Singular perturbations and interpolation—a problem in robotics
Nonlinear Analysis: Theory, Methods & Applications
Smooth interpolation of orientations with angular velocity constraints using quaternions
SIGGRAPH '92 Proceedings of the 19th annual conference on Computer graphics and interactive techniques
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
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
Spherical averages and applications to spherical splines and interpolation
ACM Transactions on Graphics (TOG)
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
On parametric smoothness of generalised B-spline curves
Computer Aided Geometric Design
Bézier curves and C2 interpolation in Riemannian manifolds
Journal of Approximation Theory
Recursive computation of Hermite spherical spline interpolants
Journal of Computational and Applied Mathematics
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The classical de Casteljau algorithm for constructing Bézier curves can be generalised to a sphere of arbitrary dimension by replacing line segments with shortest great circle arcs. The resulting spherical Bézier curves are C∞ and interpolate the endpoints of their control polygons. In the present paper, we address the problem of piecing these curves together into C2 splines. For this purpose, we compute the endpoint velocities and accelerations of a spherical Bézier curve of arbitrary degree and use the formulae to define control points that give the curve a desired initial velocity and acceleration. In addition, for uniform splines we establish a simple relationship between the control points of neighbouring curve segments that is necessary and sufficient for C2 continuity. As illustration, we solve an interpolation problem involving sparse data using both the present method and a normalised polynomial interpolant. The normalised spline exhibits large variations in speed and magnitude of acceleration, whilst the spherical Bézier spline is far better behaved. These considerations are important in applications where velocities and accelerations need to moderated or estimated, notably computer animation and rigid body trajectory planning, where interpolation in the 3-sphere is a fundamental task.