KAHIBARO
Table of Contents
Why coordinate transformations matter
A physical point in space does not change when we describe it in a different coordinate system. Only the numbers used to label that point change. Coordinate transformations are the rules that connect one set of coordinates to another.
For example, the same point can be written as $(x,y)$ in Cartesian coordinates, $(r,\theta)$ in polar coordinates, $(r,\phi,z)$ in cylindrical coordinates, or $(r,\theta,\phi)$ in spherical coordinates. A coordinate transformation tells us how to move from one description to another without changing the actual location of the point.
This is important in physics because many problems become easier in the coordinate system that matches the symmetry of the situation. Motion along a straight line often fits Cartesian coordinates well. Circular motion often fits polar coordinates better. Problems with cylinders or spheres are often easier in cylindrical or spherical coordinates.
The basic idea
Suppose a point in a plane has Cartesian coordinates $(x,y)$ and polar coordinates $(r,\theta)$. These are two different labels for the same point. The transformation is just a change in description.
A transformation can go in two directions. We can start from one coordinate system and calculate the coordinates in another system. We can also go back.
A good transformation must be clear and consistent. If two coordinate systems describe the same point, the transformed values must refer to exactly the same physical location.
A coordinate transformation changes the numerical description of a point, not the point itself.
Transformations between Cartesian and polar coordinates
In two dimensions, Cartesian and polar coordinates are closely related.
If a point has polar coordinates $(r,\theta)$, then its Cartesian coordinates are
$$
x = r\cos\theta
$$
$$
y = r\sin\theta
$$
These formulas come from right triangle geometry.
To go the other way, from Cartesian to polar, we use
$$
r = \sqrt{x^2 + y^2}
$$
and
$$
\tan\theta = \frac{y}{x}
$$
However, the angle must be chosen carefully because the tangent alone does not determine the correct quadrant. In practice, one uses the signs of $x$ and $y$ to determine the correct angle.
For Cartesian to polar conversion,
$$
r = \sqrt{x^2 + y^2}
$$
and the angle $\theta$ must be chosen in the correct quadrant.
Geometric picture in the plane
The relationship between Cartesian and polar coordinates can be seen directly from a diagram.
Transformations between Cartesian and cylindrical coordinates
Cylindrical coordinates are useful in three dimensions when there is circular symmetry around an axis, usually the $z$ axis. A point is described by $(r,\phi,z)$, where $r$ is the distance from the $z$ axis, $\phi$ is the angle around the axis, and $z$ is the same vertical coordinate used in Cartesian coordinates.
From cylindrical to Cartesian,
$$
x = r\cos\phi
$$
$$
y = r\sin\phi
$$
$$
z = z
$$
From Cartesian to cylindrical,
$$
r = \sqrt{x^2 + y^2}
$$
$$
\tan\phi = \frac{y}{x}
$$
$$
z = z
$$
Notice that cylindrical coordinates are essentially polar coordinates in the $xy$ plane, together with the usual height $z$.
Transformations between Cartesian and spherical coordinates
Spherical coordinates are useful when a problem has symmetry about a point, such as the gravitational field around a planet. A point is described by $(r,\theta,\phi)$ in one common convention, where $r$ is the distance from the origin, $\theta$ is the polar angle measured from the positive $z$ axis, and $\phi$ is the azimuthal angle in the $xy$ plane.
Using this convention, from spherical to Cartesian we have
$$
x = r\sin\theta\cos\phi
$$
$$
y = r\sin\theta\sin\phi
$$
$$
z = r\cos\theta
$$
From Cartesian to spherical,
$$
r = \sqrt{x^2 + y^2 + z^2}
$$
$$
\theta = \cos^{-1}\left(\frac{z}{r}\right)
$$
$$
\tan\phi = \frac{y}{x}
$$
Again, the angle $\phi$ must be placed in the correct quadrant.
In spherical coordinates, always check which convention is being used for the angles. Different books may swap the names of $\theta$ and $\phi$.
Geometric picture in three dimensions
The spherical coordinates of a point can be understood by first projecting the point onto the $xy$ plane.
Direct transformations between cylindrical and spherical coordinates
Sometimes it is useful to transform directly between cylindrical and spherical coordinates.
