Regular Surfaces & Tangent Planes

Parametrizations, the regularity condition, and the tangent plane.

The idea

A regular surface is the precise notion of a smooth two-dimensional object in $\mathbb{R}^{3}$, and it is the setting on which the calculus of surfaces is built. Formally, a regular surface is a set covered by smooth parametrizations $\mathbf{X}(u, v)$, each mapping a region of the $(u,v)$-plane onto a patch of the surface and each satisfying the regularity condition: the partial derivatives $\mathbf{X}_u = \frac{\partial \mathbf{X}}{\partial u}, \qquad \mathbf{X}_v = \frac{\partial \mathbf{X}}{\partial v}$ are linearly independent at every point, equivalently $\mathbf{X}_u \times \mathbf{X}_v \ne \mathbf{0}$.

The condition prevents the patch from collapsing. Hold $v$ fixed and vary $u$: this traces a curve on the surface with velocity $\mathbf{X}_u$. If the two velocities were parallel, or one of them vanished, the image would fold down to a curve or a point, leaving no two-dimensional object to do geometry on.

Where regularity holds, $\mathbf{X}_u$ and $\mathbf{X}_v$ span the tangent plane at the point: the set of velocities of all curves on the surface through it. The plane belongs to the surface, not to the parametrization — different coordinates give different partials spanning the same plane. Its unit normal $\mathbf{N} = (\mathbf{X}_u \times \mathbf{X}_v) / \lVert \mathbf{X}_u \times \mathbf{X}_v \rVert$ is determined only up to sign, and to orient the surface is to choose that sign consistently.

Ways to work on it

• Walkthrough. Partials, the regularity condition, and the normal vector.
• Practice. Compute a partial of a parametrization at a point.
• Hardest. Find where a parametrization fails to be regular.

Not sure where to start? Take the ten-question placement test.