C2 spherical Bézier splines

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Abstract

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.

References (62)

  • C. Belta et al.

    On the computation of rigid body motion

    Electron. J. Computational Kinematics

    (2002)
  • C. Belta et al.

    Euclidean metrics for motion generation on SE(3)

    J. Mech. Engrg. Sci. Part C

    (2002)
  • P. Bézier

    The Mathematical Basis of the UNISURF CAD System

    (1986)
  • S. Buss et al.

    Spherical averages and applications to spherical splines and interpolation

    ACM Trans. Graphics

    (2001)
  • Camarinha, M., 1996. The geometry of cubic polynomials on Riemannian manifolds. Ph.D. Thesis, University of Coimbra,...
  • M. Camarinha et al.

    Splines of class Ck on non-Euclidean spaces

    IMA J. Math. Control Inform.

    (1995)
  • P. Crouch et al.

    The dynamic interpolation problem: on Riemannian manifolds, Lie groups and symmetric spaces

    J. Dynam. Control Systems

    (1995)
  • P. Crouch et al.

    The de Casteljau algorithm on Lie groups and spheres

    J. Dynam. Control Systems

    (1999)
  • de Casteljau, P., 1959. Outillages méthodes de calcul. Technical Report, Andre Citroen Automobiles,...
  • M. do Carmo

    Riemannian Geometry

    (1992)
  • Duff, T., 1986. Splines in animation and modelling. In: ACM SIGGRAPH Course...
  • G. Farin

    NURBS: From Projective Geometry to Practical Use

    (1999)
  • N. Fisher

    Statistical Analysis of Circular Data

    (1993)
  • N. Fisher et al.

    Statistical Analysis of Spherical Data

    (1987)
  • Gabriel, S., Kajiya, J., 1985. Spline interpolation in curved space. In: ACM SIGGRAPH Course...
  • J. Gallier et al.

    Computing exponentials of skew-symmetric matrices and logarithms of orthogonal matrices

    Intern. J. Robotics Automation

    (2003)
  • Q. Ge et al.

    Geometric construction of Bézier motions

    Trans. ASME J. Mech. Design

    (1994)
  • Q. Ge et al.

    Computer aided geometric design of motion interpolants

    Trans. ASME J. Mech. Design

    (1994)
  • R. Giambo et al.

    An analytical theory for Riemannian cubic polynomials

    IMA J. Math. Control Inform.

    (2002)
  • R. Giambo et al.

    Optimal control on Riemannian manifolds by interpolation

    Math. Control Signals Systems

    (2004)
  • M. Hofer et al.

    Energy-minimizing splines in manifolds

    ACM Trans. Graphics

    (2004)
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