The De Casteljau algorithm on Lie groups and spheres.

*(English)*Zbl 0961.53027In the paper under review, the authors examine the De Casteljau algorithm in the context of Riemannian symmetric spaces.

The problem of synthesizing a smooth motion of a rigid body or groups of rigid bodies, that interpolates a set of configurations in space, has importance in engineering applications. In this context, a concept of dynamic interpolation, in which the usual interpolation notion is generalized to include the case where the interpolating curves are generated by dynamical systems, is developed (see, for instance, the first author and J. W. Jackson [Prog. Syst. Control Theory 8, 156-166 (1991; Zbl 0791.41003)]). Such problems turn out to be more computationally complex than the same problem in Euclidean polynomial interpolation, where computationally efficient algorithms are available to compute the interpolating polynomials. One of these algorithms is the De Casteljau algorithm. In essence, this algorithm is a geometric construction whereby two points in \({\mathbb R}^m\) are joined by a polynomial via an iterative linear interpolation process. This successive linear interpolation does indeed yield a polynomial. Since it is geometrically based, the De Casteljau algorithm can be generalized from \({\mathbb R}^m\) to other spaces, as long as the linear interpolation process is suitably redefined.

The goal of this paper is to develop details of the De Casteljau algorithm in the special cases of connected and compact Lie groups and spheres. That one could generalize the concept of the De Casteljau algorithm to arbitrary Riemannian manifolds was first pointed out by F. Park and B. Ravani [ASME J. Mechan. Design 117, 36-40 (1995)], where usual straight line segments are replaced by geodesic segments.

The first objective of the paper under review is to work out a general expression for the first two derivatives of generalized polynomial curves defined by the generalized De Casteljau algorithm, for the special case of compact Lie groups. Thus, this result holds specifically for all of the orthogonal groups \(SO(m)\), \(m\geq 3\), and, in particular, the rotation group \(SO(3)\) in \({\mathbb R}^3\). The second objective is to work out the details of the generalized De Casteljau algorithm on \(m\)-dimensional spheres \(S^m,\quad m\geq 2\). The Lie group \(SO(m+1)\) acts transitively on \(S^m\), and geodesics on \(S^m\) correspond to certain geodesics on \(SO(m+1)\), which are one-parameter groups in \(SO(m+1)\). Thus the algorithm for \(S^m\) can be based upon the somewhat simpler algorithm for \(SO(m+1)\).

The problem of synthesizing a smooth motion of a rigid body or groups of rigid bodies, that interpolates a set of configurations in space, has importance in engineering applications. In this context, a concept of dynamic interpolation, in which the usual interpolation notion is generalized to include the case where the interpolating curves are generated by dynamical systems, is developed (see, for instance, the first author and J. W. Jackson [Prog. Syst. Control Theory 8, 156-166 (1991; Zbl 0791.41003)]). Such problems turn out to be more computationally complex than the same problem in Euclidean polynomial interpolation, where computationally efficient algorithms are available to compute the interpolating polynomials. One of these algorithms is the De Casteljau algorithm. In essence, this algorithm is a geometric construction whereby two points in \({\mathbb R}^m\) are joined by a polynomial via an iterative linear interpolation process. This successive linear interpolation does indeed yield a polynomial. Since it is geometrically based, the De Casteljau algorithm can be generalized from \({\mathbb R}^m\) to other spaces, as long as the linear interpolation process is suitably redefined.

The goal of this paper is to develop details of the De Casteljau algorithm in the special cases of connected and compact Lie groups and spheres. That one could generalize the concept of the De Casteljau algorithm to arbitrary Riemannian manifolds was first pointed out by F. Park and B. Ravani [ASME J. Mechan. Design 117, 36-40 (1995)], where usual straight line segments are replaced by geodesic segments.

The first objective of the paper under review is to work out a general expression for the first two derivatives of generalized polynomial curves defined by the generalized De Casteljau algorithm, for the special case of compact Lie groups. Thus, this result holds specifically for all of the orthogonal groups \(SO(m)\), \(m\geq 3\), and, in particular, the rotation group \(SO(3)\) in \({\mathbb R}^3\). The second objective is to work out the details of the generalized De Casteljau algorithm on \(m\)-dimensional spheres \(S^m,\quad m\geq 2\). The Lie group \(SO(m+1)\) acts transitively on \(S^m\), and geodesics on \(S^m\) correspond to certain geodesics on \(SO(m+1)\), which are one-parameter groups in \(SO(m+1)\). Thus the algorithm for \(S^m\) can be based upon the somewhat simpler algorithm for \(SO(m+1)\).

Reviewer: Adela-Gabriela Oiaga (Bucureşti)

##### MSC:

53C35 | Differential geometry of symmetric spaces |

65D17 | Computer-aided design (modeling of curves and surfaces) |

70E15 | Free motion of a rigid body |

53C22 | Geodesics in global differential geometry |

70-08 | Computational methods for problems pertaining to mechanics of particles and systems |

70Q05 | Control of mechanical systems |