The Cylindrical Coordinate System

# The Cylindrical Coordinate System

Recall that when we were dealing with polar coordinates in $\mathbb{R}^2$ that we thought of every point $P$ on the plane as an ordered pair $(r, \theta)$ where $r$ was the signed length of the ray extending from the origin to $P$, and $\theta$ was the angle between the polar axis (or $x$-axis) and the ray. Furthermore, we realized that the polar coordinate system acted with parametric equations $x= r \cos \theta$ and $y = r \sin \theta$.

We will not introduce a new coordinate system for $\mathbb{R}^3$ that is analogous to the polar coordinate system in $\mathbb{R}^2$. This new coordinate system is called the Cylindrical Coordinate System that represents every point $P$ in $\mathbb{R}^3$ as an order triple $(r, \theta, z)$ where $r$ represents the signed length of the projection of the ray from the origin to $P$ onto the $xy$-plane, $\theta$ represents the angle from the positive polar axis to the projection of the ray onto the $xy$-plane, and $z$ represents the signed height of $P$ above/below the $z$-axis. This coordinate system can be represented with the following parametric equations:

(1)
\begin{align} \left\{\begin{matrix} x = r \cos \theta\\ y = r \sin \theta\\ z = z \end{matrix}\right. \end{align}

Notice that if $z = 0$ then we reduce our set of cylindrical equations back to polar equations. Furthermore, also notice that the distance from the origin to point $P$ can be computed as $\mathrm{distance} = \sqrt{r^2 + z^2} = \sqrt{x^2 + y^2 + z^2}$.

In particular, the cylindrical coordinate system is great for representing surfaces that have cylindrical symmetry to them. We will now look at some equations using the cylindrical coordinate system in $\mathbb{R}^3$.

Graph Description
Equations in the form of $r = c$ where $c \in \mathbb{R}$ represents cylinders centered around the $z$-axis and with radius $r$. This should make sense as $r^2 = x^2 + y^2$.
Equations in the form $\theta = c$ where $c \in \mathbb{R}$ represent half-planes. The projection of these half planes onto the $xy$-axis form half-lines starting at the origin and with an angle of $\theta = c$ between the line and the polar-axis.
Equations in the form $z = c$ where $c \in \mathbb{R}$ represent planes that are parallel to the $xy$-plane.