Cartesian and Polar Coordinates

Why two systems?

If you have ever used graph paper, you already know Cartesian coordinates. A point is named by how far along and how far up it is.

That is not always the most convenient way to talk about a point. Sometimes you care about distance from a centre, and the angle around that centre. That is what polar coordinates are for.

This article covers both on the flat 2D plane, then the usual 3D versions: ordinary (x, y, z), cylindrical, and spherical coordinates.

2D Cartesian coordinates

Draw two axes that cross at right angles at the origin.

• x goes right
• y goes up

A point is written (x, y). For example (3, 4) means “3 right and 4 up”.

The distance from the origin to (x, y) uses Pythagoras:

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

So for (3, 4):

$r = \sqrt{3^2 + 4^2} = 5$

2D polar coordinates

Polar coordinates ask a different pair of questions:

• how far from the origin? Call that r
• what angle from the positive x-axis? Call that θ (theta)

A point is written (r, θ). Angles are usually measured anticlockwise from the positive x-axis.

Converting between them

From Cartesian to polar:

$r = \sqrt{x^2 + y^2}$ $\theta = \operatorname{atan2}(y, x)$

From polar to Cartesian:

$x = r \cos \theta$ $y = r \sin \theta$

Use atan2(y, x) rather than plain arctan(y/x). The plain version loses which quadrant you are in.

For (3, 4) you get r = 5 and θ ≈ 53°. Same point, two labels.

Which one is nicer?

A circle of radius 5 around the origin looks like this in Cartesian:

$x^2 + y^2 = 25$

And like this in polar:

$r = 5$

That is the whole point of polar coordinates: they make round things simple.

Use Cartesian for grids, screens, and left/right/up/down motion. Use polar when there is a natural centre and rotation around it.

Into 3D

Add a third axis, z, pointing “up” out of the page. A point becomes (x, y, z).

Distance from the origin is still Pythagoras, just with one more term:

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

In 3D, “polar” splits into two common systems.

Cylindrical coordinates

Keep 2D polar in the floor plane, then add height z.

A point is written (ρ, φ, z):

• ρ (rho): distance from the z-axis
• φ (phi): angle around the z-axis
• z: height

Useful for pipes, screws, and anything with a clear axis to spin around. A vertical cylinder of radius 2 is simply ρ = 2.

Conversions:

$\rho = \sqrt{x^2 + y^2}$ $x = \rho \cos \varphi,\quad y = \rho \sin \varphi,\quad z = z$

Spherical coordinates

Now measure everything from the origin, with one radius and two angles.

A point is written (r, θ, φ) in the common physics convention:

• r: distance from the origin
• θ: angle down from the positive z-axis (0 at the “north pole”)
• φ: angle around the z-axis (same idea as in cylindrical)

Useful for spheres, sky directions, and anything measured from a centre. A sphere of radius 3 is simply r = 3.

Conversions:

$x = r \sin \theta \cos \varphi$ $y = r \sin \theta \sin \varphi$ $z = r \cos \theta$

Watch out: maths textbooks sometimes swap the names of θ and φ. Always check the diagram in whatever book or page you are reading.

A quick comparison

Situation	Natural choice
Grid, screen, room	Cartesian
Circle / spin in a plane	Polar
Pipe / screw / shaft	Cylindrical
Sphere / sky directions	Spherical

Or shorter still:

• Cartesian measures along axes
• Polar-style systems measure by radius and angle(s)

A few gotchas

• At the origin, r = 0 and the angle is undefined.
• Angles wrap around: θ and θ + 360° are the same direction.
• In cylindrical, ρ is distance from the axis. In spherical, r is distance from the origin. Those are different.

Closing thought

Coordinate systems are not competing truths about space. They are competing labels.

The geometry stays the same. What changes is which numbers make the next equation short. If r = 5 is shorter than x² + y² = 25, that is not a different universe. It is the same circle, spoken in a dialect that fits it better.