An algebraic approach to curves and surfaces on the sphere and on other quadrics
Selected papers of the international symposium on Free-form curves and free-form surfaces
Fundamentals of computer aided geometric design
Fundamentals of computer aided geometric design
A general construction scheme for unit quaternion curves with simple high order derivatives
SIGGRAPH '95 Proceedings of the 22nd annual conference on Computer graphics and interactive techniques
Smooth invariant interpolation of rotations
ACM Transactions on Graphics (TOG)
Fast construction of accurate quaternion splines
Proceedings of the 24th annual conference on Computer graphics and interactive techniques
Animating rotation with quaternion curves
SIGGRAPH '85 Proceedings of the 12th annual conference on Computer graphics and interactive techniques
Images as Embedded Maps and Minimal Surfaces: Movies, Color, Texture, and Volumetric Medical Images
International Journal of Computer Vision - Special issue on computer vision research at the Technion
Geometric partial differential equations and image analysis
Geometric partial differential equations and image analysis
Fast computation of weighted distance functions and geodesics on implicit hyper-surfaces: 730
Journal of Computational Physics
Computational Line Geometry
Motion of curves constrained on surfaces using a level-set approach
Journal of Computational Physics
Geometric design of motions constrained by a contacting surface pair
Computer Aided Geometric Design
Energy-minimizing splines in manifolds
ACM SIGGRAPH 2004 Papers
Existence of set-interpolating and energy-minimizing curves
Computer Aided Geometric Design
Convergence and C1 analysis of subdivision schemes on manifolds by proximity
Computer Aided Geometric Design - Special issue: Geometric modelling and differential geometry
Computer Aided Geometric Design
Constructing spherical curves by interpolation
Advances in Engineering Software
Bézier curves and C2 interpolation in Riemannian manifolds
Journal of Approximation Theory
Subdivision Curves on Surfaces and Applications
CIARP '08 Proceedings of the 13th Iberoamerican congress on Pattern Recognition: Progress in Pattern Recognition, Image Analysis and Applications
Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion
Computer Aided Geometric Design
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Given an m-dimensional surface Φ in Rn, we characterize parametric curves in Φ which interpolate or approximate a sequence of given points pi ∈ Φ and minimize a given energy functional. As energy functionals we study familiar functionals from spline theory, which are linear combinations of L2 norms of certain derivatives. The characterization of the solution curves is similar to the well-known unrestricted case. The counterparts to cubic splines on a given surface, defined as interpolating curves minimizing the L2 norm of the second derivative, are C2; their segments possess fourth derivative vectors, which are orthogonal to Φ; at an end point, the second derivative is orthogonal to Φ. Analogously, we characterize counterparts to splines in tension, quintic C4 splines and smoothing splines. On very special surfaces, some spline segments can be determined explicitly. In general, the computation has to be based on numerical optimization. Applications of splines on surfaces go beyond geometric modeling, and arise for example also in motion planning and image processing.