If spherical coordinates are $(r,\theta,\phi)$ and cylindrical coordinates are $(\rho,\phi,z)$, then
$$
\rho = r\sin\theta
$$
$$
z = r\cos\theta
$$
$$
\phi = \phi
$$
Also,
$$
r = \sqrt{\rho^2 + z^2}
$$
$$
\theta = \cos^{-1}\left(\frac{z}{r}\right)
$$
Here $\rho$ is often used for cylindrical radius to avoid confusion with spherical radius $r$.
Summary table of common transformations
| From | To | Transformation |
|---|---|---|
| Polar $(r,\theta)$ | Cartesian $(x,y)$ | $x=r\cos\theta,\ y=r\sin\theta$ |
| Cartesian $(x,y)$ | Polar $(r,\theta)$ | $r=\sqrt{x^2+y^2},\ \tan\theta=y/x$ |
| Cylindrical $(r,\phi,z)$ | Cartesian $(x,y,z)$ | $x=r\cos\phi,\ y=r\sin\phi,\ z=z$ |
| Cartesian $(x,y,z)$ | Cylindrical $(r,\phi,z)$ | $r=\sqrt{x^2+y^2},\ \tan\phi=y/x,\ z=z$ |
| Spherical $(r,\theta,\phi)$ | Cartesian $(x,y,z)$ | $x=r\sin\theta\cos\phi,\ y=r\sin\theta\sin\phi,\ z=r\cos\theta$ |
| Cartesian $(x,y,z)$ | Spherical $(r,\theta,\phi)$ | $r=\sqrt{x^2+y^2+z^2},\ \theta=\cos^{-1}(z/r),\ \tan\phi=y/x$ |
Transforming equations
Coordinate transformations are not only for points. They also let us rewrite equations.
For example, the circle centered at the origin with radius $a$ has Cartesian equation
$$
x^2 + y^2 = a^2
$$
Since in polar coordinates $x^2 + y^2 = r^2$, the same circle becomes simply
$$
r = a
$$
This shows why choosing a good coordinate system can simplify a problem.
As another example, the sphere centered at the origin with radius $a$ has Cartesian equation
$$
x^2 + y^2 + z^2 = a^2
$$
In spherical coordinates this becomes
$$
r = a
$$
The geometry is the same, but the equation is much simpler.
A smart choice of coordinates can turn a complicated equation into a very simple one.
Common difficulties
One common difficulty is angle convention. In two dimensions, angles are usually measured from the positive $x$ axis. In spherical coordinates, the polar angle is often measured from the positive $z$ axis, but some sources use different notation.
Another difficulty is that more than one angle can describe the same direction if full rotations are allowed. For instance, $\theta$ and $\theta + 2\pi$ represent the same direction in polar coordinates.
A third difficulty is the origin. At the origin, some angular coordinates are not defined, because a point at zero distance has no unique direction.
At the origin, angular coordinates are undefined. Only the radial distance is zero.
A simple worked example
Suppose a point has Cartesian coordinates $(3,4)$.
Its polar radius is
$$
r = \sqrt{3^2 + 4^2} = 5
$$
Its angle satisfies
$$
\tan\theta = \frac{4}{3}
$$
so
$$
\theta = \tan^{-1}\left(\frac{4}{3}\right)
$$
because the point lies in the first quadrant.
So the polar coordinates are $(5,\theta)$ with
$$
\theta = \tan^{-1}\left(\frac{4}{3}\right)
$$
Now consider a three-dimensional point $(x,y,z) = (3,4,12)$.
Its spherical radius is
$$
r = \sqrt{3^2 + 4^2 + 12^2} = \sqrt{169} = 13
$$
Its azimuthal angle is
$$
\phi = \tan^{-1}\left(\frac{4}{3}\right)
$$
Its polar angle is
$$
\theta = \cos^{-1}\left(\frac{12}{13}\right)
$$
So its spherical coordinates are $(13,\theta,\phi)$.
Physical meaning
In physics, coordinate transformations help us describe the same reality in the most useful way. A planet orbiting a star, the electric field around a charged sphere, and fluid flow through a pipe may all be easier to understand after changing coordinates.
The important idea is that physics does not depend on the labels we choose. Coordinates are tools. Transformations let us move from one tool to another while describing the same physical world.