Curvature continuous curves and surfaces
Computer Aided Geometric Design
Improperly parametrized rational curves
Computer Aided Geometric Design
Geometric and solid modeling: an introduction
Geometric and solid modeling: an introduction
A geometric characterization of parametric cubic curves
ACM Transactions on Graphics (TOG)
Detecting Cusps and Inflection Points in Curves
Detecting Cusps and Inflection Points in Curves
Geometric continuity: a parametrization independent measure of continuity for computer aided geometric design (curves, surfaces, splines)
Algorithms for intersecting parametric and algebraic curves I: simple intersections
ACM Transactions on Graphics (TOG)
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We present an algorithm for determining whether a given rational parametric curve, defined as vector valued function over a finite domain, has a regular parametrization. A curve has a regular parametrization if it has no cusps in its defining interval. It has been known that the vanishing of the derivative vector is a necessary condition for the existence of cusps. We show that if a curve is properly parametrized, then the vanishing of the derivative vector is a necessary and sufficient condition for the existence of cusps. If a curve has no cusps in its defining interval, its proper parametrization is a regular parametrization. We present a simple algorithm to compute the proper parametrization of a polynomial parametric curve which is used to analyze for cusps and later on reduce the problem of detecting cusps in a rational curve to that of a polynomial curve.