Outer product factorization in Clifford algebra

Clifford algebra plays an important role in mathematics and physics, and has various applications in geometric reasoning, computer vision and robotics. When applying Clifford algebra to geometric problems, an important technique is parametric representation of geometric entities, such as planes and spheres in Euclidean and spherical spaces, which occur in the form of homogeneous multivectors. Computing a parametric representation of a geometric entity is equivalent to factoring a homogeneous multivector into an outer product of vectors. Such a factorization is called outer product factorization. When no signature constraint is imposed on the vectors whose outer product equals the homogeneous multivector, a classical result can be found in the book of Hodge and Pedoe (1953), where a sufficient and necessary condition for the factorability is given and called quadratic Plucker relations (p-relations). However, the p-relations are generally algebraically dependent and contain redundancy. In this paper we construct a Ritt-Wu basis of the p-relations, which serves as a much simplified criterion on the factorability. When there are signature constraints on the vectors, we propose an algorithm that can judge whether the constraints are satisfiable, and if so, produce a required factorization.