The equivalence problem for vectors in the two-dimensional Minkowski spacetime and its application to Bezier curves

Abstract

Let M(1,1) be the group of all transformations of the 2-dimensional Minkowski spacetime M generated by  all pseudo-orthogonal transformations and parallel translations of M. Let  SM(1,1) is the proper subgroup of M(1,1) and SL(1,1) is the ortochoronous proper subgroup of M(1,1). In this paper, conditions for the equivalence of two systems of vectors {x_{1},x_{2},..., x_{m}} and {y_{1},y_{2},..., y_{m}} are obtained for groups G=M(1,1), SM(1,1), SL(1,1). Finally, we present a necessary and sufficient conditions for judging whether  Bezier curves in M of degree m are G-equivalent.