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Curves on a surface which minimize length between the endpoints are called geodesics; they are the shape that an elastic band stretched between the two points would take. Mathematically they are described using ordinary differential equations and the calculus of variations. The differential geometry of surfaces revolves around the study of ...
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, ... For a surface in R 3, ...
In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of R 3. It permits the calculation of curvature and metric properties of a surface such as length and area in a manner consistent with the ambient space.
The classical nineteenth century approach to the differential geometry of surfaces, due in large part to Carl Friedrich Gauss, has been reworked in this modern framework, which provides the natural setting for the classical theory of the moving frame as well as the Riemannian geometry of higher-dimensional Riemannian manifolds.
In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by (read "two").
This is a list of differential geometry topics. See also glossary of differential and metric geometry and list of Lie group topics . Differential geometry of curves and surfaces
Pages in category "Differential geometry of surfaces" The following 47 pages are in this category, out of 47 total. ... Translation surface (differential geometry) U.
In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology. In the simplest application, the case of a triangle on a plane, the sum of its angles is 180 degrees. [1]