A Laguerre-Type Action for the Solution of Geometric Constraint Problems

A well-known idea is to identify spheres, points, and hyperplanes in Euclidean space Rn with points in real projective space. To address geometric constraints such as intersections, tangencies, and angle requirements, it is important to also encode the orientations of hyperplanes and spheres. The natural space for encoding such geometric objects is the real projective quadric with signature (n+1,2). In this article, we first provide a general formula for calculating the angles formed by the geometric objects encoded by the points of the quadric. The main result is the determination of a very simple parametrization of a Laguerre-type subgroup that acts transitively on the quadric while preserving the geometric nature of its points. That is, points of the quadric representing oriented spheres, points, and oriented hyperplanes in Rn are mapped by the action to points of the same geometric type. We also provide simple parametrizations of the isotropies of the action. The action described in this article provides the foundation for an effective solution to geometric constraint problems.