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

• Main Author: Muhsin Incesu
• Format: Article
• Language: English
• Published: AIMS Press 2020-01-01
• Series: AIMS Mathematics
• Subjects: linear similarity; ls(3)-equivalence; points systems; generator invariants; bézier curves; bézier surfaces; font design
• Online Access: https://www.aimspress.com/article/10.3934/math.2020084/fulltext.html

Let <em>G</em> be a transformation group and act on <em>X</em>. Any elements <em>x, y</em> ∈ <em>X</em> are called the <em>G</em>-equivalent elements if there exist a transformation <em>g</em> ∈ <em>G</em> such that <em>y</em> = <em>gx</em> is satisfied. Similarly let <em>A</em> = {<em>x</em>₁, <em>x</em>₂, …, <em>x</em>_<em>n</em>} and <em>B</em> = {<em>y</em>₁, <em>y</em>₂, …, <em>y</em>_<em>n</em>} be any two subspaces of <em>X</em> with <em>n</em>-elements. Then the subspaces <em>A</em> and <em>B</em> are called the <em>G</em>-equivalent subspaces if there exist a transformation <em>g</em> ∈ <em>G</em> such that <em>y</em>_<em>i</em> = <em>gx</em>_<em>i</em> is satisfied for every <em>i</em> = 1, 2, …, <em>n</em>.<br /> The linear similarity transformations’ group in 3 dimensional Euclidean space will be denoted by <em>LS</em> (3). This paper presents the <em>G</em>-equivalence conditions of the subspaces <em>A</em> and <em>B</em> of 3-dimensional Euclidean space <em>E</em>³ with <em>m</em>-elements where the transformation group <em>G</em> = <em>LS</em> (3) is the linear similarity transformation group in <em>E</em>³. Later the <em>G</em> = <em>LS</em> (3)-equivalence conditions of Bézier curves and surfaces are studied in terms of the rational <em>G</em> = <em>LS</em> (3) invariants of their control points. Finally by using quadratic Bézier 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 <em>G</em> = <em>LS</em> (3)-equivalent to designed letter “S”.