On organizing principles of Discrete Differential Geometry. Geometry of spheres

Abstract

Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. In this survey we discuss the following two fundamental Discretization Principles: the transformation group principle (smooth geometric objects and their discretizations are invariant with respect to the same transformation group) and the consistency principle (discretizations of smooth parametrized geometries can be extended to multidimensional consistent nets). The main concrete geometric problem discussed in this survey is a discretization of curvature line parametrized surfaces in Lie geometry. We find a discretization of curvature line parametrization which unifies the circular and conical nets by systematically applying the Discretization Principles.

57 pages, 18 figures; In the second version the terminology is slightly changed and umbilic points are discussed

References in corpus (2)

• Discrete differential geometry. Consistency as integrability
• Two-dimensional Dirac operator and surface theory

Cited by in corpus (20)

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• Curvature line parametrization from circle patterns
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• Holographic coordinates
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