Coordinate systems are tools that let us use algebraic methods to
understand geometry. While the
**rectangular** (also called
**Cartesian**) coordinates that we
have been discussing are the most common, some problems are easier to
analyze in alternate coordinate systems.

A coordinate system is a scheme that allows us to identify any point in the plane or in three-dimensional space by a set of numbers. In rectangular coordinates these numbers are interpreted, roughly speaking, as the lengths of the sides of a rectangular "box.''

In two dimensions you may already be familiar with an alternative,
called
**polar coordinates**.
In this system, each
point in the plane is identified by a pair of numbers $(r,\theta)$.
The number $\theta$ measures the angle between the positive
$x$-axis and a vector with tail at the origin and head at the
point, as shown in figure 12.6.1; the number
$r$ measures the distance from the origin to the
point. Either of these may be negative; a negative $\theta$ indicates
the angle is measured clockwise from the positive
$x$-axis instead of counter-clockwise, and a negative $r$ indicates
the point at distance $|r|$ in the opposite of the direction given by
$\theta$.
Figure 12.6.1 also shows the point with
rectangular coordinates $\ds (1,\sqrt3)$ and polar coordinates
$(2,\pi/3)$, 2 units from the origin and $\pi/3$ radians from the
positive $x$-axis.

We can extend polar coordinates to three dimensions simply by adding a
$z$ coordinate; this is called
**cylindrical coordinates**
.
Each point in three-dimensional space is represented by three
coordinates $(r,\theta,z)$ in the obvious way: this point is $z$ units
above or below the point $(r,\theta)$ in the $x$-$y$ plane,
as shown in figure 12.6.2. The point
with rectangular coordinates $\ds (1,\sqrt3, 3)$ and cylindrical
coordinates $(2,\pi/3,3)$ is also indicated
in figure 12.6.2.

Some figures with relatively complicated equations in rectangular coordinates will be represented by simpler equations in cylindrical coordinates. For example, the cylinder in figure 12.6.3 has equation $\ds x^2+y^2=4$ in rectangular coordinates, but equation $r=2$ in cylindrical coordinates.

Given a point $(r,\theta)$ in polar coordinates, it is easy to see (as in figure 12.6.1) that the rectangular coordinates of the same point are $(r\cos\theta,r\sin\theta)$, and so the point $(r,\theta,z)$ in cylindrical coordinates is $(r\cos\theta,r\sin\theta,z)$ in rectangular coordinates. This means it is usually easy to convert any equation from rectangular to cylindrical coordinates: simply substitute $$\eqalign{ x&=r\cos\theta\cr y&=r\sin\theta\cr} $$ and leave $z$ alone. For example, starting with $\ds x^2+y^2=4$ and substituting $x=r\cos\theta$, $y=r\sin\theta$ gives $$\eqalign{ r^2\cos^2\theta+r^2\sin^2\theta&=4\cr r^2(\cos^2\theta+\sin^2\theta)&=4\cr r^2&=4\cr r&=2.\cr }$$ Of course, it's easy to see directly that this defines a cylinder as mentioned above.

Cylindrical coordinates are an obvious extension of polar coordinates
to three dimensions, but the use of the $z$ coordinate means they are
not as closely analogous to polar coordinates as another standard
coordinate system. In polar coordinates, we identify a point by a
direction and distance from the origin; in three dimensions we can do
the same thing, in a variety of ways. The question is: how do we
represent a direction? One way is to give the angle of rotation,
$\theta$, from the positive $x$ axis, just as in cylindrical
coordinates, and also an angle of rotation, $\phi$, from the positive
$z$ axis. Roughly speaking, $\theta$ is like longitude and $\phi$ is
like latitude. (Earth longitude is measured as a positive or negative
angle from the prime meridian, and is always between 0 and 180
degrees, east or west; $\theta$ can be any positive or negative angle,
and we use radians except in informal circumstances.
Earth latitude is measured north or south
from the equator; $\phi$ is measured from the north pole down.) This
system is called
**spherical coordinates**
;
the coordinates are listed in the order
$(\rho,\theta,\phi)$, where $\rho$ is the distance from the
origin, and like $r$ in cylindrical coordinates it may be negative.
The general case and an
example are pictured in figure 12.6.4; the
length marked $r$ is the $r$ of cylindrical coordinates.

