Mean curvature flow for generating discrete surfaces with piecewise constant mean curvatures

• Published in: Computer aided geometric design Vol. 101; p. 102169
• Main Authors: Hayashi, Kazuki, Jikumaru, Yoshiki, Ohsaki, Makoto, Kagaya, Takashi, Yokosuka, Yohei
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.04.2023
• Subjects: Constant mean curvature (CMC); Discrete differential geometry; Energy minimization; Form-finding; Mean curvature flow; Energy minimization; Discrete differential geometry; Constant mean curvature (CMC); Form-finding; Mean curvature flow
• ISSN: 0167-8396; 1879-2332
• DOI: 10.1016/j.cagd.2023.102169

Piecewise constant mean curvature (P-CMC) surfaces are generated using the mean curvature flow (MCF). As an extension of the known fact that a CMC surface is the stationary point of an energy functional, a P-CMC surface can be obtained as the stationary point of an energy functional of multiple patch surfaces and auxiliary surfaces between them. A new formulation is presented for the MCF as the negative gradient flow of the energy functional for multiple patch continuous surfaces, which are further discretized so as to determine the change in the vertex positions of triangular meshes on the surface as well as along the internal boundaries between patches. Numerical examples show that multiple patch surfaces approximately reach the specified mean curvatures through the proposed method, which can diversify the options for the shape design using CMC surfaces. •Piecewise constant mean curvature (P-CMC) discrete surfaces are generated by mean curvature flow (MCF).•The internal boundary shape that is equilibrium in its perpendicular direction is automatically determined.•Line search and Nesterov momentum are effective in improving convergence property.