ｐ120・３(１−ζ) ζ2＋ｘ11ζ3・・・(13）
ｘ(0,η,１−η)＝ｘ00(１−η)3＋Ｐ012・３(１−η)2η＋Ｐ021・３(１−η)η2＋ｘ11η3・・・(14）
なお、式(12)ではν＝ζ＝０なのでμ＝η、式(13)ではω＝１−η＝０なのでη＝１、μ＝１−ζ、式(14)ではμ＝η−ζ＝０なのでν＝ζ＝ηである。
Here, the correspondence with the three vertices of the Nagata patch is P 003 = x 00 , P 300 = x 10 , P 030 = x 11 . The correspondence with the parameters is μ = η−ζ, ν = ζ, ω = 1−η (0 ≦ ζ ≦ η ≦ 1), and each frame line has a cubic Bezier determined by the above-described method. Fit the curve as follows.
x (η, 0,1-η) = x 00 (1-η) 3 + P 102 · 3 (1-η) 2 η + P 201 · 3 (1-η) η 2 + x 10 η 3 (12)
x (1-ζ, ζ, 0) = x 10 (1-ζ) 3 + P 210 · 3 (1-ζ) 2 ζ + P 120 · 3 (1-ζ) ζ 2 + x 11 ζ 3 (13)
x (0, η, 1-η) = x 00 (1-η) 3 + P 012 · 3 (1-η) 2 η + P 021 · 3 (1-η) η 2 + x 11 η 3 (14)
In equation (12), ν = ζ = 0, so μ = η, in equation (13), ω = 1−η = 0, so η = 1, μ = 1−ζ, and in equation (14), μ = η−ζ. Since = 0, ν = ζ = η.
上記式(12)〜(14)により、３次三角ベジエ曲面の１０個の制御点のうち、曲線三角形領域の中央に位置する制御点Ｐ111を除く９個は３本の枠線から定まる。一方、中央の制御点Ｐ111については、これを決める方法が別途必要である。以下に、この方法を説明する。 Of the 10 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). On the other hand, for the central control point P 111 , a method for determining this is separately required. This method will be described below.
３本の枠線は、式(12)〜(14)から以下のように表すことができる。
ｘ(η,0,１−η)＝Ｐ003(１−η)3＋Ｐ102・３(１−η)2η＋Ｐ201・３(１−η)η2＋Ｐ300η3・・・(15）
ｘ(１−ζ,ζ,0)＝Ｐ300(１−ζ)3＋Ｐ210・３(１−ζ)2ζ＋Ｐ120・３(１−ζ) ζ2＋Ｐ030ζ3・・・(16）
ｘ(0,η,１−η)＝Ｐ003(１−η)3＋Ｐ012・３(１−η)2η＋Ｐ021・３(１−η)η2＋Ｐ030η3・・・(17）
ここで、式(15)はν＝０とした枠線、式(16)はω＝０とした枠線、式(17)はμ＝０とした枠線を示している。
The three frame lines can be expressed as follows from the equations (12) to (14).
x (η, 0,1-η) = P 003 (1-η) 3 + P 102 · 3 (1-η) 2 η + P 201 · 3 (1-η) η 2 + P 300 η 3 (15)
x (1-ζ, ζ, 0) = P 300 (1-ζ) 3 + P 210 · 3 (1-ζ) 2 ζ + P 120 · 3 (1-ζ) ζ 2 + P 030 ζ 3 (16)
x (0, η, 1-η) = P 003 (1-η) 3 + P 012 · 3 (1-η) 2 η + P 021 · 3 (1-η) η 2 + P 030 η 3 (17)
Here, Expression (15) shows a frame line with ν = 0, Expression (16) shows a frame line with ω = 0, and Expression (17) shows a frame line with μ = 0.
