Jay P. Fillmore

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 Shoemakes 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.

A Note on Rotation Matrices

Properly establishing the relation between linear algebra and geometry makes it easier to obtain the three-by-three orthogonal matrix that describes a specified rotation.

On Backtracking: A Combinatorial Description of the Algorithm

A basic algorithm for solving many discrete problems is the so-called “backtracking” algorithm. The basic problem is that of generating the elements of a subset

A linear algebra setting for the rota-mullin theory of polynomials of binomial type

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Matrix Exponentials---Another Approach

of a finite set in an efficient manner. If a group G acts on

Möbius groups over general fields using Clifford algebras associated with spheres

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The fifteen-parameter conformal group

, then one might wish to obtain only nonisomorphic elements of

A note on split dilations defined by higher residues

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New euclidean theorems by the use of Laguerre transformations — Some geometry of Minkowski (2+1)-space

. In this paper the basic backtracking algorithm is described in terms of chains of partitions on the set S. The corresponding isomorph rejection problem is described in terms of G-invariant chains of partitions on S. Examples and flow charts are given.

Planar sections of the quadric of Lie cycles and their Euclidean interpretations

The linear algebra and combinatorial aspects of the Rota-Mullin theory of polynomials of binomial type are separated and the former is developed in terms of shift operators on infinite dimensional vector spaces with a view towards application in the calculus of finite differences.