Most parameterization techniques focus on how to "flatten out" the surface into the plane while maintaining some properties as best as possible (such as area). These techniques are used to produce the mapping between the manifold and the surface.

What is the purpose of parameterization?

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.

What does it mean to parameterize a vector?

Every vector-valued function provides a parameterization of a curve. ... In , a parameterization of a curve is a set of three equations , x = x ( t ) , , y = y ( t ) , and z = z ( t ) that describes the coordinates of a point ( x , y , z ) on the curve in terms of a parameter .