2.7 Cylindrical and spherical coordinates

The Cartesian coordinate system provides a straightforward way to describe the location of points in space. Some surfaces, however, can be difficult to model with equations based on the Cartesian system. This is a familiar problem; recall that in two dimensions, polar coordinates often provide a useful alternative system for describing the location of a point in the plane, particularly in cases involving circles. In this section, we look at two different ways of describing the location of points in space, both of them based on extensions of polar coordinates. As the name suggests, cylindrical coordinates are useful for dealing with problems involving cylinders, such as calculating the volume of a round water tank or the amount of oil flowing through a pipe. Similarly, spherical coordinates are useful for dealing with problems involving spheres, such as finding the volume of domed structures.

Cylindrical coordinates

When we expanded the traditional Cartesian coordinate system from two dimensions to three, we simply added a new axis to model the third dimension. Starting with polar coordinates, we can follow this same process to create a new three-dimensional coordinate system, called the cylindrical coordinate system. In this way, cylindrical coordinates provide a natural extension of polar coordinates to three dimensions.

In the xy -plane, the right triangle shown in [link] provides the key to transformation between cylindrical and Cartesian, or rectangular, coordinates.

As when we discussed conversion from rectangular coordinates to polar coordinates in two dimensions, it should be noted that the equation $\text{tan}\theta =\frac{y}{x}$ has an infinite number of solutions. However, if we restrict $\theta$ to values between $0$ and $2\pi ,$ then we can find a unique solution based on the quadrant of the xy -plane in which original point $\left(x,y,z\right)$ is located. Note that if $x=0,$ then the value of $\theta$ is either $\frac{\pi }{2},\frac{3\pi }{2},$ or $0,$ depending on the value of $y.$

Notice that these equations are derived from properties of right triangles. To make this easy to see, consider point $P$ in the xy -plane with rectangular coordinates $\left(x,y,0\right)$ and with cylindrical coordinates $\left(r,\theta ,0\right),$ as shown in the following figure.