Determination of planar Bézier curves under linear pseudo-similarity transformations

Let be the 2-dimensional pseudo-Euclidean space of index 1. The group of all linear pseudo-similarities of and the group of all orientation-preserving linear pseudo-similarities of are introduced and its denoted by LPSim(E_1^2 ) and 〖LPSim〗^+ (E_1^2 ), respectively. Let G=〖LPSim〗^ (E_1^2 ) or G= 〖LPSim〗^+ (E_1^2 ). G-invariants of m-tuple points in and the conditions of G-similarity of two systems of m-tuple points are given. L-pseudo regular Bézier curve is defined. The solutions of problems of G-similarity for two Bézier curves of degree m≥1 and two L-pseudo-regular Bézier curves of degree m≥1 are found in terms of G-invariants of m-tuple points in . For two Bézier curves in with the common G-invariants, evident forms of all linear pseudo-similarity tansformations, carrying one to other, are obtained.