IBM Journal of Research and Development
An algebraic approach to curves and surfaces on the sphere and on other quadrics
Selected papers of the international symposium on Free-form curves and free-form surfaces
Determination and classification of triangular quadric patches
Computer Aided Geometric Design
Curves and surfaces for CAGD: a practical guide
Curves and surfaces for CAGD: a practical guide
Bipolar and Multipolar Coordinates
Proceedings of the 9th IMA Conference on the Mathematics of Surfaces
Rational quadratic circles are parametrized by chord length
Computer Aided Geometric Design
Curves with rational chord-length parametrization
Computer Aided Geometric Design
Curves with chord length parameterization
Computer Aided Geometric Design
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
Complex rational Bézier curves
Computer Aided Geometric Design
Computer Aided Geometric Design
Surfaces with rational chord length parameterization
GMP'10 Proceedings of the 6th international conference on Advances in Geometric Modeling and Processing
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The investigation of rational varieties with chord length parameterization (shortly RCL varieties) was started by Farin (2006) who observed that rational quadratic circles in standard Bezier form are parametrized by chord length. Motivated by this observation, general RCL curves were studied. Later, the RCL property was extended to rational triangular Bezier surfaces of an arbitrary degree for which the distinguishing property is that the ratios of the three distances of a point to the three vertices of an arbitrary triangle inscribed to the reference circle and the ratios of the distances of the parameter point to the three vertices of the corresponding domain triangle are identical. In this paper, after discussing rational tensor-product surfaces with the RCL property, we present a general unifying approach and study the conditions under which a k-dimensional rational variety in d-dimensional Euclidean space possesses the RCL property. We analyze the entire family of RCL varieties, provide their general parameterization and thoroughly investigate their properties. Finally, the previous observations for curves and surfaces are presented as special cases of the introduced unifying approach.