一类C^2连续的单位四元数插值样条曲线  被引量:13

A C^2-Continuous Unit Quaternion Interpolatory Spline Curve

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作  者:邢燕[1] 樊文[1] 檀结庆[1] 许任政 Xing Yan;Fan Wen;Tan Jieqing;Xu Renzheng(School of Mathematics, Hefei University of Technology, Hefei 230009)

机构地区:[1]合肥工业大学数学学院,合肥230009

出  处:《计算机辅助设计与图形学学报》2017年第1期45-51,共7页Journal of Computer-Aided Design & Computer Graphics

基  金:国家自然科学基金(61472466;11601115)

摘  要:为了能够快速生成光滑的单位四元数插值样条曲线,实时控制三维物体关键帧动画,提出一种基于四次多项式的单位四元数插值样条曲线,并证明了它的插值性和C^2连续性.首先选择适当的四次多项式调配函数,使其生成的样条曲线能够插值给定的型值点列并达到C^2连续;然后通过累和形式的基函数作指数、代表关键帧方向的单位四元数为底的指数函数的连乘形式把欧氏空间中的样条曲线推广到单位四元数空间S^3中.文中构造的曲线能够自动通过给定的关键帧朝向序列,避免经典的B样条单位四元数曲线从型值点反求控制顶点过程中求解非线性方程组带来的时间开销,提高了计算效率.实验结果表明,该方案可以有效地应用于三维关键帧动画.In order to be able to quickly generate smooth unit quaternion interpolatory spline curves and realtimelycontrol3D object keyframe animations,this paper presents a quartic-polynomial-based unit quaternioninterpolatory spline curve,and proves its interpolatory property and C2continuity.First,proper quarticpolynomial blending functions are chosen to generate a C2-continuous spline curve in Euclidean space whichcan interpolate a given sequence of data points.Then,via the product of several exponential functions wherethe exponents are the cumulative forms of the blending functions,and the bases are constant unit quaternionsrepresenting the given keyframe orientations,the spline curve in Euclidean space is extended to one in unitquaternion space S3.The proposed curve can automatically pass through a given sequence of keyframe orientationsaccurately,and avoid the iterative process in solving the quaternionic nonlinear system of equationswhen obtaining the spline control points from given data points,as used in classic B-spline unit quaternioninterpolatory curves,and improve the computational efficiency.Experimental results demonstratedthe effectiveness of the proposed scheme in the application of3D keyframe animations.

关 键 词:单位四元数 插值样条 型值点 S3空间 

分 类 号:TP391.72[自动化与计算机技术—计算机应用技术]

 

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