1.3.3. Cylindrical Coordinates

Table of Contents

Idea and Definition

Cylindrical coordinates are a way to describe the position of a point in three dimensional space using one circular part and one vertical part. They are especially useful when a situation has symmetry around an axis, such as pipes, cylinders, rotating objects, and fields around wires.

In cylindrical coordinates, a point is written as $(r,\phi,z)$.

Here, $r$ is the distance from the point to the $z$ axis, $\phi$ is the angle around the $z$ axis, and $z$ is the same vertical coordinate used in Cartesian coordinates.

So, instead of saying where a point is with $(x,y,z)$, we describe it by how far it is from the axis, how much it has rotated around the axis, and how high it is.

Geometric Meaning

Imagine looking at a point from above, down the $z$ axis. In the horizontal plane, the point has polar coordinates $(r,\phi)$. Then the coordinate $z$ tells how high or low the point is.

This is why cylindrical coordinates can be thought of as polar coordinates in the $xy$ plane, plus a vertical coordinate.

Rendering scientific sketch…

Cylindrical coordinates of a point

Relation to Cartesian Coordinates

The connection between cylindrical and Cartesian coordinates comes from the polar coordinate relations in the $xy$ plane.

If a point has cylindrical coordinates $(r,\phi,z)$, then its Cartesian coordinates are

$$
x = r\cos\phi
$$

$$
y = r\sin\phi
$$

$$
z = z
$$

The reverse conversion is

$$
r = \sqrt{x^2 + y^2}
$$

$$
\phi = \tan^{-1}\left(\frac{y}{x}\right)
$$

$$
z = z
$$

The angle formula needs care, because the correct quadrant matters. In practice, one uses the angle that matches the signs of $x$ and $y$.

Important relations for cylindrical coordinates:
$$
x = r\cos\phi, \qquad y = r\sin\phi, \qquad z = z
$$
$$
r = \sqrt{x^2+y^2}
$$
The coordinate $r$ is always nonnegative, and $\phi$ gives the direction around the $z$ axis.

Coordinate Surfaces

Each cylindrical coordinate creates a special kind of surface when it is held constant.

If $r$ is constant, the surface is a circular cylinder centered on the $z$ axis.

If $\phi$ is constant, the surface is a vertical half plane passing through the $z$ axis.

If $z$ is constant, the surface is a horizontal plane.

This is one reason the name "cylindrical coordinates" is used.

Constant coordinate	Surface produced
$r = \text{constant}$	Circular cylinder
$\phi = \text{constant}$	Vertical half plane
$z = \text{constant}$	Horizontal plane

Example of Conversion

Suppose a point has cylindrical coordinates

$$
(r,\phi,z) = (2,\tfrac{\pi}{3},5)
$$

Then

$$
x = 2\cos\left(\frac{\pi}{3}\right) = 2 \cdot \frac{1}{2} = 1
$$

$$
y = 2\sin\left(\frac{\pi}{3}\right) = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3}
$$

and

$$
z = 5
$$

So the Cartesian coordinates are

$$
(1,\sqrt{3},5)
$$

Now suppose instead we start from Cartesian coordinates $(3,3,4)$. Then

$$
r = \sqrt{3^2+3^2} = \sqrt{18} = 3\sqrt{2}
$$

$$
\phi = \tan^{-1}\left(\frac{3}{3}\right) = \frac{\pi}{4}
$$

$$
z = 4
$$

So the cylindrical coordinates are

$$
(3\sqrt{2},\tfrac{\pi}{4},4)
$$

How to Read the Coordinates

A cylindrical coordinate description can be understood step by step.

First move out from the $z$ axis by a distance $r$. Then rotate by angle $\phi$ in the $xy$ plane. Then move vertically to height $z$.

This makes cylindrical coordinates very natural for shapes like tubes, coils, and rotating systems.

When Cylindrical Coordinates Are Useful

Cylindrical coordinates are often the best choice when the system has rotational symmetry around one axis. Examples include a straight wire, a round pipe, a cylindrical tank, or the motion of an object around an axis.

In such cases, equations and geometry often become simpler in cylindrical coordinates than in Cartesian coordinates.

Use cylindrical coordinates when the problem involves symmetry about an axis, especially the $z$ axis.

Comparison with Other Systems

Cylindrical coordinates are not the same as polar coordinates, although they are closely related. Polar coordinates describe points in a plane using $(r,\phi)$. Cylindrical coordinates extend this idea to three dimensions by adding $z$.

They are also different from spherical coordinates, where position is described using a distance from the origin and two angles.

System	Coordinates	Best for
Cartesian	$(x,y,z)$	Box shaped geometry
Polar	$(r,\phi)$	Flat circular geometry
Cylindrical	$(r,\phi,z)$	Axial symmetry in 3D
Spherical	$(r,\theta,\phi)$	Full radial symmetry

Important Notes

Different points can sometimes be written in more than one angular form because angles that differ by $2\pi$ represent the same direction. For example, $\phi$ and $\phi + 2\pi$ describe the same line around the axis.

Also, when $r=0$, the point lies on the $z$ axis, and the angle $\phi$ does not matter physically, because all directions around the axis are equivalent there.

At $r=0$, the angular coordinate $\phi$ is not uniquely defined.
Angles that differ by $2\pi$ represent the same physical direction.

Visual Picture

Rendering scientific sketch…

Coordinate surfaces in cylindrical coordinates

Summary

Cylindrical coordinates describe a point in space as $(r,\phi,z)$. The coordinate $r$ measures distance from the $z$ axis, $\phi$ measures angle around that axis, and $z$ gives vertical position. They are ideal for three dimensional problems with axial symmetry, and they connect to Cartesian coordinates through

$$
x=r\cos\phi, \qquad y=r\sin\phi, \qquad z=z
$$

and

$$
r=\sqrt{x^2+y^2}, \qquad \phi=\tan^{-1}\left(\frac{y}{x}\right)
$$

They provide a natural description for cylindrical shapes and many important physical systems.

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