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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Centripetal Catmull–Rom spline ： ウィキペディア英語版 Centripetal Catmull–Rom spline In computer graphics, centripetal Catmull–Rom spline is a variant form of Catmull-Rom spline 〔E. Catmull and R. Rom. A class of local interpolating splines. Computer Aided Geometric Design, pages 317-326, 1974.〕 formulated according to the work of Barry and Goldman.〔P. J. Barry and R. N. Goldman. A recursive evaluation algorithm for a class of Catmull–Rom splines. SIGGRAPH Computer Graphics, 22(4):199-204, 1988.〕 It is a type of interpolating spline (a curve that goes through its control points) defined by four control points $\backslash mathbf\_0,\; \backslash mathbf\_1,\; \backslash mathbf\_2,\; \backslash mathbf\_3$, with the curve drawn only from $\backslash mathbf\_1$ to $\backslash mathbf\_2$. ==Definition== Let $\backslash mathbf\_i\; =\; (\backslash quad\; y\_i)^T$ denote a point. For a curve segment $\backslash mathbf$ defined by points $\backslash mathbf\_0,\; \backslash mathbf\_1,\; \backslash mathbf\_2,\; \backslash mathbf\_3$ and knot sequence $t\_0,\; t\_1,\; t\_2,\; t\_3$, the centripetal Catmull-Rom spline can be produced by: : $\backslash mathbf\; =\; \backslash frac\backslash mathbf\_1+\backslash frac\backslash mathbf\_2$ where : $\backslash mathbf\_1\; =\; \backslash frac\backslash mathbf\_1+\backslash frac\backslash mathbf\_2$ : $\backslash mathbf\_2\; =\; \backslash frac\backslash mathbf\_2+\backslash frac\backslash mathbf\_3$ : $\backslash mathbf\_1\; =\; \backslash frac\backslash mathbf\_0+\backslash frac\backslash mathbf\_1$ : $\backslash mathbf\_2\; =\; \backslash frac\backslash mathbf\_1+\backslash frac\backslash mathbf\_2$ : $\backslash mathbf\_3\; =\; \backslash frac\backslash mathbf\_2+\backslash frac\backslash mathbf\_3$ and :$t\_\; =\; \backslash left()^\; +\; t\_i$ in which $\backslash alpha$ ranges from 0 to 1 for knot parameterization, and $i\; =\; 0,1,2,3$ with $t\_0\; =\; 0$. For centripetal Catmull-Rom spline, the value of $\backslash alpha$ is $0.5$. When $\backslash alpha\; =\; 0$, the resulting curve is the standard Catmull-Rom spline (uniform Catmull-Rom spline); when $\backslash alpha\; =\; 1$, the product is a chordal Catmull-Rom spline. Plugging $t\; =\; t\_1$ into the spline equations $\backslash mathbf\_1,\; \backslash mathbf\_2,\; \backslash mathbf\_3,\; \backslash mathbf\_1,\; \backslash mathbf\_2,$ and $\backslash mathbf$ shows that the value of the spline curve at $t\_1$ is $\backslash mathbf\; =\; \backslash mathbf\_1$. Similarly, substituting $t\; =\; t\_2$ into the spline equations shows that $\backslash mathbf\; =\; \backslash mathbf\_2$ at $t\_2$. This is true independent of the value of $\backslash alpha$ since the equation for $t\_$ is not needed to calculate the value of $\backslash mathbf$ at points $t\_1$ and $t\_2$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Centripetal Catmull–Rom spline」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.