Straightening Euclidean invariants

Abstract

The Gröbner basis method is a powerful tool in automated geometry theorem proving. Normally, one works in the ring of coordinates of the points in a particular configuration. Tim Havel has suggested using instead the ring of interpoint squared distances because it is the invariant subring under the group of Euclidean isometries. One difficulty with this approach is that it is not always clear how to express some invariants in terms of squared distances. To that end, we present a new straightening algorithm for Euclidean invariants. We will also prove the first and second fundamental theorems of vector invariants for the group of Euclidean isometries (that the invariant subring is a finitely generated algebra over the reals, and that it can be expressed as a polynomial ring modulo finitely generated ideal, respectively. Another difficulty is that the ring of interpoint squared distances must be represented as the quotient of a polynomial ring by an ideal. Unfortunately, no canonical Gröbner basis for this ideal is known. We will present a candidate for such a basis and prove that it is a basis in some cases.