LS (3)-equivalence conditions of control points and application to spatial Bezier curves and surfaces

Abstract

Let G be a transformation group and act on X. Any elements x, y epsilon X are called the G-equivalent elements if there exist a transformation g epsilon G such that y = gx is satisfied. Similarly let A = {x(1), x(2), . . . , x(n)} and B = {y(1), y(2), . . . , y(n)} be any two subspaces of X with n-elements. Then the subspaces A and B are called the G-equivalent subspaces if there exist a transformation g epsilon G such that y(i) = gx(i) is satisfied for every i = 1, 2, . . . , n. The linear similarity transformations' group in 3 dimensional Euclidean space will be denoted by LS (3). This paper presents the G-equivalence conditions of the subspaces A and B of 3-dimensional Euclidean space E-3 with m-elements where the transformation group G = LS (3) is the linear similarity transformation group in E-3. Later the G = LS (3)-equivalence conditions of Bezier curves and surfaces are studied in terms of the rational G = LS (3) invariants of their control points. Finally by using quadratic Bezier curves, a simple letter "S" is designed and two di fferent shadow curves of this letter (composite curves) are obtained. Then it is emphasized that these shadow curves are G = LS (3)-equivalent to designed letter "S".