図７にて説明したように、クーンズ曲面は、１つの枠線とそれに対向する枠線とから１次式補間される線織面を全組合せについて足し合わせ、境界条件の余分を調整して定義される曲面である。これを図８の例に適用すると、以下のようになる。すなわち、ν＝０枠線とμ＝０枠線とから１次式補間される第１の線織面をＳ1、μ＝０枠線とω＝０枠線とから１次式補間される第２の線織面をＳ2、ν＝０枠線とω＝０枠線とから１次式補間される第３の線織面をＳ3、３頂点から１次補間で定義される１次三角ベジエ曲面をＳ4とした場合、求める三角クーンズ曲面Ｓは、以下の通りとなる。
Ｓ＝（Ｓ1＋Ｓ2＋Ｓ3−Ｓ4）／２・・・(18)
As explained with reference to FIG. 7, the Coons curved surface is defined by adding the ruled surface interpolated from one frame line and the opposite frame line for all combinations and adjusting the excess of boundary conditions. Is a curved surface. Applying this to the example of FIG. That is, the first ruled surface that is linearly interpolated from the ν = 0 frame line and the μ = 0 frame line is linearly interpolated from the S 1 , μ = 0 frame line and the ω = 0 frame line. The second ruled surface is defined as S 2 , and the third ruled surface that is linearly interpolated from the ν = 0 frame line and the ω = 0 frame line is defined as S 3 , and 1 is defined by the primary interpolation from the three vertices. If the next triangular Bezier surface was S 4, it obtains triangular Coons curved surface S is as follows.
S = (S 1 + S 2 + S 3 −S 4 ) / 2 (18)
ここで、第１の線織面Ｓ1は、図９に示すように、ν＝０枠線の中間制御点Ｐ201とμ＝０枠線の中間制御点Ｐ021とを結ぶ線分の等分点を制御点とする三角ベジエ曲面であり、以下のように表される。
Ｓ1(μ，ν，ω)＝Ｐ003・ω3＋Ｐ102・３μω2＋Ｐ201・３μ2ω＋Ｐ300・μ3＋Ｐ012・３νω2＋｛（Ｐ201＋Ｐ021）／２｝・６μνω＋｛（２Ｐ300＋Ｐ030）／３｝・３μ2ν＋Ｐ021・３ν2ω＋｛（Ｐ300＋２Ｐ030）／３｝・３μν2＋Ｐ030・ν3・・・(19)
Here, as shown in FIG. 9, the first ruled surface S 1 is a line segment connecting the intermediate control point P 201 of the ν = 0 frame line and the intermediate control point P 021 of the μ = 0 frame line. A triangular Bezier curved surface with a minute point as a control point, and is expressed as follows.
S 1 (μ, ν, ω) = P 003 · ω 3 + P 102 · 3 μω 2 + P 201 · 3 μ 2 ω + P 300 · μ 3 + P 012 · 3νω 2 + {(P 201 + P 021 ) / 2} · 6 μνω + {( 2P 300 + P 030 ) / 3} · 3μ 2 ν + P 021 · 3ν 2 ω + {(P 300 + 2P 030 ) / 3} · 3 μν 2 + P 030 · ν 3 (19)
同様に、第２の線織面Ｓ2は、μ＝０枠線の中間制御点Ｐ012とω＝０枠線の中間制御点Ｐ210とを結ぶ線分の等分点を制御点とする三角ベジエ曲面であり、以下のように表される。
Ｓ2(μ，ν，ω)＝Ｐ003・ω3＋｛（２Ｐ003＋Ｐ300）／３｝・３μω2＋｛（Ｐ003＋２Ｐ300）／３｝・３μ2ω＋Ｐ300・μ3＋Ｐ012・３νω2＋｛（Ｐ012＋Ｐ210）／２｝・６μνω＋Ｐ210・３μ2ν＋Ｐ021・３ν2ω＋Ｐ120・３μν2＋Ｐ030・ν3・・・(20)
Similarly, the second ruled surface S 2 has a control point as an equidistant point of a line segment connecting the intermediate control point P 012 of μ = 0 frame line and the intermediate control point P 210 of ω = 0 frame line. It is a triangular Bezier curved surface and is expressed as follows.
