Regular Patch

A regular patch is a Patch ${\bf x}:U\to\Bbb{R}^n$ for which the Jacobian $J({\bf x})(u,v)$ has rank 2 for all $(u,v)\in U$. A Patch is said to be regular at a point $(u_0, v_0)\in U$ provided that its Jacobian has rank 2 at $(u_0, v_0)$. For example, the points at $\phi=\pm \pi/2$ in the standard parameterization of the Sphere $(\cos\theta\sin\phi, \sin\theta\sin\phi, \cos\phi)$ are not regular.

An example of a Patch which is regular but not Injective is the Cylinder defined parametrically by $(\cos u, \sin u, v)$ with $u\in(-\infty, \infty)$ and $v\in (-2,2)$. However, if ${\bf x}:U\to\Bbb{R}^n$ is an injective regular patch, then x maps $U$ diffeomorphically onto ${\bf x}(U)$.

See also Injective Patch, Patch, Regular Surface

References

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, p. 187, 1993.