Evolution of plane curves with a curvature adjusted tangential velocity

• Published in: Japan journal of industrial and applied mathematics Vol. 28; no. 3; pp. 413 - 442
• Main Authors: Ševčovič, Daniel, Yazaki, Shigetoshi
• Format: Journal Article
• Language: English
• Published: Japan Springer Japan 01.10.2011; Springer Nature B.V
• Subjects: Applications of Mathematics; Computational Mathematics and Numerical Analysis; Mathematics; Mathematics and Statistics; Original Paper; Flowing finite volume method; Semi-implicit scheme; 65N40; Curvature driven flow of a plane curve; Crystalline curvature flow equation; Curvature adjusted tangential velocity; 53C80; 35K65
• ISSN: 0916-7005; 1868-937X
• DOI: 10.1007/s13160-011-0046-9

We study evolution of a closed embedded plane curve with the normal velocity depending on the curvature, the orientation and the position of the curve. We propose a new method of tangential redistribution of points by curvature adjusted control in the tangential motion of evolving curves. The tangential velocity may not only distribute grid points uniformly along the curve but also produce a suitable concentration and/or dispersion depending on the curvature. Our study is based on solutions to the governing system of nonlinear parabolic equations for the position vector, tangent angle and curvature of a curve. We furthermore present a semi-implicit numerical discretization scheme based on the flowing finite volume method. Several numerical examples illustrating capability of the new tangential redistribution method are also presented in this paper.