Accurate High-Order Derivatives of Geodesic Paths on Smooth Surfaces. (November 2021)

• Record Type: Journal Article
• Authors: Scholz, Felix; Maekawa, Takashi
• Is Part Of: Computer aided design. Volume 140(2021)
• Publication Date: 2021-11
• Subjects: Differential geometry -- Geodesic curves -- Developable surfaces -- Robotics -- Numerical differentiation; Computer-aided design -- Periodicals; Engineering design -- Data processing -- Periodicals; Computer graphics -- Periodicals; Conception technique -- Informatique -- Périodiques; Infographie -- Périodiques; Computer graphics; Engineering design -- Data processing; Periodicals; Electronic journals; 620.00420285
• DOI: 10.1016/j.cad.2021.103082
• Languages: English
• ISSNs: 0010-4485

Abstract

Abstract: We propose a new approach for the accurate numerical computation of high-order derivatives along geodesic curves on surfaces. The method is based on the observation that for geodesics the Darboux frame and the Frenet–Serret frame are locally equal up to a constant rotation around the tangent. It computes derivatives of arbitrary order from the result of the numerical method employed for computing the geodesic. Since it does not rely on finite difference approximations, no additional discretization errors are introduced. Applications of the method include motion planning of autonomous vehicles and geometric modeling with developable surfaces. Highlights: We introduce a novel method for the accurate numerical computation of the high-order derivatives of geodesic curves on surfaces. Unlike finite difference approaches, we do not introduce further discretization errors when computing derivatives of arbitrary order. The problem of computing the high-order derivatives is reduced to a linear system. The method can be integrated into methods for the design of developable surfaces and motion planning of autonomous vehicles.