Abstract
Riemannian quadratics are C 1 curves on Riemannian manifolds, obtained by performing the quadratic recursive deCastlejeau algorithm in a Riemannian setting. They are of interest for interpolation problems in Riemannian manifolds, such as trajectory-planning for rigid body motion. Some interpolation properties of Riemannian quadratics are analysed when the ambient manifold is a sphere or projective space, with the usual Riemannian metrics.
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A.H. Barr, B. Currin, S. Gabriel and J.F. Hughes, Smooth interpolation of orientations with angular velocity constraints using quaternions, Computer Graphics 26(2) (1992) 313-320.
A.S. Cavaretta, W. Dahmen and C.A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93 (453) 1991.
G. de Rham, Un peu mathématiques á propos d'une courbe plane, Revue de Mathématiques Élémentaires II (1947).
G. de Rham, Sur certaines équations fonctionelles, in: Ouvrage Publié á l'Occasion de son Centenaire (l'École Polytechnique de l'Université de Lausanne, 1953) pp. 95-97.
G. de Rham, Sur une courbe plane, J. Math. Pures Appl. 35 (1956) 25-42.
G. de Rham, Sur les courbes limites des polygones obtenus par trisection, L'Enseignement Mathématique 5 (1959) 29-43.
G. de Rham, Sur quelques fonctions différentiables dont toutes les valeurs sont des valeurs critiques, in: Celebrazioni Archimedee del Secolo XX, Siracusa, II, 11-16 April 1961, pp. 61-65.
T. Duff, Quaternion splines for animating orientations, in: Second Computer Graphics Workshop, Monterey, CA, USA, 12-13 December 1985 (Usenix Association) pp. 54-62.
S.A. Gabriel and J.T. Kajiya, Spline interpolation in curved manifolds (1985) (unpublished).
R.N. Goldman, Recursive triangles, in: Computation of Curves and Surfaces, eds. W. Dahmen, M. Gasca and C.A. Micchelli, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, Vol. 307 (Kluwer Academic, Dordrecht, 1989) pp. 27-72.
H.B. Keller, Numerical Methods for Two-Point Boundary-Value Problems (Blaisdell, New York, 1968).
J. Milnor, Morse Theory, Annals of Mathematics Studies, Vol. 51 (Princeton Univ. Press, Princeton, 1963).
L. Noakes, Riemannian quadratics, in: Curves and Surfaces with Applications in CAGD, Vol. 1, eds. A. Le Méhauté, C. Rabut and L.L. Schumaker (Vanderbilt Univ. Press, Nashville/London, 1997) pp. 319-328.
L. Noakes, Nonlinear corner-cutting, Adv. Comput. Math. 8 (1998) 165-177.
L. Noakes, A global algorithm for geodesics, J. Australian Math. Soc. Series A 64 (1998) 37-50.
L. Noakes, Accelerations of Riemannian quadratics, Proc. Amer. Math. Soc. 127 (1999) 1827-1836.
L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved spaces, IMAJ. Math. Control Inform. 6 (1989) 465-473.
K. Shoemake, Animating rotation with quaternion curves, SIGGRAPH 19(3) (1985) 245-254.
G.P. Paternain, Geodesic Flows (Birkhäuser, Basel, 1999).
N. Steenrod, The Topology of Fibre Bundles (Princeton Univ. Press, Princeton, 1951).
H. von Koch, Sur une courbe continue sans tangente obtenue par une construction géometrique élémentaire, Arkiv. Mat. Astronomik Fysik 1 (1904) 681-702.
J.H.C. Whitehead, Convex regions in the geometry of paths, Quart. J. Math. Oxford 3 (1932) 33-42.
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Noakes, L. Quadratic Interpolation on Spheres. Advances in Computational Mathematics 17, 385–395 (2002). https://doi.org/10.1023/A:1016277023669
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DOI: https://doi.org/10.1023/A:1016277023669