S 2 (μ, ν, ω ) = P 003 · ω 3 + {(2P 003 + P 300) / 3} · 3μω 2 + {(P 003 + 2P 300) / 3} · 3μ 2 ω + P 300 · μ 3 + P 012・ 3νω 2 + {(P 012 + P 210 ) / 2} • 6 μνω + P 210 • 3 μ 2 ν + P 021 • 3ν 2 ω + P 120 • 3 μν 2 + P 030 • ν 3 (20)
同様に、第３の線織面Ｓ3は、ν＝０枠線の中間制御点Ｐ102とω＝０枠線の中間制御点Ｐ120とを結ぶ線分の等分点を制御点とする三角ベジエ曲面であり、以下のように表される。
Ｓ3(μ，ν，ω)＝Ｐ003・ω3＋Ｐ102・３μω2＋Ｐ201・３μ2ω＋Ｐ300・μ3＋｛（２Ｐ003＋Ｐ030）／３｝・３νω2＋｛（Ｐ102＋Ｐ120）／２｝・６μνω＋Ｐ210・３μ2ν＋｛（Ｐ003＋２Ｐ030）／３｝・３ν2ω＋Ｐ120・３μν2＋Ｐ030・ν3・・・(21)
Similarly, the third ruled surface S 3 has a control point that is an equally divided point connecting the intermediate control point P 102 of the ν = 0 frame line and the intermediate control point P 120 of the ω = 0 frame line. It is a triangular Bezier curved surface and is expressed as follows.
S 3 (μ, ν, ω) = P 003 · ω 3 + P 102 · 3 μω 2 + P 201 · 3 μ 2 ω + P 300 · μ 3 + {(2P 003 + P 030 ) / 3} · 3νω 2 + {(P 102 + P 120) / 2} · 6μνω + P 210 · 3μ 2 ν + {(P 003 + 2P 030) / 3} · 3ν 2 ω + P 120 · 3μν 2 + P 030 · ν 3 ··· (21)
これらの式(18)〜(21)から、３次三角ベジエ曲面の中央制御点Ｐ111を計算すると、次の式(22)となる。
Ｐ111＝｛（Ｐ201＋Ｐ021）／４｝＋｛（Ｐ012＋Ｐ210）／４｝＋｛（Ｐ102＋Ｐ120）／４｝−｛（Ｐ003＋Ｐ300＋Ｐ030）／６｝・・・(22)
From these equations (18) to (21), when calculating the cubic triangular Bezier surface of the central control point P 111, the following equation (22).
P 111 = {(P 201 + P 021 ) / 4} + {(P 012 + P 210 ) / 4} + {(P 102 + P 120 ) / 4} − {(P 003 + P 300 + P 030 ) / 6}. ·(twenty two)
このように、本実施形態によれば、与えられた３頂点Ｐ003，Ｐ300，Ｐ030を補間する曲面パッチとして、３次多項式(3)により曲線平面と３次ベジエ曲線とを当て嵌めてν＝０枠線、ω＝０枠線およびμ＝０枠線を補間し、これらの各枠線から式(18)の定義式による三角クーンズ曲面を構成して３次三角ベジエ曲面を得ることができる。 Thus, according to the present embodiment, 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. Interpolate the ν = 0 frame line, ω = 0 frame line, and μ = 0 frame line, and form a triangular Coons surface according to the definition of equation (18) from these frame lines to obtain a cubic triangular Bezier surface. Can do.
なお、上記実施形態では、多角形メッシュの一例として三角形メッシュをとりあげて説明をしたが、本発明はこれに限定されない。例えば、四角形メッシュに本発明を適用することも可能である。 In the above embodiment, a triangular mesh has been described as an example of a polygonal mesh, but the present invention is not limited to this. For example, the present invention can be applied to a quadrilateral mesh.
その他、上記実施形態は、何れも本発明を実施するにあたっての具体化の一例を示したものに過ぎず、これによって本発明の技術的範囲が限定的に解釈されてはならないものである。すなわち、本発明はその要旨、またはその主要な特徴から逸脱することなく、様々な形で実施することができる。 In addition, each of the above-described embodiments is merely an example of implementation in carrying out the present invention, and the technical scope of the present invention should not be construed in a limited manner. That is, the present invention can be implemented in various forms without departing from the gist or the main features thereof.
