On invariants of curves in centro-affine geometry

Abstract

Let GL(n, R) be the general linear group of n × n real matrices. Definitions of GL(n, R)-equivalence and the centro-affine type of curves are introduced. All possible centro-affine types are founded. For every centro affine type all invariant parametrizations of a curve are described. The problem of GL(n, R)-equivalence of curves is reduced to that of paths. A generating system of the differential field of invariant differential rational functions of a path is described. They can be viewed as centro-affine curvatures of a path. Global conditions of GL(n, R)equivalence of curves are given in terms of the centro-affine type and the generating differential invariants. Independence of elements of the generating differential invariants is proved.

Feature Matching and Heat Flow in Centro-Affine Geometry

“…Several methods have been developed to construct differential invariants and other invariant quantities in such Klein geometries, [14,34,35,39,45]. In particular, invariants can be straightforwardly and algorithmically obtained by the method of equivariant moving frames introduced in [12].…”

On invariants of null curves in the pseudo-Euclidean geometry

“…(See [16,17].) A J 1 -path x(t) and a J 2 -path y(r) in E n p will be called D-equivalent if a C ∞ -diffeomorphism ϕ : J 2 → J 1 exists such that ϕ (r) > 0 and y(r) = x(ϕ(r)) for all r ∈ J 2 .…”

“…(See [16,17].) Two curves α and β in E n p are called G-equivalent if β = F α for some F ∈ G. This being the case, we write α…”

On Invariants of Immersions of an n-Dimensional Manifold in an n-Dimensional Pseudo-Euclidean Space*

“…We use methods of the invariant theory and the theory of differential equations. A similar approach to the theory of curves was used in the book [12] and papers [13,18].…”