三次Bezier曲线与Catmull-Rom样条曲线的转换
Conversion Between Cubic Bezier Curves and Catmull-Rom Splines
摘要：
样条曲线是表示复杂形状的主要方法之一，已成为计算机图形学（CG）和计算机辅助几何设计（CAGD）领域中复杂曲面造型的主要技术。其中，Bézier和Catmull-Rom样条是工程子领域中最常见的。在本文中，我们专注于三次Bézier和Catmull-Rom曲线段之间的转换，而不是通过他们的性质。通过推导转换方程，我们的目标是将Catmull-Rom或Bézier三次曲线的原始控制点集转换为一个新的控制点集，当将其视为另一曲线的控制点集时，该控制点集与原始曲线的形状大致相同。由于提供了简单的控制点的线性变换，该方法是非常简单，高效，易于实现，这是本文进一步验证了一些数值和可视化的例子。
英文摘要：
Splines are one of the main methods of mathematically representing complicated shapes, which have become the primary technique in the fields of Computer Graphics (CG) and Computer-Aided Geometric Design (CAGD) for modeling complex surfaces. Among all, Bézier and Catmull-Rom splines are the most common in the sub-fields of engineering. In this paper, we focus on conversion between cubic Bézier and Catmull-Rom curve segments, rather than going through their properties. By deriving the conversion equations, we aim at converting the original set of the control points of either of the Catmull-Rom or Bézier cubic curves to a new set of control points, which corresponds to approximately the same shape as the original curve, when considered as the set of the control points of the other curve. Due to providing simple linear transformations of control points, the method is very simple, efficient, and easy to implement, which is further validated in this paper using some numerical and visual examples.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