１ 曲線演算部
２ 曲面演算部
３ 曲線描画部
４ 曲面描画部
５ 表示制御部
DESCRIPTION OF SYMBOLS 1 Curve calculation part 2 Curved surface calculation part 3 Curve drawing part 4 Curved surface drawing part 5 Display control part
Claims (6)
上記線分の両端点をＰ30，Ｐ03、上記中間制御点をＰ21，Ｐ12、上記両端点の位置ベクトルをｘ0，ｘ1、上記曲線の曲率パラメータをｃ、０≦ζ≦１をパラメータとして、次の３次多項式によって上記線分を曲線補間する曲線演算部と、
ｘ(ζ)＝Ｐ30(１−ζ)3＋Ｐ21・３(１−ζ) 2ζ＋Ｐ12・３(１−ζ)ζ2＋Ｐ03ζ3
ここで、
Ｐ30＝ｘ0 ，Ｐ03＝ｘ1
Ｐ21＝(ｘ0＋２Ｐ11)／３，Ｐ21＝(２Ｐ11＋ｘ1)／３
Ｐ11＝(ｘ0＋ｘ1)／２−ｃ／２
上記曲線演算部により求められた上記曲線を描画する曲線描画部とを備えたことを特徴とする曲面描画装置。 A curved surface drawing apparatus that draws a curved line obtained by smoothly interpolating the sides of the polygonal mesh from each vertex and intermediate control points of the polygonal mesh or a curved surface obtained by smoothly interpolating the inside,
Both 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 both end points are x 0 and x 1 , the curvature parameter of the curve is c, and 0 ≦ ζ ≦ 1 As a parameter, a curve calculation unit for curve interpolating the line segment by the following cubic polynomial,
x (ζ) = P 30 (1−ζ) 3 + P 21 · 3 (1−ζ) 2 ζ + P 12 · 3 (1−ζ) ζ 2 + P 03 ζ 3
here,
P 30 = x 0 , P 03 = x 1
P 21 = (x 0 + 2P 11 ) / 3, P 21 = (2P 11 + x 1 ) / 3
P 11 = (x 0 + x 1 ) / 2−c / 2
A curved surface drawing apparatus comprising: a curve drawing unit for drawing the curve obtained by the curve calculation unit.
上記曲面演算部により求められた上記クーンズ曲面を描画する曲面描画部とを更に備えたことを特徴とする請求項１に記載の曲面描画装置。 One frame using a plurality of frame lines calculated by the curve calculation unit for each line segment of the polygon mesh, each vertex of the polygon mesh, and the intermediate control point regarding each line segment between the vertexes The Beesier surface that is linearly interpolated from the combination of the line and the opposite frame line is added for all the combinations, and the Coons surface is obtained by subtracting the Bezier surface defined by the linear interpolation from each vertex of the polygon mesh. A curved surface calculation unit for calculating
The curved surface drawing apparatus according to claim 1, further comprising a curved surface drawing unit that draws the Coons curved surface obtained by the curved surface calculation unit.
ｘ(η,0,１−η)＝Ｐ003(１−η)3＋Ｐ102・３(１−η)2η＋Ｐ201・３(１−η)η2＋Ｐ300η3
ｘ(１−ζ,ζ,0)＝Ｐ300(１−ζ)3＋Ｐ210・３(１−ζ)2ζ＋Ｐ120・３(１−ζ) ζ2＋Ｐ030ζ3
ｘ(0,η,１−η)＝Ｐ003(１−η)3＋Ｐ012・３(１−η)2η＋Ｐ021・３(１−η)η2＋Ｐ030η3
上記曲面演算部は、上記３式のうち第１式と第３式との組合せから１次式補間される第１のベジエ曲面をＳ1、上記３式のうち第２式と第３式との組合せから１次式補間される第２のベジエ曲面をＳ2、上記３式のうち第１式と第２式との組合せから１次式補間される第３のベジエ曲面をＳ3、上記３頂点から１次補間で定義されるベジエ曲面をＳ4、上記三角形メッシュの中央に位置する制御点をＰ111とした場合、
Ｓ＝（Ｓ1＋Ｓ2＋Ｓ3−Ｓ4）／２
Ｐ111＝｛（Ｐ201＋Ｐ021）／４｝＋｛（Ｐ012＋Ｐ210）／４｝＋｛（Ｐ102＋Ｐ120）／４｝−｛（Ｐ003＋Ｐ300＋Ｐ030）／６｝
により上記クーンズ曲面Ｓを算出することを特徴とする請求項３に記載の曲面描画装置。 The polygonal mesh is a triangular mesh, and the three vertices of the triangular mesh are P 003 , P 300 , P 030 , and three sets of the intermediate control points relating to the three line segments connecting the three vertices are (P 102 , P 201 ), (P 210 , P 120 ), (P 012 , P 021 ), 0 ≦ ζ ≦ η ≦ 1, and the curve calculation unit calculates three frame lines according to the following three formulas:
x (η, 0,1-η) = P 003 (1-η) 3 + P 102 · 3 (1-η) 2 η + P 201 · 3 (1-η) η 2 + P 300 η 3
