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Metamorphosis of Planar Parametric Curves via Curvature Interpolation

Metamorphosis of Planar Parametric Curves via Curvature Interpolation. Tatiana Surazhsky , Gershon Elber International Journal of Shape Modeling 8(2) : (2002). Author: Tatiana Surazhsky PostDoc Researcher ， Department of Computer S cience Technion, Israel.

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Metamorphosis of Planar Parametric Curves via Curvature Interpolation

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  1. Metamorphosis of Planar Parametric Curves via Curvature Interpolation Tatiana Surazhsky , Gershon Elber International Journal of Shape Modeling 8(2) : (2002)

  2. Author: • Tatiana Surazhsky PostDoc Researcher，Department of Computer Science Technion, Israel. Institute of Technology, Haifa, Isreal. Research interests: Geometric modeling, computer graphics. • Gershon Elber Department of Computer Science Technion, Israel Institute of Technology, Haifa, Isreal. Research interests:Computer Graphics, Computer Aided Geometric Design.

  3. 问题的提出： Morphing：给定源曲线 ，目标曲线 ，求作中间曲线 ，使之从源曲线光滑过渡到目标曲线。

  4. 主要考虑两个问题: (1)：源曲线和目标曲线的顶点对应问题 (2)：源曲线和目标曲线的对应顶点插值问题 • 本文主要介绍问题(2)，即假定顶点对应已经给定

  5. Related work 1. 对曲线上点或者控制顶点的线性插值，可能产生曲线收缩，自交问题.(fig.1,fig.2) 2. Sederberg和王国瑾老师等人利用多边形内在量(多边形的角度和边长)，对这些内在量线性插值，对于多边形不封闭现象，利用Lagrange乘子法对插值后多边形的边长进行伸缩，达到封闭多边形的目的。但是可能产生自交现象 (fig.3) T. W. Sederberg, P. Gao, G. Wang, and H. Mu, 2D shape blending: an antrinsic solution to the vertex path problem, Computer Graphics SIGGRAPH'93, 27 (1993), pp. 15-18. 3. 杨文武等人提出了基于特征分解的2-D多边形渐变，将源多边形和目标多边形按 照视觉特征进行同构特征分解,得到若干对对应的特征子多边形.在渐变过程中,每个源特征子多边形光滑地过渡到目标特征子多边形(fig.4) 软件学报，杨文武, 冯结青, 金小刚, 彭群生

  6. 1.Curvature interpolation: 1.1 重新参数化 给定平面曲线 ，其曲率表示 为 . 下面引入重新参数化，使得曲线参数变为弧长参数： 设曲线 的总长度为 ，令 ，即 ，原曲线转化为 。其曲率表示为 。 这儿因为对于弧长参数 。

  7. 1.2 Curvature interpolation 有微分几何基本理论知： 在刚体运动下，曲率唯一的表示了一条曲线，即两条曲线如果具有相同的曲 率，则这两条曲线经过刚体运动(旋转，平移)可以完全重合。 下面我们用线性插值的曲率重建曲线： 这儿 分别为曲线 的弧长参数, 表示在时刻t 中间曲线 的弧长， 为中间过渡曲线的弧长参数

  8. 2. Reconstruction from Curvature 下面用生成的 重建曲线 , 令 即： 令： ，得到

  9. 考虑到 为一数量场，而所求的曲线 为向量函数，由于 与 的特殊关系，有 记 得： 这样就有曲率 重建了曲线

  10. 2.1 Restruction for

  11. 2.2 Restruction for 明显，对于一个简单的曲率场 ，上述积分没有分析解

  12. 2.3 Computational Aspect of General Curvature Signature 一般来说，对于 ，积分方程没有分析解 我们对 利用多项式函数逼近，设为 得到 一般来说 不可积，我们再次对 作多项式逼近， 然后对逼近的多项式积分，由逼近论可知，利用多项式可以得到原始曲线的任意 误差的逼近效果。

  13. 3. Metamorphosis Using Curvature Interpolation 3.1 piecewise interpolation Given curves and ,choosing parameters , Subdivided curves into N+1 segments. We use the curvature interpolation for each pair of segments, and Then reconstruction the whole intermediate curve from its segments Using rotation and transformation to achieving continuity.

  14. 3.2 Metamorphosis of Closed Curves For closed curves ,, is always unclosed. We have to adjust points on the curve to ensure that the intermediate curve is closed. Here we distribute the error evenly by To ensure that the intermediate curve is continuity, we let We have and

  15. Examples

  16. That’s all，Thanks!

  17. withered

  18. Self-intersection

  19. The source and destination polygons are decomposed into ten pairs of corresponding sub-polygons by choosing ten corresponding feature-decomposition vertices interactively

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