22.1 Parametric representations of surfaces

Parametric Surfaces

Parametric Equations and Surface Patches

A parametric surface works by using two parameters to trace out a surface in 3D space. You define a vector-valued function $\vec{r}(u,v) = \langle x(u,v),\, y(u,v),\, z(u,v) \rangle$ that takes each point $(u,v)$ in a 2D domain and maps it to a point in $\mathbb{R}^3$.

Each point on the surface corresponds to a pair of parameter values $(u,v)$ within a specified domain. A surface patch is the portion of the surface you get by restricting $u$ and $v$ to a bounded range. Think of it as cutting out a piece of the full surface by limiting the parameter window.

Example (unit sphere): Set $x = \cos u \sin v$, $y = \sin u \sin v$, $z = \cos v$, with $0 \leq u < 2\pi$ and $0 \leq v \leq \pi$. Here $u$ sweeps around the equator (longitude) and $v$ sweeps from the north pole to the south pole (colatitude). Fixing $v$ and varying $u$ traces out a circle of latitude; fixing $u$ and varying $v$ traces a meridian.

One thing to watch: parametric representations aren't always one-to-one everywhere. The sphere parametrization above collapses to a single point at $v = 0$ (north pole) and $v = \pi$ (south pole) regardless of $u$. These are not defects of the sphere itself, just artifacts of the parametrization.

Parameter Space and Coordinate Functions

The parameter space (or parameter domain) is the region $D$ in the $uv$-plane over which $u$ and $v$ range. It's typically a rectangle, but it can be any connected region.

The three coordinate functions $x(u,v)$, $y(u,v)$, $z(u,v)$ are the scalar components of $\vec{r}$. They do the actual work of placing each $(u,v)$ pair at a specific location in space.

Example (torus): A torus with major radius $R$ (center of the tube to the center of the torus) and minor radius $r$ (radius of the tube) has the parametrization:

• $x = (R + r\cos v)\cos u$
• $y = (R + r\cos v)\sin u$
• $z = r\sin v$

with $0 \leq u < 2\pi$ and $0 \leq v < 2\pi$. The parameter $u$ controls rotation around the central axis of the torus, while $v$ controls rotation around the tube itself. The parameter domain is a rectangle in the $uv$-plane, but opposite edges get identified, which is why the torus is topologically a "rolled-up" rectangle.

Coordinate Systems

Cartesian, Cylindrical, and Spherical Coordinates

Different coordinate systems make certain parametrizations much easier to write down. Choosing the right system for a given surface can save significant work.

Cartesian coordinates $(x, y, z)$ use three perpendicular axes. They're the default, but they aren't always the most natural choice for curved surfaces.

Cylindrical coordinates $(r, \theta, z)$ use a radial distance $r$ from the $z$-axis, an angle $\theta$ in the $xy$-plane measured from the positive $x$-axis, and the usual height $z$.

• Conversion to Cartesian: $x = r\cos\theta$, $y = r\sin\theta$, $z = z$

Cylindrical coordinates are the natural choice for surfaces with axial symmetry about the $z$-axis, such as cylinders and cones. For instance, the cylinder $x^2 + y^2 = 4$ is simply $r = 2$ in cylindrical coordinates, giving the parametrization $\vec{r}(\theta, z) = \langle 2\cos\theta,\, 2\sin\theta,\, z \rangle$.

Spherical coordinates $(\rho, \theta, \phi)$ use a distance $\rho$ from the origin, an azimuthal angle $\theta$ in the $xy$-plane, and a polar angle $\phi$ measured from the positive $z$-axis.

• Conversion to Cartesian: $x = \rho\sin\phi\cos\theta$, $y = \rho\sin\phi\sin\theta$, $z = \rho\cos\phi$

Spherical coordinates are ideal for surfaces centered at the origin, like spheres and cones defined by a fixed polar angle.

Worked example: Convert the spherical point $(\rho, \theta, \phi) = \left(1, \frac{\pi}{4}, \frac{\pi}{3}\right)$ to Cartesian.

1. $x = 1 \cdot \sin\!\left(\frac{\pi}{3}\right)\cos\!\left(\frac{\pi}{4}\right) = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{6}}{4}$
2. $y = 1 \cdot \sin\!\left(\frac{\pi}{3}\right)\sin\!\left(\frac{\pi}{4}\right) = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{6}}{4}$
3. $z = 1 \cdot \cos\!\left(\frac{\pi}{3}\right) = \frac{1}{2}$

So the Cartesian point is $\left(\frac{\sqrt{6}}{4},\, \frac{\sqrt{6}}{4},\, \frac{1}{2}\right)$.

Surface Representations

Level Surfaces and Implicit Equations

Not every surface starts life as a parametric equation. A level surface is defined implicitly by an equation $f(x,y,z) = c$, where $c$ is a constant. Every point $(x,y,z)$ satisfying that equation lies on the surface.

Example: The equation $x^2 + y^2 + z^2 = 1$ defines the unit sphere. Here $f(x,y,z) = x^2 + y^2 + z^2$, and the sphere is the level surface at $c = 1$. Choosing different values of $c$ gives concentric spheres of radius $\sqrt{c}$.

Implicit equations define surfaces without giving you an explicit parametrization. This is useful for visualization (plotting several level surfaces of $f$ at different $c$-values reveals the structure of a scalar field), but for computing surface integrals or areas you'll usually need to convert to a parametric form.

Example: The level surfaces of $f(x,y,z) = x^2 + y^2 - z$ are the surfaces $z = x^2 + y^2 - c$. Each one is a circular paraboloid (bowl opening upward) shifted vertically by $-c$. Stacking several of these for different $c$-values shows how the function varies through space.

Going between representations: You can often convert an implicit surface to a parametric one. For the paraboloid $z = x^2 + y^2$, set $u = x$ and $v = y$ to get $\vec{r}(u,v) = \langle u,\, v,\, u^2 + v^2 \rangle$. Alternatively, using cylindrical coordinates gives $\vec{r}(r,\theta) = \langle r\cos\theta,\, r\sin\theta,\, r^2 \rangle$, which is often cleaner for integration because it exploits the surface's rotational symmetry.