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A parametric surface is a surface in the Euclidean space which is defined by a parametric equation with two parameters :. Parametric representation is a very general way to specify a surface, as well as implicit representation .
In the case of a single parameter, parametric equations are commonly used to express the trajectory of a moving point, in which case, the parameter is often, but not necessarily, time, and the point describes a curve, called a parametric curve. In the case of two parameters, the point describes a surface, called a parametric surface.
Parametrization is a mathematical process consisting of expressing the state of a system, process or model as a function of some independent quantities called parameters. The state of the system is generally determined by a finite set of coordinates , and the parametrization thus consists of one function of several real variables for each ...
It is possible to extend the formula to 3, 4, or n dimensions, by means of the spherical product of superformulas. For example, the 3D parametric surface is obtained by multiplying two superformulas r 1 and r 2.
Let n(u, v) be the unit normal vector and let (E, F, G) and (L, M, N) be the coefficients of the first and second fundamental forms of this surface, respectively. The Weingarten equation gives the first derivative of the unit normal vector n at point P in terms of the tangent vectors r u and r v :
The parameter domain is the surface that the mesh is mapped onto. Parameterization was mainly used for mapping textures to surfaces. Recently, it has become a powerful tool for many applications in mesh processing. [citation needed] Various techniques are developed for different types of parameter domains with different parameterization properties.
Catmull–Clark level-3 subdivision of a cube with the limit subdivision surface shown below. (Note that although it looks like the bi-cubic interpolation approaches a sphere, an actual sphere is quadric.) Visual difference between sphere (green) and Catmull-Clark subdivision surface (magenta) from a cube
The points on the surface may be parameterized in polar coordinates by a pair of numbers (r, θ).Each such pair corresponds to a point in three dimensions according to the parametric equations [2] (,) = (,) = ( +) (,) = / (). The surface can also be expressed as the solution to a polynomial equation of order 16 in the Cartesian coordinates of the three-dimensional space.