x (1-ζ, ζ, 0) = P 300 (1-ζ) 3 + P 210 · 3 (1-ζ) 2 ζ + P 120 · 3 (1-ζ) ζ 2 + P 030 ζ 3
x (0, η, 1-η) = P 003 (1-η) 3 + P 012 · 3 (1-η) 2 η + P 021 · 3 (1-η) η 2 + P 030 η 3
The curved surface calculation unit is a first Bezier curved surface that is linearly interpolated from a combination of the first and third expressions among the three expressions, S 1 , and among the three expressions, the second and third expressions of S 2 a second Bezier curved surface is a linear expression interpolation of a combination, the first equation and the third Bezier curved surface S 3 which is combined from a linear equation interpolation with the second type of the above three equations, the When a Bezier surface defined by linear interpolation from three vertices is S 4 and a control point located at the center of the triangular mesh is P 111 ,
S = (S 1 + S 2 + S 3 −S 4 ) / 2
P 111 = {(P 201 + P 021 ) / 4} + {(P 012 + P 210 ) / 4} + {(P 102 + P 120 ) / 4} − {(P 003 + P 300 + P 030 ) / 6}
The curved surface drawing apparatus according to claim 3, wherein the Coons curved surface S is calculated by:
上記線分の両端点をＰ30，Ｐ03、上記中間制御点をＰ21，Ｐ12、上記両端点の位置ベクトルをｘ0，ｘ1、上記曲線の曲率パラメータをｃ、０≦ζ≦１をパラメータとして、次の３次多項式によって上記線分を曲線補間する曲線演算手段、および
ｘ(ζ)＝Ｐ30(１−ζ)3＋Ｐ21・３(１−ζ) 2ζ＋Ｐ12・３(１−ζ)ζ2＋Ｐ03ζ3
ここで、
Ｐ30＝ｘ0 ，Ｐ03＝ｘ1
Ｐ21＝(ｘ0＋２Ｐ11)／３，Ｐ21＝(２Ｐ11＋ｘ1)／３
Ｐ11＝(ｘ0＋ｘ1)／２−ｃ／２
上記曲線演算手段により求められた上記曲線を描画する曲線描画部手段
としてコンピュータを機能させるための曲面描画用プログラム。 A surface drawing program for drawing a curved surface obtained by smoothly interpolating the sides of the polygonal mesh from each vertex and intermediate control points of the polygonal mesh or a curved surface obtained by smoothly interpolating the inside.
Both 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 both end points are x 0 and x 1 , the curvature parameter of the curve is c, and 0 ≦ ζ ≦ 1 As a parameter, curve calculation means for curve interpolation of the line segment by the following cubic polynomial, and x (ζ) = P 30 (1-ζ) 3 + P 21 · 3 (1-ζ) 2 ζ + P 12 · 3 ( 1-ζ) ζ 2 + P 03 ζ 3
here,
P 30 = x 0 , P 03 = x 1
P 21 = (x 0 + 2P 11 ) / 3, P 21 = (2P 11 + x 1 ) / 3
P 11 = (x 0 + x 1 ) / 2−c / 2
A curved surface drawing program for causing a computer to function as a curve drawing unit that draws the curve obtained by the curve calculation unit.
上記曲面演算手段により求められた上記クーンズ曲面を描画する曲面描画手段
としてコンピュータを更に機能させるための請求項５に記載の曲面描画用プログラム。 One frame using a plurality of frame lines calculated by the curve calculation unit for each line segment of the polygon mesh, each vertex of the polygon mesh, and the intermediate control point regarding each line segment between the vertexes The Beesier surface that is linearly interpolated from the combination of the line and the opposite frame line is added for all the combinations, and the Coons surface is obtained by subtracting the Bezier surface defined by the linear interpolation from each vertex of the polygon mesh. 6. The curved surface drawing program according to claim 5, wherein the computer further functions as curved surface calculating means for calculating the curve, and curved surface drawing means for drawing the Coons curved surface obtained by the curved surface calculating means.
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