Gamma四元数样条曲线相关问题研究

Research on the Correlation Problems of Gamma Quaternion Spline Curves

摘要: 随着计算机技术的发展，计算机图形学的重要性逐渐被体现出来，样条曲线也随之迅速发展起来。其研究领域也涉及诸多内容，主要有各阶次样条曲线的不同扩展，光顺的新方法研究以及造型与形状调整。本文将四元数方法与样条理论相结合，以Bézier四元数曲线和Boehm的Gamma样条曲线理论为基础，构建了Gamma四元数样条曲线并对其性质进行研究。首先，将欧氏空间中的Gamma样条曲线推广到四元数空间上，定义了球面Gamma四元数样条曲线。然后，对其光滑性进行了研究，证明了该样条曲线是C¹连续的，并且在满足一定条件时，该曲线就是C²连续的。最后，对上述方法进行了应用。该构造方法引入了可以改变样条曲线形状的参数，从而使得构造的样条曲线具有很强的灵活性。

Abstract: With the development of computer technology, the importance of computer graphics is gradually reflected, and the spline curve also develops rapidly. Its research field also involves a lot of contents, including different extension of spline curves of various orders, new methods of smoothing and modeling and shape adjustment. Based on Bézier quaternion curve and Boehm’s Gamma spline curve theory, this paper combines quaternion method with spline theory, constructs Gamma qua-ternion spline curve and studies its properties. Firstly, the Gamma quaternion spline curve in Eu-clidean space is extended to quaternion space, and the spherical Gamma quaternion spline curve is defined. Then, it is proved that the spline curve is C¹ continuous, and it is C² continuous when cer-tain conditions are satisfied. Finally, the above method is applied. The parameters that can change the shape of the spline curve are introduced in this construction method so that the spline curve can be constructed with strong flexibility.