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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.
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.
Generally, the minimum number of parameters required to describe a model or geometric object is equal to its dimension, and the scope of the parameters—within their allowed ranges—is the parameter space. Though a good set of parameters permits identification of every point in the object space, it may be that, for a given parametrization ...
Assume that f is a scalar, vector, or tensor field defined on a surface S. To find an explicit formula for the surface integral of f over S, we need to parameterize S by defining a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. Let such a parameterization be r(s, t), where (s, t) varies in some region T in ...
Often it is used to create a 3D surface from a point cloud through either downsampling or upsampling. In numerical analysis to handle contributions of geometry where it is difficult to obtain discretizations, the moving least squares methods have also been used and generalized to solve PDEs on curved surfaces and other geometries.