Spherical quadratic Bézier triangles with chord length parameterization and tripolar coordinates in space

We consider special rational triangular Bézier surfaces of degree two on the sphere in standard form and show that these surfaces are parameterized by chord length. More precisely, it is shown that the ratios of the three distances of a point to the patch vertices and the ratios of the distances of the parameter point to the three vertices of the (suitably chosen) domain triangle are identical. This observation extends an observation of Farin (2006) about rational quadratic curves representing circles to the case of surfaces. In addition, we discuss the relation to tripolar coordinates.

Research highlights► Study of special triangular Bézier surfaces of degree two on the sphere. ► Proof that these spherical surfaces can be parameterized by chord lengths. ► Extension of an observation by Farin (2006) for quadratic curves to surfaces. ► Introduction of tripolar coordinates. ► Relation between surfaces parameterized by chord length and tripolar coordinates.