Parametric Surfaces

Main idea

A curve is traced by one moving parameter; a surface needs two, like latitude and longitude on a globe: $\vec r(u, v) = \langle x(u, v), y(u, v), z(u, v)\rangle$ . Nudging each parameter gives two tangent directions, and their cross product gives the normal and the area of each tiny patch.

\vec r(u, v), \qquad \vec n = {\htmlData{sym=0}{\vec r_u\times\vec r_v}}, \qquad {\htmlData{sym=1}{dS}} = \|{\htmlData{sym=0}{\vec r_u\times\vec r_v}}\|\,du\,dv

What the symbols mean

$\vec r_u\times\vec r_v$
Normal vector
Perpendicular to the surface: the cross product of the two tangent directions.
$dS$
Surface area element
The area of the tiny parallelogram spanned by r_u du and r_v dv.

Explanation

A parametric surface is simply a recipe that turns two numbers into a point in space. A globe is the classic example: give me a latitude and a longitude and I'll tell you exactly where on the sphere that is. The notation $\vec r(u, v) = \langle x(u, v), y(u, v), z(u, v)\rangle$ says the same thing, with $u$ and $v$ as the two input numbers and three formulas for where the point lands. Think of it as taking a flat rectangle of $(u, v)$ values and bending it into a shape. That's all a parametric surface is.

Take a look at parametric curves: $\vec r(t)$ uses one parameter, so the point can only move forward or back along a line of travel. A surface has room to move in two independent directions, so it needs two parameters.

Examples:

Graph $z = f(x, y)$ : $\vec r = \langle x, y, f(x, y)\rangle$ . Any graph is a parametric surface, with $x$ and $y$ as the parameters.
Cylinder of radius $a$ : $\langle a\cos\theta, a\sin\theta, z\rangle$ . It's the polar circle, stacked up in $z$ .
Sphere of radius $a$ : $\langle a\sin\varphi\cos\theta, a\sin\varphi\sin\theta, a\cos\varphi\rangle$ , spherical coordinates with $\rho = a$ fixed, with $\|\vec r_\varphi\times\vec r_\theta\| = a^2\sin\varphi$ .

Tangent vectors. Hold $v$ fixed and vary $u$ : the point traces a curve on the surface, and its velocity is $\vec r_u$ , tangent to the surface. Holding $u$ fixed instead gives another tangent, $\vec r_v$ . On a globe, these are moving along a meridian and along a circle of latitude.

Why the cross product. Both tangents lie in the surface, so their cross product $\vec r_u\times\vec r_v$ sticks straight out of it: a normal vector. And its length is the area of the parallelogram they span. A tiny $du$ by $dv$ rectangle of parameters lands on a tiny patch with sides $\vec r_u\,du$ and $\vec r_v\,dv$ , so the patch has area $\|\vec r_u\times\vec r_v\|\,du\,dv$ . It makes sense that this is the same idea as the Jacobian in change of variables: both measure how much a map stretches area. The surface area formula from multiple integrals is the special case of a graph, where $\|\vec r_x\times\vec r_y\| = \sqrt{1 + f_x^2 + f_y^2}$ .

A common mistake. It's tempting to take the cross product's length to be $1$ because the parameters are "nice." It almost never is. On the sphere it's $a^2\sin\varphi$ , which shrinks to zero at the poles, where the circles of latitude collapse to a point.

See it

A sphere of radius 1.5 as a parametric surface, with a point running around the circle φ = 1 as θ sweeps. At every spot the two tangent vectors lie flat against the sphere, and their cross product sticks straight out. Notice its length stays at about 1.89, which is a² sin φ: every patch on this circle gets stretched the same amount. Drag to turn it.

Worked examples

Example 1

Find the area of the cylinder $\langle\cos\theta, \sin\theta, z\rangle$ , $0 \le \theta \le 2\pi$ , $0 \le z \le 3$ .

1
The tangent vectors are $\vec r_\theta = \langle -\sin\theta, \cos\theta, 0\rangle$ and $\vec r_z = \langle 0, 0, 1\rangle$ . Their cross product is $\langle\cos\theta, \sin\theta, 0\rangle$ , which points straight out from the axis and has length $1$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Find the point $\vec r(1, 2)$ on $\vec r(u, v) = \langle u, v, u^2 + v^2\rangle$ .

Problem 2+5 XP

Find $\vec r_u\times\vec r_v$ for the plane $\vec r = \langle u, v, 2u + 3v\rangle$ .

Problem 3+5 XP

What surface is $\langle 2\cos\theta, 2\sin\theta, z\rangle$ ?

Problem 4+5 XP

For the sphere of radius $a$ , what is $\|\vec r_\varphi\times\vec r_\theta\|$ ?

Problem 5+5 XP

Find the area of the cylinder $\langle\cos\theta, \sin\theta, z\rangle$ , $0 \le z \le 3$ .

Problem 6+5 XP

The side of a cylindrical water tank, with radius 2 m and height 5 m, needs paint. Parametrize it as $\vec r(\theta, z) = \langle 2\cos\theta, 2\sin\theta, z\rangle$ and find its area, in square meters.

Problem 7+5 XP

Find the area of the cone $\vec r(u, v) = \langle u\cos v, u\sin v, u\rangle$ for $0 \le u \le 2$ , $0 \le v \le 2\pi$ .

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

A torus (the surface of a donut) is $\vec r(\theta, \varphi) = \langle(3 + \cos\varphi)\cos\theta,\; (3 + \cos\varphi)\sin\theta,\; \sin\varphi\rangle$ for $0 \le \theta, \varphi \le 2\pi$ . Find its surface area.

Where it's used

3D modeling software represents curved surfaces (car bodies, characters, aircraft skins) as parametric patches like NURBS, with normals from $\vec r_u\times\vec r_v$ for lighting.

Curl and Divergence Surface Integrals