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)
Animating rotation with quaternion curves
SIGGRAPH '85 Proceedings of the 12th annual conference on Computer graphics and interactive techniques
Curves and surfaces for CAGD: a practical guide
Curves and surfaces for CAGD: a practical guide
NURBS: From Projective Geometry to Practical Use
NURBS: From Projective Geometry to Practical Use
The Mathematical Basis of the UNISURF CAD System
The Mathematical Basis of the UNISURF CAD System
The De Casteljau Algorithm on Lie Groups and Spheres
Journal of Dynamical and Control Systems
Energy-minimizing splines in manifolds
ACM SIGGRAPH 2004 Papers
Convergence and C1 analysis of subdivision schemes on manifolds by proximity
Computer Aided Geometric Design - Special issue: Geometric modelling and differential geometry
A variational approach to spline curves on surfaces
Computer Aided Geometric Design - Special issue: Geometric modelling and differential geometry
Computer Aided Geometric Design
A Theoretical Development for the Computer Generation and Display of Piecewise Polynomial Surfaces
IEEE Transactions on Pattern Analysis and Machine Intelligence
Elastic Morphing of 2D and 3D Objects on a Shape Manifold
ICIAR '09 Proceedings of the 6th International Conference on Image Analysis and Recognition
Fitting smoothing splines to time-indexed, noisy points on nonlinear manifolds
Image and Vision Computing
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In a connected Riemannian manifold, generalised Bezier curves are C^~ 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 C^2 splines, and thereby allow C^2 interpolation of a sequence of data points. For the case of uniform splines in symmetric spaces, we show that C^2 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.