樣條之CatmullRom

所謂樣條曲線是指給定一組控制點而得到一條曲線，曲線的大致形狀由這些點予以控制，一般可分為插值樣條和逼近樣條兩種，插值樣條通常用於數字化繪圖或動畫的設計，逼近樣條一般用來構造物體的表面。CatmullRom樣條與上一節所講的B樣條很相似，不同在於CatmullRom樣條的曲線會經過其每一個控制點。

The centripetal Catmull–Rom is a subclass of cubic Hermite spline that extends the Catmull–Rom implementation by allowing each of the four control points to be associated with an arbitrary time interval in the computation of a value on the curve. This modifies the behavior of the curve. The curve is an interpolation, and will intersect with all but the first and last control points. If the time intervals are uniform, the result will be the same as that of the original Catmull–Rom spline curve. Thechordal

curve uses the two dimensional Euclidean distance between control points to provide the time elements, while thecentripetalcurve uses the square root of the Euclidean distance. The principal reason for using the centripetal version is that it has been shown to be free of cusps and self-intersections.

關於插值與樣條的介紹請看：http://www.cnblogs.com/WhyEngine/p/4020294.html

核心程式碼：

 1 void    YcCatmullRomSpline::BuildWeights()
 2 {
 3     ClearWeights();
 4 
 5     for (Yuint i = 0; i < 4; i++)
 6     {
 7         m_splineWeights[i] = (Yreal*)malloc((m_subD)*sizeof(Yreal));
 8         m_tangentWeights[i] = (Yreal*)malloc((m_subD)*sizeof(Yreal));
 9     }
10 
11     Yreal u, u_2, u_3;
12     for (Yuint i = 0; i < m_subD; i++)
13     {
14         u = (float)i / m_subD;
15         u_2 = u * u;
16         u_3 = u_2 * u;
17 
18         // 參見"遊戲程式設計精粹1"P333
19         m_splineWeights[0][i] = (-1.0f*u_3 + 2.0f*u_2 - 1.0f*u + 0.0f)*0.5f;
20         m_splineWeights[1][i] = ( 3.0f*u_3 - 5.0f*u_2 + 0.0f*u + 2.0f)*0.5f;
21         m_splineWeights[2][i] = (-3.0f*u_3 + 4.0f*u_2 + 1.0f*u + 0.0f)*0.5f;
22         m_splineWeights[3][i] = ( 1.0f*u_3 - 1.0f*u_2 + 0.0f*u + 0.0f)*0.5f;
23 
24         m_tangentWeights[0][i] = (-3.0f*u_2 +  4.0f*u - 1.0f)*0.5f;
25         m_tangentWeights[1][i] = ( 9.0f*u_2 - 10.0f*u + 0.0f)*0.5f;
26         m_tangentWeights[2][i] = (-9.0f*u_2 +  8.0f*u + 1.0f)*0.5f;
27         m_tangentWeights[3][i] = ( 3.0f*u_2 -  2.0f*u + 0.0f)*0.5f;
28     }
29 }

切圖：

相關軟體的下載地址為：http://files.cnblogs.com/WhyEngine/TestSpline.zip