Self-intersections of rational Bézier curves

• Published in: Graphical models Vol. 76; no. 5; pp. 312 - 320
• Main Authors: Zhu, Chun-Gang, Zhao, Xuan-Yi
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.09.2014; Elsevier
• Subjects: Applied sciences; Computer aided design; Computer science; control theory; systems; Control polygon; Curve fitting; Curves (geometry); Equivalence; Exact sciences and technology; Injectivity; Mathematical analysis; Mathematical models; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Polygons; Rational Bézier curve; Sciences and techniques of general use; Self-intersection; Software; Control polygon; Injectivity; Self-intersection; Rational Bézier curve; Intersection; Geometrical model; Bézier curve; Rational interpolation; Polynomial interpolation; Degeneration; Polygon; Curve fitting; Computer aided design
• ISSN: 1524-0703; 1524-0711
• DOI: 10.1016/j.gmod.2014.04.001

[Display omitted] •We define the well-posedness of control polygon of the rational Bezier curve.•The Bezier curve is injective if and only if its control polygon is well-posed.•We present a geometric method to determine the injectivity of the Bezier curve. Rational Bézier curves provide a curve fitting tool and are widely used in Computer Aided Geometric Design, Computer Aided Design and Geometric Modeling. The injectivity (one-to-one property) of rational Bézier curve as a mapping function is equivalent to the curve without self-intersections. We present a geometric condition on the control polygon which is equivalent to the injectivity of rational Bézier curve with this control polygon for all possible choices of weights. The proof is based on the degree elevation and toric degeneration of rational Bézier curve.