Parametric equation

A parametric equation expresses coordinates as functions of one or more independent parameters. For example, $x=\cos t$, $y=\sin t$ describes a circle; multiple parameters describe surfaces. A smooth curve is a differentiable single-parameter realization. A reparametrization changes the parameter without changing the geometric image.

The domain of a function $U$ used by a parametrization $\mathbf{r}$ is its parameter domain. A parametrized surface may require several such domains, and duplicated images must be accounted for in a surface integral.