1.13 - Cylindrical Coordinates

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Introduction

Cylindrical coordinates $(\rho, \theta, z)$ extend polar coordinates into three dimensions by adding a vertical coordinate. They are particularly useful for problems with cylindrical or rotational symmetry, such as those involving circular pipes, rotating objects, or systems with axial symmetry. In cylindrical coordinates, a point is described by its distance from the $z$-axis, the angle it makes with the positive $x$-axis in the $xy$-plane, and its height along the $z$-axis.

Cylindrical Coordinates Definition:

In three dimensions, a point $P$ can be represented using cylindrical coordinates $(\rho, \theta, z)$, where:

  • $\rho$ (the radial coordinate): the distance from the $z$-axis to the projection of the point onto the $xy$-plane
  • $\theta$ (the azimuthal angle): the angle measured counterclockwise from the positive $x$-axis to the projection of the point onto the $xy$-plane
  • $z$ (the vertical coordinate): the height of the point along the $z$-axis (same as in Cartesian coordinates)

By convention:

  • $\rho \geq 0$ (distance is always non-negative)
  • $\theta$ is typically measured in radians, with $\theta \in [0, 2\pi)$ or $\theta \in (-\pi, \pi]$
  • $z$ can be any real number
  • The $z$-axis corresponds to $\rho = 0$, where $\theta$ is undefined

Interactive Cylindrical Coordinate Plot

Use the interactive 3D plot below to explore cylindrical coordinates. Adjust the sliders to change $\rho$, $\theta$, and $z$, and see how the point moves in 3D space. The plot shows the point, the position vector from the origin, and the projection onto the $xy$-plane.

Cylindrical Coordinate 3D Visualization

120
50°
90
245°
20°
Cylindrical: (ρ=120, θ=50°, z=90) | Cartesian: (x=77.1, y=91.9, z=90)

Transformation Between Cylindrical and Cartesian Coordinates

Converting between cylindrical $(\rho, \theta, z)$ and Cartesian $(x, y, z)$ coordinates is very similar to converting polar coordinates (see Chapter 1.11 Polar Coordinates).

Cylindrical to Cartesian Transformation:
\begin{aligned}x &= \rho\cos\theta\\ y &= \rho\sin\theta\\ z&= z \end{aligned}

The $x$ and $y$ transformations are identical to those in polar coordinates, while $z$ remains unchanged.

Cartesian to Cylindrical Transformation:
\begin{aligned}\rho &= \sqrt{x^2 + y^2}\\ \theta &= \arctan\left(\frac{y}{x}\right)\\ z &= z \end{aligned}

As with polar coordinates, care must be taken when determining $\theta$ using the $\text{atan2}(y, x)$ function to handle all quadrants correctly.

Important Notes:
  • Cylindrical coordinates are essentially polar coordinates in the $xy$-plane plus a $z$-coordinate.
  • The relationship between cylindrical and Cartesian coordinates is not one-to-one: the point $(\rho, \theta, z)$ and $(\rho, \theta + 2\pi n, z)$ for any integer $n$ represent the same Cartesian point.
  • For points on the $z$-axis ($\rho = 0$), $\theta$ is undefined (can be any angle).
  • Cylindrical coordinates are particularly useful for problems with cylindrical symmetry, where physical properties depend only on $\rho$ and $z$, but not on $\theta$.

Basis Vectors in Cylindrical Coordinates

Just like basis vectors in Polar coordinates, the Cylindrical basis vectors change directions if $\theta$ changes. The derivation of the basis vectors is almost identical to the derivation of the basis vectors in Polar coordinates (see Chapter 1.11 Polar Coordinates). We leave the derivation to the reader.

Cylindrical Unit Vectors:
\begin{aligned}\hat{\rho} &= \cos\theta\hat{x} + \sin\theta\hat{y}\\ \hat{\theta} &= -\sin\theta\hat{x} + \cos\theta\hat{y}\\ \hat{z} &= \hat{z} \end{aligned}

Position Vector in Cylindrical Coordinates

To project the position vector $\mathbf{r}$ onto the Cylindrical basis, we follow the same method used for Polar coordinates. We first write the position vector in Cartesian coordinates, and then use the transformation equations:

\begin{aligned}\mathbf{r} &= x\hat{x} + y\hat{y} + z\hat{z} \\ &= \rho\cos\theta\hat{x} + \rho\sin\theta\hat{y} + z\hat{z} \end{aligned}

The first two terms are just $\rho \hat{\rho}$. After making this substitution, the position vector becomes:

Position Vector in Cylindricl Coordinates: $$\mathbf{r} = \rho\hat{\rho} + z\hat{z}$$

Note that $\hat{\rho}$ and $\hat{\theta}$ depend on the angle $\theta$ and therefore change direction as we move around the cylinder.

Examples

Example 1: Converting from Cylindrical to Cartesian

Convert the cylindrical coordinates $(\rho, \theta, z) = (3, \pi/4, 5)$ to Cartesian coordinates.

Solution:

Using the transformation equations:

$$x = \rho\cos\theta = 3\cos(\pi/4) = 3 \cdot \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2}$$
$$y = \rho\sin\theta = 3\sin(\pi/4) = 3 \cdot \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2}$$
$$z = z = 5$$

Therefore, the Cartesian coordinates are $\left(\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2}, 5\right)$.

Example 2: Converting from Cartesian to Cylindrical

Convert the Cartesian coordinates $(x, y, z) = (4, -3, 7)$ to cylindrical coordinates.

Solution:

First, calculate $\rho$:

$$\rho = \sqrt{x^2 + y^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$$

For $\theta$, since $x > 0$ and $y < 0$, the point is in the fourth quadrant. Using $\text{atan2}(y, x)$:

$$\theta = \arctan\left(\frac{-3}{4}\right) \approx -0.644 \text{ radians} \approx -36.87°$$

And $z = 7$. Therefore, the cylindrical coordinates are approximately $(5, -0.644, 7)$.