Spherical averages and applications to spherical splines and interpolation

This article introduces a method for computing weighted averages on spheres based on least squares minimization that respects spherical distance. We prove existence and uniqueness properties of the weighted averages, and give fast iterative algorithms with linear and quadratic convergence rates. Our methods are appropriate to problems involving averages of spherical data in meteorological, geophysical, and astronomical applications. One simple application is a method for smooth averaging of quaternions, which generalizes Shoemake's spherical linear interpolation.The weighted averages methods allow a novel method of defining Bézier and spline curves on spheres, which provides direct generalization of Bézier and B-spline curves to spherical spline curves. We present a fast algorithm for spline interpolation on spheres. Our spherical splines allow the use of arbitrary knot positions; potential applications of spherical splines include smooth quaternion curves for applications in graphics, animation, robotics, and motion planning.

[1]  B. Joe,et al.  Orientation interpolation in quaternion space using spherical biarcs , 1993 .

[2]  Bahram Ravani,et al.  Computer aided geometric design of motion interpolants , 1994 .

[3]  Sung Yong Shin,et al.  A general construction scheme for unit quaternion curves with simple high order derivatives , 1995, SIGGRAPH.

[4]  Robert L. Parker,et al.  Interpolation of unit vectors , 1979 .

[5]  Lawrence Charles Paulson,et al.  Quaternions , 1873, Nature.

[6]  Bert Jüttler,et al.  Visualization of moving objects using dual quaternion curves , 1994, Comput. Graph..

[7]  Numérisation de documents anciens mathématiques,et al.  Mathematical modelling and numerical analysis : Modélisation mathématique et analyse numérique. , 1985 .

[8]  Vijay Kumar,et al.  Planning of smooth motions on SE(3) , 1996, Proceedings of IEEE International Conference on Robotics and Automation.

[9]  John C. Hart,et al.  Visualizing quaternion rotation , 1994, TOGS.

[10]  Reinhard Wolf.  On the average distance property of spheres in Banach spaces , 1994 .

[11]  Roy Thompson,et al.  Statistical comparison of palaeomagnetic directional records from lake sediments , 1984 .

[12]  Gerold Wagner,et al.  On means of distances on the surface of a sphere. II. (Upper bounds) , 1990 .

[13]  P. Jupp,et al.  Fitting Smooth Paths to Spherical Data , 1987 .

[14]  Kenneth S. Roberts,et al.  Smooth interpolation of rotational motions , 1988, Proceedings CVPR '88: The Computer Society Conference on Computer Vision and Pattern Recognition.

[15]  H. Coxeter.  Quaternions and Reflections , 1946 .

[16]  Nicholas I. Fisher,et al.  A Note on Spherical Splines , 1985 .

[17]  Larry L. Schumaker,et al.  Bernstein-Bézier polynomials on spheres and sphere-like surfaces , 1996, Comput. Aided Geom. Des..

[18]  G. S. Watson.  Statistics on Spheres , 1983 .

[19]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[20]  Emanuele Casini,et al.  Centroids, Centers, Medians: What is the Difference? , 1997 .

[21]  Roy Thompson,et al.  A robust least-squares Gondwanan apparent polar wander path and the question of palaeomagnetic assessment of Gondwanan reconstruction , 1982 .

[22]  O. Gross.  4. The Rendezvous Value of a Metric Space , 1964 .

[23]  Frank Chongwoo Park,et al.  Smooth invariant interpolation of rotations , 1997, TOGS.

[24]  J. Milnor.  Topology from the differentiable viewpoint , 1965 .

[25]  Ken Shoemake,et al.  Quaternion calculus and fast animation , 1987 .

[26]  Myung-Soo Kim,et al.  Hermite Interpolation of Solid Orientations with Circular Blending Quaternion Curves , 1996, Comput. Animat. Virtual Worlds.

[27]  Erik B. Dam,et al.  Quaternions, Interpolation and Animation , 2000 .

[28]  D. Sattinger,et al.  Calculus on Manifolds , 1986 .

[29]  Ravi Ramamoorthi,et al.  Fast construction of accurate quaternion splines , 1997, SIGGRAPH.

[30]  Ken Shoemake,et al.  Animating rotation with quaternion curves , 1985, SIGGRAPH.

[31]  James Arvo,et al.  Stratified sampling of spherical triangles , 1995, SIGGRAPH.

[32]  Ravi Ramamoorthi,et al.  Dynamic Splines with Constraints for Animation , 1997 .

[33]  Steve Hollasch,et al.  Advanced animation and rendering techniques , 1994, Comput. Graph..

[34]  John F. Hughes,et al.  Smooth interpolation of orientations with angular velocity constraints using quaternions , 1992, SIGGRAPH.

[35]  Elijah Polak,et al.  Optimization: Algorithms and Consistent Approximations , 1997 .

[36]  M. G. Wagner,et al.  Computer-Aided Design With Spatial Rational B-Spline Motions , 1996 .

[37]  Myung-Soo Kim,et al.  Interpolating solid orientations with circular blending quaternion curves , 1995, Comput. Aided Des..

[38]  Lyle Noakes,et al.  Cubic Splines on Curved Spaces , 1989 .

[39]  J. L. Brown,et al.  Problems with defining barycentric coordinates for the sphere , 1992 .

[40]  J. Munkres,et al.  Calculus on Manifolds , 1965 .

[41]  Kee-Won Nam,et al.  Hermite Interpolation of Solid Orientations with Circular Blending Quaternion Curves , 1996 .