Discrete Gaussian Curvature Flow for Piecewise Constant Gaussian Curvature Surface. (May 2021) - Hayashi, Kazuki; Jikumaru, Yoshiki; Ohsaki, Makoto; Kagaya, Takashi; Yokosuka, Yohei
Abstract: A method is presented for generating a discrete piecewise constant Gaussian curvature (CGC) surface. An energy functional is first formulated so that its stationary point is the linear Weingarten (LW) surface, which has a property such that the weighted sum of mean and Gaussian curvatures is constant. The CGC surface is obtained using the gradient derived from the first variation of a special type of the energy functional of the LW surface and updating the surface shape based on the Gaussian curvature flow. A filtering method is incorporated to prevent oscillation and divergence due to unstable property of the discretized Gaussian curvature flow. Two techniques are proposed to generate a discrete piecewise CGC surface with preassigned internal boundaries. The step length of Gaussian curvature flow is adjusted by introducing a line search algorithm to minimize the energy functional. The effectiveness of the proposed method is demonstrated through numerical examples of generating various shapes of CGC surfaces. Highlights: Generating constant Gaussian curvature surfaces using Gaussian curvature flow. Discontinuity of Gaussian curvature along the prescribed internal boundary. Derivation of gradients as the first variation of a discretized energy functional. Algorithm fully compatible to minimization of the energy functional. Stabilization and acceleration of algorithm using filtering and line search.