Bezier curves and C2 interpolation in Riemannian manifolds

by: Tomasz Popiel, Lyle Noakes

Journal of Approximation Theory, Vol. 148, No. 2. (October 2007), pp. 111-127.

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Abstract

In a connected Riemannian manifold, generalised Bezier curves are C[infinity] curves defined by a generalisation, in which line segments are replaced by minimal geodesics, of the classical de Casteljau algorithm. As in Euclidean space, these curves join their first and last control points. We compute the endpoint velocities and (covariant) accelerations of a generalised Bezier curve of arbitrary degree and use the formulae to express the curve's control points in terms of these quantities. These results allow generalised Bezier curves to be pieced together into C2 splines, and thereby allow C2 interpolation of a sequence of data points. For the case of uniform splines in symmetric spaces, we show that C2 continuity is equivalent to a simple relationship, involving the global symmetries at knot points, between the control points of neighbouring curve segments. We also present some examples in hyperbolic 2-space.

@article{citeulike:2637725, abstract = {In a connected Riemannian manifold, generalised Bezier curves are C[infinity] curves defined by a generalisation, in which line segments are replaced by minimal geodesics, of the classical de Casteljau algorithm. As in Euclidean space, these curves join their first and last control points. We compute the endpoint velocities and (covariant) accelerations of a generalised Bezier curve of arbitrary degree and use the formulae to express the curve's control points in terms of these quantities. These results allow generalised Bezier curves to be pieced together into C2 splines, and thereby allow C2 interpolation of a sequence of data points. For the case of uniform splines in symmetric spaces, we show that C2 continuity is equivalent to a simple relationship, involving the global symmetries at knot points, between the control points of neighbouring curve segments. We also present some examples in hyperbolic 2-space.}, author = {Popiel, Tomasz and Noakes, Lyle }, citeulike-article-id = {2637725}, doi = {10.1016/j.jat.2007.03.002}, journal = {Journal of Approximation Theory}, keywords = {curve, manifold}, month = {October}, number = {2}, pages = {111--127}, posted-at = {2008-04-07 14:22:39}, priority = {2}, title = {Bezier curves and C2 interpolation in Riemannian manifolds}, url = {http://dx.doi.org/10.1016/j.jat.2007.03.002}, volume = {148}, year = {2007} }

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TY - JOUR ID - citeulike:2637725 L3 - citeulike-article-id:2637725 TI - Bezier curves and C2 interpolation in Riemannian manifolds JF - Journal of Approximation Theory VL - 148 IS - 2 SP - 111 EP - 127 N2 - In a connected Riemannian manifold, generalised Bezier curves are C[infinity] curves defined by a generalisation, in which line segments are replaced by minimal geodesics, of the classical de Casteljau algorithm. As in Euclidean space, these curves join their first and last control points. We compute the endpoint velocities and (covariant) accelerations of a generalised Bezier curve of arbitrary degree and use the formulae to express the curve's control points in terms of these quantities. These results allow generalised Bezier curves to be pieced together into C2 splines, and thereby allow C2 interpolation of a sequence of data points. For the case of uniform splines in symmetric spaces, we show that C2 continuity is equivalent to a simple relationship, involving the global symmetries at knot points, between the control points of neighbouring curve segments. We also present some examples in hyperbolic 2-space. KW - curve KW - manifold AU - Popiel, Tomasz AU - Noakes, Lyle PY - 2007/10// UR - http://dx.doi.org/10.1016/j.jat.2007.03.002 ER -

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