Parametrization

Parametrization in 2D-Spaces

2D Regions can be parametrized using a one-dimensional parametrization $r(t)=f(t)i+g(t)j+h(t)k$ with $a\le t\le b$

Parametrization in 3D-Spaces

3D Surfaces can be parametrized using two substitution variables such that: $f(x,y,z)=r(u,v)$ where $r(u,v)=f(u,v)i+g(u,v)j+h(u,v)k$ with $a\le u\le b$ and $c\le v\le d$

Parametrization can be conveniently done by coordinate conversion or identities.

Smooth Parametrization

A parametrized surface $r(u,v)=f(u,v)i+g(u,v)j+h(u,v)k$ is smooth if the partial derivatives $r\_u$ and $r\_v$ are continuous and if $r\_u\times r\_v$ is never zero on the parameter’s domain