Curves and surfaces for computer aided geometric design
Curves and surfaces for computer aided geometric design
Spherical averages and applications to spherical splines and interpolation
ACM Transactions on Graphics (TOG)
The De Casteljau Algorithm on Lie Groups and Spheres
Journal of Dynamical and Control Systems
A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices
SIAM Journal on Matrix Analysis and Applications
On the Geometry of Rolling and Interpolation Curves on Sn, SOn, and Grassmann Manifolds
Journal of Dynamical and Control Systems
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In this paper, we present a generalization of the classical least squares problem on Euclidean spaces, introduced by Lagrange, to more general Riemannian manifolds. Using the variational definition of Riemannian polynomials, we formulate a higher-order variational problem on a manifold equipped with a Riemannian metric, which depends on a smoothing parameter and gives rise to what we call smoothing geometric splines. These are curves with a certain degree of smoothness that best fit a given set of points at given instants of time and reduce to Riemannian polynomials when restricted to each subinterval.We show that the Riemannian mean of the given points is achieved as a limiting process of the above. Also, when the Riemannian manifold is an Euclidean space, our approach generates, in the limit, the unique polynomial curve which is the solution of the classical least squares problem. These results support our belief that the approach presented in this paper is the natural generalization of the classical least squares problem to Riemannian manifolds.