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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.
Below are some examples of how differential geometry is applied to other fields of science and mathematics. In physics, differential geometry has many applications, including: Differential geometry is the language in which Albert Einstein's general theory of relativity is expressed.
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).
Ruled surface generated by two Bézier curves as directrices (red, green) A surface in 3-dimensional Euclidean space is called a ruled surface if it is the union of a differentiable one-parameter family of lines. Formally, a ruled surface is a surface in is described by a parametric representation of the form
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
In mathematics, regular surface may refer to: Regular surface (differential geometry) Non-singular algebraic variety of dimension two This page was last edited on 6 ...
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.
A surface (for example a roof) can be manufactured using a jig for curve and several identical jigs of curve . The jigs can be designed without any knowledge of mathematics. By positioning the jigs the rules of a translation surface have to be respected only. Descriptive geometry
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