As with cylindrical coordinates, we can easily convert equations in rectangular coordinates to the equivalent in spherical coordinates, though it is a bit more difficult to discover the proper substitutions. Figure 12.6.5 shows the typical point in spherical coordinates from figure 12.6.4, viewed now so that the arrow marked $r$ in the original graph appears as the horizontal "axis'' in the left hand graph. From this diagram it is easy to see that the $z$ coordinate is $\rho\cos\phi$, and that $r=\rho\sin\phi$, as shown. Thus, in converting from rectangular to spherical coordinates we will replace $z$ by $\rho\cos\phi$. To see the substitutions for $x$ and $y$ we now view the same point from above, as shown in the right hand graph. The hypotenuse of the triangle in the right hand graph is $r=\rho\sin\phi$, so the sides of the triangle, as shown, are $x=r\cos\theta=\rho\sin\phi\cos\theta$ and $y=r\sin\theta=\rho\sin\phi\sin\theta$. So the upshot is that to convert from rectangular to spherical coordinates, we make these substitutions: $$\eqalign{ x&=\rho\sin\phi\cos\theta\cr y&=\rho\sin\phi\sin\theta\cr z&=\rho\cos\phi.\cr} $$

Example 12.6.1 As the cylinder had a simple equation in cylindrical coordinates, so does the sphere in spherical coordinates: $\rho=2$ is the sphere of radius 2. If we start with the Cartesian equation of the sphere and substitute, we get the spherical equation: $$\eqalign{ x^2+y^2+z^2&=2^2\cr \rho^2\sin^2\phi\cos^2\theta+ \rho^2\sin^2\phi\sin^2\theta+\rho^2\cos^2\phi&=2^2\cr \rho^2\sin^2\phi(\cos^2\theta+\sin^2\theta)+\rho^2\cos^2\phi&=2^2\cr \rho^2\sin^2\phi+\rho^2\cos^2\phi&=2^2\cr \rho^2(\sin^2\phi+\cos^2\phi)&=2^2\cr \rho^2&=2^2\cr \rho&=2\cr }$$

Example 12.6.2 Find an equation for the cylinder $\ds x^2+y^2=4$ in spherical coordinates.

Proceeding as in the previous example: $$\eqalign{ x^2+y^2&=4\cr \rho^2\sin^2\phi\cos^2\theta+ \rho^2\sin^2\phi\sin^2\theta=4\cr \rho^2\sin^2\phi(\cos^2\theta+\sin^2\theta)&=4\cr \rho^2\sin^2\phi&=4\cr \rho\sin\phi&=2\cr \rho&={2\over\sin\phi}\cr }$$

## Exercises 12.6

**Ex 12.6.1**
Convert the following points in rectangular coordinates to
cylindrical and spherical coordinates:

a. $(1,1,1)$

b. $(7,-7,5)$

c. $(\cos(1),\sin(1),1)$

d. $(0,0,-\pi)$ (answer)

**Ex 12.6.2**
Find an equation for the sphere $\ds x^2+y^2+z^2=4$ in
cylindrical coordinates.
(answer)

**Ex 12.6.3**
Find an equation for the $y$-$z$ plane in cylindrical
coordinates.
(answer)

**Ex 12.6.4**
Find an equation equivalent to $\ds x^2+y^2+2z^2+2z-5=0$ in
cylindrical coordinates.
(answer)

**Ex 12.6.5**
Suppose the curve $\ds \ds z=e^{-x^2}$ in the $x$-$z$ plane is
rotated around the $z$ axis. Find an equation for the resulting
surface in cylindrical coordinates.
(answer)

**Ex 12.6.6**
Suppose the curve $\ds z=x$ in the $x$-$z$ plane is
rotated around the $z$ axis. Find an equation for the resulting
surface in cylindrical coordinates.
(answer)

**Ex 12.6.7**
Find an equation for the plane $y=0$ in
spherical coordinates.
(answer)

**Ex 12.6.8**
Find an equation for the plane $z=1$ in
spherical coordinates.
(answer)

**Ex 12.6.9**
Find an equation for the sphere with radius 1 and center at
$(0,1,0)$ in spherical coordinates.
(answer)

**Ex 12.6.10**
Find an equation for the cylinder $\ds x^2+y^2=9$ in
spherical coordinates.
(answer)

**Ex 12.6.11**
Suppose the curve $\ds z=x$ in the $x$-$z$ plane is
rotated around the $z$ axis. Find an equation for the resulting
surface in spherical coordinates.
(answer)

**Ex 12.6.12**
Plot the polar equations $r=\sin(\theta)$ and $r=\cos(\theta)$
and comment on their similarities. (If you get stuck on how to plot
these, you can multiply both sides of each equation by $r$ and convert
back to rectangular coordinates).

**Ex 12.6.13**
Extend exercises 6
and
11 by rotating the curve $z=mx$
around the $z$ axis and converting to both cylindrical and spherical
coordinates.
(answer)

**Ex 12.6.14**
Convert the spherical formula $\rho=\sin \theta \sin \phi$ to
rectangular coordinates and describe the surface defined by the
formula (Hint: Multiply both sides by $\rho$.)
(answer)

**Ex 12.6.15**
We can describe points in the first octant by $x >0$, $y>0$ and
$z>0$. Give similar inequalities for the first octant in cylindrical
and spherical coordinates.
(answer)