Exact sphere representations over Platonic solids based on rational multisided Bezier patches

The paper presents several possibilities for exact representation of the sphere obtained by composing rational multisided Bezier patches known as S-patches. We first derive a general method that utilizes the (inverse) stereographic projection and enables exact representation of a sphere section in terms of an S-patch over a regular polygon. Then, we apply the method to the faces of all five Platonic solids inscribed into the sphere. Depending on the Platonic solid, the obtained S-patches are defined over triangular, square or pentagonal domains and unify two previously known constructions based on triangular and tensor product Bezier patches. (C) 2022 Elsevier B.V. All rights reserved.

Automated summary

The paper presents several possibilities for exact representation of the sphere using rational multisided Bezier patches known as S-patches. The authors derive a general method using the stereographic projection to represent a sphere section with an S-patch over a regular polygon. They then apply the method to the faces of Platonic solids inscribed into the sphere, resulting in S-patches defined over triangular, square, or pentagonal domains. This approach unifies previously known constructions based on triangular and tensor product Bezier patches.