<i>LS</i> (3)-equivalence conditions of control points and application to spatial Bézier curves and surfaces

Abstract

Let $G$ be a transformation group and act on $X$. Any elements $x, y \in X$ are called the $G-$equivalent elements if there exist a transformation $g \in G$ such that $y = gx$ is satisfied. Similarly let $A = \left\{ x_{1}, x_{2}, ..., x_{n} \right\}$ and $B = \left\{ y_{1}, y_{2}, ..., y_{n} \right\}$ 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 $g \in G$ 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 different 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.