Parametrized Surfaces

A parametrized curve is a map from a single number t into space — one knob, one dimension of freedom, so you trace out a line. A surface is the natural next step: give yourself two knobs, u and v, and let a point in space depend smoothly on both. Sweep them and you paint out a whole two-dimensional sheet.

\sigma(u,v) = \big(x(u,v),\ y(u,v),\ z(u,v)\big).

The pair (u,v) lives in a flat patch of the plane — the parameter domain — and \sigma bends, stretches and drapes that flat patch over some shape in \mathbb{R}^3. A globe-maker does exactly this in reverse: they take the curved Earth and try to flatten it back onto a rectangular chart of latitude and longitude. Here we go the other way — from the flat (u,v)-rectangle to the curved surface.

Coordinate curves: freeze one knob

The quickest way to see a parametrization is to hold one parameter still and let the other run. Freeze v = v_0 and let u vary: you get a curve lying in the surface, the u-curve. Freeze u = u_0 instead and you get a v-curve. Together these coordinate curves weave the familiar grid you see printed on a globe: the lines of longitude are one family, the lines of latitude the other.

u \mapsto \sigma(u, v_0) \quad\text{and}\quad v \mapsto \sigma(u_0, v).

Through every point of the surface there runs one curve of each family, and — this is the key idea — each coordinate curve has a velocity vector, found by differentiating with respect to the moving parameter while the other stays pinned. Those are exactly the partial derivatives.

\sigma_u = \frac{\partial \sigma}{\partial u}, \qquad \sigma_v = \frac{\partial \sigma}{\partial v}.

\sigma_u is tangent to the u-curve and \sigma_v is tangent to the v-curve. Both are vectors in 3D, both live at the point \sigma(u_0,v_0), and both are genuinely tangent to the sheet there.

The tangent plane and the unit normal

Two vectors at a point, pointing in different directions, span a plane. That plane — \operatorname{span}\{\sigma_u, \sigma_v\} — is the tangent plane to the surface, the surface's best flat approximation at that point, just as the tangent plane of a graph is its best linear fit. Every direction you can travel along the surface from the point is a combination a\,\sigma_u + b\,\sigma_v.

Perpendicular to that plane is a single line — the normal direction. The cross product \sigma_u \times \sigma_v points along it (perpendicular to both factors by construction), and normalising gives the unit normal:

\mathbf{n} = \frac{\sigma_u \times \sigma_v}{\lVert \sigma_u \times \sigma_v \rVert}.

This little vector does an enormous amount of work later: it orients the surface, it detects which way is "out", and its rate of turning is the whole secret of curvature.

A parametrization \sigma(u,v) is regular at a point when:

Four surfaces you should carry in your head

1) The plane. The simplest of all: \sigma(u,v) = (u, v, 0). Then \sigma_u = (1,0,0), \sigma_v = (0,1,0), and \sigma_u \times \sigma_v = (0,0,1) — a constant unit normal pointing straight up. Flat everywhere, as it should be.

2) A graph z = f(x,y). Any function of two variables is a surface via \sigma(u,v) = (u, v, f(u,v)). Differentiating,

\sigma_u = (1, 0, f_u), \quad \sigma_v = (0, 1, f_v), \quad \sigma_u \times \sigma_v = (-f_u,\ -f_v,\ 1).

The normal is (-f_u, -f_v, 1) — the same combination that appears in every tangent-plane formula. Its length is \sqrt{f_u^2 + f_v^2 + 1}, which is never zero, so a graph is regular everywhere: no function can fold back on itself.

3) The cylinder of radius R: \sigma(u,v) = (R\cos u,\ R\sin u,\ v). Here \sigma_u = (-R\sin u,\ R\cos u,\ 0) runs around the circle and \sigma_v = (0,0,1) runs straight up the axis. They are always perpendicular, and \lVert \sigma_u \rVert = R.

4) The sphere of radius R, in colatitude v and longitude u:

\sigma(u,v) = \big(R\cos u \sin v,\ R\sin u \sin v,\ R\cos v\big), \quad 0 \le u \le 2\pi,\ 0 \le v \le \pi.

The u-curves are lines of latitude (horizontal circles) and the v-curves are lines of longitude (meridian half-circles) — precisely the grid on the globe. We will meet this parametrization again and again.

See the grid and the tangents

Here is the sphere \sigma(u,v) above. One line of latitude (a u-curve, constant v) and one line of longitude (a v-curve, constant u) are picked out in colour, and they cross at a single point. The two arrows there are \sigma_u (tangent to the latitude) and \sigma_v (tangent to the longitude). They point in different directions, so they span the tangent plane — the surface is regular at that point. Drag to rotate.

Vignettes

Our sphere parametrization looks perfect — until you look hard at the poles. At the north pole (v = 0) every value of the longitude u lands on the same point. The whole edge v=0 of the flat (u,v)-rectangle gets crushed to one spot. That is why every flat atlas of the Earth does something strange at the top and bottom (Greenland balloons, the poles smear into lines).

The deep fact is that no single smooth, regular parametrization can cover a whole sphere — the topology won't allow it. The cure is to use several overlapping patches, each regular on its own piece, and stitch them together. A collection of such patches is called an atlas, and each patch a chart — the language of manifolds. A surface is a shape you can cover with charts; the sphere simply needs at least two.

A smooth surface and a smooth parametrization of it are not the same thing. The sphere is perfectly smooth and round at its poles — nothing special happens to the shape there. But the parametrization \sigma(u,v) breaks down at the poles: plug in v = 0 and you find

\sigma_u = (-R\sin u \sin v,\ R\cos u \sin v,\ 0)\big|_{v=0} = (0,0,0),

so \sigma_u \times \sigma_v = \mathbf{0}. There is no tangent plane from this map and no unit normal — a singular point of the parametrization. Do not conclude the surface is singular! The failure is in your chosen coordinates, not in the geometry. Switch to a patch that puts the pole in its interior and everything is smooth again. The moral: "regular" is a property of the parametrization, and a bad singular point may just be a bad choice of (u,v).