Quadratic Trigonometric Spline Curves with Multiple Shape Parameters

Abstract

Quadratic trigonometric spline curves with multiple shape parameters are presented in this paper. Analogous to the cubic B-spline curves, each trigonometric spline curve segment is generated by four consecutive control points. The trigonometric spline curves with a non–uniform knot vector are C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> continuous. With a uniform knot vector, the trigonometric spline curves are C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> continuous when all shape parameter λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> =1. Taking different values of the shape parameters, one can globally or locally adjust the shapes of the curves, so that the trigonometric spline curves can be close to the cubic B-spline curves or closer to the given control polygon than the cubic B-spline curves. The trigonometric spline curves also can represent ellipse and generate a family of ellipse with the same control points. A quadratic trigonometric Bézier curves are also introduced as a special case of the given trigonometric spline curves.