Curves and surfaces with rational chord length parameterization

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.