Discrete differential geometry. Consistency as integrability

TL;DR

Discrete differential geometry bridges smooth and discrete geometry, developing discrete analogs of classical surface theory, with recent advances revealing deep connections to integrable systems and applications in computer graphics.

Contribution

This work systematically presents recent progress in discrete differential geometry, highlighting its foundational structures and links to integrability and applications.

Findings

Enhanced understanding of classical geometric structures through discrete analogs

Revealed connections between discrete geometry and integrable systems

Implications for computer graphics and geometric modeling

Abstract

A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studies geometric shapes with finite number of elements (such as polyhedra), the discrete differential geometry aims at the development of discrete equivalents of notions and methods of smooth surface theory. Current interest in this field derives not only from its importance in pure mathematics but also from its relevance for other fields like computer graphics. Recent progress in discrete differential geometry has lead, somewhat unexpectedly, to a better understanding of some fundamental structures lying in the basis of the classical differential geometry and of the theory of integrable systems. The goal of this book is to give a systematic presentation of current achievements in this field.