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Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus , Stokes' theorem and the divergence theorem , are frequently given in a parametric form.
For example, one sphere that is described in Cartesian coordinates with the equation x 2 + y 2 + z 2 = c 2 can be described in spherical coordinates by the simple equation r = c. (In this system—shown here in the mathematics convention—the sphere is adapted as a unit sphere, where the radius is set to unity and then can generally be ignored ...
A sphere (from Greek σφαῖρα, sphaîra) [1] is a geometrical object that is a three-dimensional analogue to a two-dimensional circle.Formally, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. [2]
In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation. The inverse process is called implicitization. [1] "
For a plane, a sphere, and a torus there exist simple parametric representations. This is not true for the fourth example. The implicit function theorem describes conditions under which an equation (,,) = can be solved (at least implicitly) for x, y or z. But in general the solution may not be made explicit.
A sphere is the surface of a solid ball, here having radius r. In mathematics, a surface is a mathematical model of the common concept of a surface.It is a generalization of a plane, but, unlike a plane, it may be curved; this is analogous to a curve generalizing a straight line.
ellipsoid as an affine image of the unit sphere. The key to a parametric representation of an ellipsoid in general position is the alternative definition: An ellipsoid is an affine image of the unit sphere. An affine transformation can be represented by a translation with a vector f 0 and a regular 3 × 3 matrix A:
In the case of two parameters, the point describes a surface, called a parametric surface. In all cases, the equations are collectively called a parametric representation, [2] or parametric system, [3] or parameterization (also spelled parametrization, parametrisation) of the object. [1] [4] [5]