Geometric conditions for injectivity of 3D Bezier volumes

The one-to-one property of injectivity is a crucial concept in computer-aided design, geometry, and graphics. The injectivity of curves (or surfaces or volumes) means that there is no self-intersection in the curves (or surfaces or volumes) and their images or deformation models. Bezier volumes are a special class of Bezier polytope in which the lattice polytope equals square(m,n,l), (m, n, l epsilon Z). Piecewise 3D Bezier volumes have a wide range of applications in deformation models, such as for face mesh deformation. The injectivity of 3D Bezier volumes means that there is no self-intersection. In this paper, we consider the injectivity conditions of 3D Bezier volumes from a geometric point of view. We prove that a 3D Bezier volume is injective for any positive weight if and only if its control points set is compatible. An algorithm for checking the injectivity of 3D Bezier volumes is proposed, and several explicit examples are presented.

Automated summary

This paper discusses the concept of injectivity of 3D Bezier volumes, and proves that a Bezier volume is injective if and only if its control points set is compatible. An algorithm for checking injectivity is proposed, along with several explicit examples.