Discrete differential geometry of curves and surfaces
This book is a result of a 2-hours-a-week course the author gave at the faculty of mathematics at Kyushu University, Fukuoka in summer 2008. It is an introduction to discrete differential geometry, a rapidly growing field of modern geometry that was originated in the 1950s (see, e.g. [\textit{R. Sauer}, Differenzengeometrie. Berlin-Heidelberg-New York: Springer-Verlag (1970; Zbl 0199.25001)]) and got a new life in the 1990s (see, e.g. [\textit{A. I. Bobenko} and \textit{Yu. B. Suris}, Discrete differential geometry. Integrable structure. Providence, RI: American Mathematical Society (AMS) (2008; Zbl 1158.53001)]).NEWLINENEWLINEDiscrete differential geometry investigates discrete analogs of objects from smooth differential geometry. In some sense, it can be considered more fundamental than differential geometry because the latter can be obtained from the former as a limit. Moreover, discrete differential geometry is richer since it deals with more ingredients (like combinatorics) and since some structures get lost in the smooth limit (e.g., the tangential flow on a smooth regular curve is trivial while on a discrete curve it is not). Recent interest in discrete differential geometry is partly due to its applications in computer graphics, architecture, numerical solution of partial differential equations, simulations (flows, deformations) and physics (lattice models). Additional information about the book can be obtained from its contents:NEWLINENEWLINEPreface.NEWLINENEWLINE1. Introduction.NEWLINENEWLINE2. Discrete curves in \(\mathbb R^2\) and \(\mathbb{CP}^1\)NEWLINENEWLINE{}\quad 2.1. Basic notions.NEWLINENEWLINE{}\quad 2.2. Curvature.NEWLINENEWLINE{}\quad 2.3. New curves from old ones; evolutes and involutes.NEWLINENEWLINE{}\quad 2.4. Four vertex theorems.NEWLINENEWLINE{}\quad 2.5. Curves in \(\mathbb{CP}^1\).NEWLINENEWLINE{}\quad 2.6. Darboux transformations and time discrete flows.NEWLINENEWLINE3. Discrete curves in \(\mathbb R^3\)NEWLINENEWLINE{}\quad 3.1. \(\mathbb R^3\) and quaternions.NEWLINENEWLINE{}\quad 3.2. Möbius transformations in higher dimensions.NEWLINENEWLINE4. Discrete surfaces in \(\mathbb R^3\)NEWLINENEWLINE{}\quad 4.1. Isothermic surfaces.NEWLINENEWLINE{}\quad 4.2. The classical model for Möbius geometry.NEWLINENEWLINE{}\quad 4.3. \(s\)-Isothermic surfaces.NEWLINENEWLINE{}\quad 4.4. Curvatures and the Steiner formula.NEWLINENEWLINEReferences.
scientific article; zbMATH DE number 6076958