tangent plane to a parametric surface

The flat plane that best approximates a parametric surface at a point, spanned by the two partial-derivative vectors $\mathbf{r}_u$ and $\mathbf{r}_v$ .

The two partial derivative vectors $\mathbf{r}_u = \langle x_u,\,y_u,\,z_u\rangle$ and $\mathbf{r}_v = \langle x_v,\,y_v,\,z_v\rangle$ , each tangent to the surface along its own grid direction, span the tangent plane at a point, and their cross product $\mathbf{r}_u\times\mathbf{r}_v$ gives the normal the tangent plane needs. The requirement $\mathbf{r}_u\times\mathbf{r}_v\neq\mathbf{0}$ is the parametric analogue of "the partials exist" — at a point where the two tangent vectors are parallel (a cone's apex, a sphere's pole under some parametrizations), the surface has no well-defined tangent plane there.

Examples

For $\mathbf{r}(u,v)=\langle u\cos v,\,u\sin v,\,u\rangle$ at $(u,v)=(1,0)$ : the tangent plane is $-x+z=0$ , i.e. $z=x$ .