Geometric constraints on quadratic Bezier curves using minimal length and energy

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:255

Article

Ahn, Young Joon¹  Hoffmann, Christoph²  Rosen, Paul³ 

[1] Chosun Univ, Dept Math Educ, Kwangju 501759, South Korea

[2] Purdue Univ, Dept Comp Sci, W Lafayette, IN 47907 USA

[3] Univ Utah, Dept Comp Sci, Salt Lake City, UT 84112 USA

关键词: Quadratic Bezier curve;    Geometric constraint solving;    Arc length;    Bending energy;    Optimization;    GPU implementation;   

DOI  :  10.1016/j.cam.2013.07.005

来源: Elsevier

【 摘 要 】

This paper derives expressions for the arc length and the bending energy of quadratic Bezier curves. The formulas are in terms of the control point coordinates. For fixed start and end points of the Bezier curve, the locus of the middle control point is analyzed for curves of fixed arc length or bending energy. In the case of arc length this locus is convex. For bending energy it is not. Given a line or a circle and fixed end points, the locus of the middle control point is determined for those curves that are tangent to a given line or circle. For line tangency, this locus is a parallel line. In the case of the circle, the locus can be classified into one of six major types. In some of these cases, the locus contains circular arcs. These results are then used to implement fast algorithms that construct quadratic Bezier curves tangent to a. given line or circle, with given end points, that minimize bending energy or arc length. (C) 2013 Elsevier B.V. All rights reserved.

【 授权许可】

Free