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A sphere of radius r has surface area 4πr 2.. The surface area (symbol A) of a solid object is a measure of the total area that the surface of the object occupies. [1] The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with ...
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.
The graph of a continuous function of two variables, defined over a connected open subset of R 2 is a topological surface. If the function is differentiable, the graph is a differentiable surface. A plane is both an algebraic surface and a differentiable surface. It is also a ruled surface and a surface of revolution.
The simplest type of parametric surfaces is given by the graphs of functions of two variables: = (,), (,) = (,, (,)). A rational surface is a surface that admits parameterizations by a rational function. A rational surface is an algebraic surface. Given an algebraic surface, it is commonly easier to decide if it is rational than to compute its ...
Computational algebraic geometry is an area that has emerged at the intersection of algebraic geometry and computer algebra, with the rise of computers. It consists mainly of algorithm design and software development for the study of properties of explicitly given algebraic varieties.
In geometry, the convex hull, convex envelope or convex closure [1] of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset.
The most familiar examples arise as boundaries of solid objects in ordinary three-dimensional Euclidean space R 3, such as spheres. The exact definition of a surface may depend on the context. Typically, in algebraic geometry, a surface may cross itself (and may have other singularities), while, in topology and differential geometry, it may not.
In differential geometry, the Gauss map of a surface is a function that maps each point in the surface to a unit vector that is orthogonal to the surface at that point. Namely, given a surface X in Euclidean space R 3 , the Gauss map is a map N : X → S 2 (where S 2 is the unit sphere ) such that for each p in X , the function value N ( p ) is ...