Cylindrical Coordinate System

Cylindrical Coordinates

Cylindrical coordinates are a natural extension of the polar coordinate system to three dimensions. They identify each point in the Euclidean space ${\mathbb{R}}^3$ by its distance from the $z$-axis, the angle $\varphi$ its projection in the $xy$-plane makes with the $x$-axis and its $z$ component.

Definition: Radial Distance

The number $\rho$ is known as the radial distance.

WARNING

This is not the same as the radial distance used in spherical coordinates.

Definition: Azimuthal Angle (Azimuth)

The number $\varphi$ is known as the azimuthal angle or azimuth.

Definition: Axial Coordinate (Height)

The number $z$ is known as the axial coordinate or height.

NOTE

Despite the word “height”, the axial coordinate’s algebraic sign is crucial.

Conventions

Some people prefer to use $\theta$ or $\phi$ for the azimuth.

Additionally, a single point has infinitely many possible values for its azimuth, since adding or subtracting a multiple of $2\pi$ to an angle has no effect. However, in order to have a coordinate system, coordinates must be unique. This means that the possible values for $\varphi$ must be restricted. Common conventions for the range of $\varphi$ are $[0;2\pi )$ and $(-\pi ,\pi ]$.