Spherical coordinates are useful in analyzing systems that have some degree of symmetry about a point, such as the volume of the space inside a domed stadium or wind speeds in a planet’s atmosphere. A sphere that has Cartesian equation
has the simple equation
in spherical coordinates.
In geography, latitude and longitude are used to describe locations on Earth’s surface, as shown in
[link] . Although the shape of Earth is not a perfect sphere, we use spherical coordinates to communicate the locations of points on Earth. Let’s assume Earth has the shape of a sphere with radius
mi. We express angle measures in degrees rather than radians because latitude and longitude are measured in degrees.
In the latitude–longitude system, angles describe the location of a point on Earth relative to the equator and the prime meridian.
Let the center of Earth be the center of the sphere, with the ray from the center through the North Pole representing the positive
z -axis. The prime meridian represents the trace of the surface as it intersects the
xz -plane. The equator is the trace of the sphere intersecting the
xy -plane.
Converting latitude and longitude to spherical coordinates
The latitude of Columbus, Ohio, is
N and the longitude is
W, which means that Columbus is
north of the equator. Imagine a ray from the center of Earth through Columbus and a ray from the center of Earth through the equator directly south of Columbus. The measure of the angle formed by the rays is
In the same way, measuring from the prime meridian, Columbus lies
to the west. Express the location of Columbus in spherical coordinates.
The radius of Earth is
mi, so
The intersection of the prime meridian and the equator lies on the positive
x -axis. Movement to the west is then described with negative angle measures, which shows that
Because Columbus lies
north of the equator, it lies
south of the North Pole, so
In spherical coordinates, Columbus lies at point
Cylindrical and spherical coordinates give us the flexibility to select a coordinate system appropriate to the problem at hand. A thoughtful choice of coordinate system can make a problem much easier to solve, whereas a poor choice can lead to unnecessarily complex calculations. In the following example, we examine several different problems and discuss how to select the best coordinate system for each one.
Choosing the best coordinate system
In each of the following situations, we determine which coordinate system is most appropriate and describe how we would orient the coordinate axes. There could be more than one right answer for how the axes should be oriented, but we select an orientation that makes sense in the context of the problem.
Note : There is not enough information to set up or solve these problems; we simply select the coordinate system (
[link] ).
Find the center of gravity of a bowling ball.
Determine the velocity of a submarine subjected to an ocean current.
Calculate the pressure in a conical water tank.
Find the volume of oil flowing through a pipeline.
Determine the amount of leather required to make a football.
(credit: (a) modification of work by scl hua, Wikimedia, (b) modification of work by DVIDSHUB, Flickr, (c) modification of work by Michael Malak, Wikimedia, (d) modification of work by Sean Mack, Wikimedia, (e) modification of work by Elvert Barnes, Flickr)
Clearly, a bowling ball is a sphere, so spherical coordinates would probably work best here. The origin should be located at the physical center of the ball. There is no obvious choice for how the
x -,
y - and
z -axes should be oriented. Bowling balls normally have a weight block in the center. One possible choice is to align the
z -axis with the axis of symmetry of the weight block.
A submarine generally moves in a straight line. There is no rotational or spherical symmetry that applies in this situation, so rectangular coordinates are a good choice. The
z -axis should probably point upward. The
x - and
y -axes could be aligned to point east and north, respectively. The origin should be some convenient physical location, such as the starting position of the submarine or the location of a particular port.
A cone has several kinds of symmetry. In cylindrical coordinates, a cone can be represented by equation
where
is a constant. In spherical coordinates, we have seen that surfaces of the form
are half-cones. Last, in rectangular coordinates, elliptic cones are quadric surfaces and can be represented by equations of the form
In this case, we could choose any of the three. However, the equation for the surface is more complicated in rectangular coordinates than in the other two systems, so we might want to avoid that choice. In addition, we are talking about a water tank, and the depth of the water might come into play at some point in our calculations, so it might be nice to have a component that represents height and depth directly. Based on this reasoning, cylindrical coordinates might be the best choice. Choose the
z -axis to align with the axis of the cone. The orientation of the other two axes is arbitrary. The origin should be the bottom point of the cone.
A pipeline is a cylinder, so cylindrical coordinates would be best the best choice. In this case, however, we would likely choose to orient our
z -axis with the center axis of the pipeline. The
x -axis could be chosen to point straight downward or to some other logical direction. The origin should be chosen based on the problem statement. Note that this puts the
z -axis in a horizontal orientation, which is a little different from what we usually do. It may make sense to choose an unusual orientation for the axes if it makes sense for the problem.
A football has rotational symmetry about a central axis, so cylindrical coordinates would work best. The
z -axis should align with the axis of the ball. The origin could be the center of the ball or perhaps one of the ends. The position of the
x -axis is arbitrary.
