Cylindrical Coordinates

Cylindrical coordinates are just the extension of polar coordinates into three dimensional space. Before, we described every point on the plane with two variables r and theta, now we will describe every point in R³ with three variables: r, theta, and z.

We define r and theta in such a way that the old formulas hold:

Finally, we define z to be the same as it is in cartesian coordinates: the distance from the point to the xy-plane. Every point in space now has a triplet (r, theta, z) of cylindrical coordinates, and if we restrict 0 <= theta < 2 pi, then every point not on the z-axis (i.e, r > 0) has an unique cylindrical coordinate.

Cylindrical coordinates are useful in describing geometric objects with (surprise) cylindrical symmetry: rotational symmetry about the z-axis. For example, the implicit equation r = 3 describes an infinite cylinder with radius 3 about the z-axis. An object with cylindrical symmetry will not have any mention of theta when you convert it to cylindrical coordinates.

Often, the best way to convert equations from cylindrical coordinates to cartesian coordinates or vice-versa is to just blindly substitute and not think very much. For example, if I wanted to translate the sphere x² + y² + z² = 1 into cylindrical, I could just replace every x with r cos(theta) and every y with r sin(theta) to get r² cos² (theta) + r² sin² (theta) + z² = 1. Then, if I had some extra time, I could simplify to r² + z² = 1.

The following table summarizes all the conversions you will need:

Exercises:

1. What does the set of all points with r = 5 look like?
2. What does the set of all points with theta = pi / 5 look like?
3. What does a plane look like a cylindrical coordinates? What about a line?
4. Let's say I define a new coordinate system called cylindrical-x. I want cylindrical-x to be just like cylindrical coordinates, but I want r to be the distance from the x-axis instead of the distance from the z-axis. What should the other two variables in the cylindrical-x system be called, and how should they be defined?
5. Convert the point with cartesian coordinates (1,2,3) to cylindrical coordinates.
6. Convert the point with cylindrical coordinates (r = 2, theta = pi, z = 5) to cartesian coordinates.