A geometric characterization of parametric cubic curves
ACM Transactions on Graphics (TOG)
Detecting cusps and inflection points in curves
Computer Aided Geometric Design
The NURBS book
Identification of inflection points and cusps on rational curves
Computer Aided Geometric Design
An efficient and robust algorithm for solving the foot point problem
Proceedings of the international conference on Mathematical methods for curves and surfaces II Lillehammer, 1997
Inflection points and singularities on planar rational cubic curve segments
Computer Aided Geometric Design
Singularities of rational Bézier curves
Computer Aided Geometric Design
Inflection points and singularities on C-curves
Computer Aided Geometric Design
C-curves: An extension of cubic curves
Computer Aided Geometric Design
GMP'08 Proceedings of the 5th international conference on Advances in geometric modeling and processing
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We consider parametric curves that are represented by combination of control points and basis functions. We let a control point vary while the rest is held fixed. We show that the locus of the moving control point that yields a zero curvature point on the curve is a developable surface, the regression curve of which is the locus that guarantees a cusp on the curve. We also specify the surface that is described by those positions of the moving control point that yield a loop on the curve. Then we apply this approach to detect cusps, inflection points and loops of C-Bezier curves. Finally, we compare cubic Bezier, cubic rational Bezier and C-Bezier curves from singularity point of view.