New Aspects of Geometrical Calculus With Invariants

In this paper, we show how to treat projective con gurations of points in an invariant way. By this, we mean that we do not want to take any basis, but compute directly with these geometric objects and with coe cients which are invariant by a change of basis. We show, in a rst part, how to de ne two operations \sums" and \intersection" in the exterior algebra ^E of a linear space E. These generalization of the Cayley algebra operators are achieved by extending the coe cients to the ring S(^E) (symmetric algebra of ^E). It allows us to transform two projective conditions on a point in a single one equivalent to the rst. In a second part, we apply these techniques to show how a special irreducible component (called the generic component) of the variety asssociated to a triangular projective con guration of points can be described by rules and characterized by the dimension of its projections. This leads to an algorithm which reduces an element of a \formal" space ^SF to zero if and only if the corresponding geometric property is true on the generic component. It can be used to test properties of non-degenerated con gurations as shown on examples, whatever the dimension of the ambiant space may be.