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Simple examples. A simple example of a regular surface is given by the 2-sphere {(x, y, z) | x 2 + y 2 + z 2 = 1}; this surface can be covered by six Monge patches (two of each of the three types given above), taking h(u, v) = ± (1 − u 2 − v 2) 1/2. It can also be covered by two local parametrizations, using stereographic projection.
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, ... (for example a surface in an ambient space of ...
Translation surface: definition In differential geometry a translation surface is a surface that is generated by translations: For two space curves c 1 , c 2 {\displaystyle c_{1},c_{2}} with a common point P {\displaystyle P} , the curve c 1 {\displaystyle c_{1}} is shifted such that point P {\displaystyle P} is moving on c 2 {\displaystyle c ...
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
Typically, in algebraic geometry, a surface may cross itself (and may have other singularities), while, in topology and differential geometry, it may not. A surface is a topological space of dimension two; this means that a moving point on a surface may move in two directions (it has two degrees of freedom).
For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space. Topological surfaces are sometimes equipped with additional information, such as a Riemannian metric or a complex structure, that connects them to other disciplines within mathematics, such as differential geometry and complex analysis.
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold M (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there.
When a surface has a constant positive Gaussian curvature, then the geometry of the surface is spherical geometry. Spheres and patches of spheres have this geometry, but there exist other examples as well, such as the lemon / American football. When a surface has a constant negative Gaussian curvature, then it is a pseudospherical surface and ...