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

tan θ = \frac{y}{x}

then we can find a unique solution based on the quadrant of the xy-plane in which original point

(x, y, z)

Notice that these equations are derived from properties of right triangles. To make this easy to see, consider point

P

Let’s consider the differences between rectangular and cylindrical coordinates by looking at the surfaces generated when each of the coordinates is held constant. If

c

are all planes. Planes of these forms are parallel to the yz-plane, the xz-plane, and the xy-plane, respectively. When we convert to cylindrical coordinates, the z-coordinate does not change. Therefore, in cylindrical coordinates, surfaces of the form

z = c

are planes parallel to the xy-plane. Now, let’s think about surfaces of the form

r = c .

The points on these surfaces are at a fixed distance from the z-axis. In other words, these surfaces are vertical circular cylinders. Last, what about

θ = c ?

are at a fixed angle from the x-axis, which gives us a half-plane that starts at the z-axis ([link] and [link]).

If this process seems familiar, it is with good reason. This is exactly the same process that we followed in Introduction to Parametric Equations and Polar Coordinates to convert from polar coordinates to two-dimensional rectangular coordinates.

The use of cylindrical coordinates is common in fields such as physics. Physicists studying electrical charges and the capacitors used to store these charges have discovered that these systems sometimes have a cylindrical symmetry. These systems have complicated modeling equations in the Cartesian coordinate system, which make them difficult to describe and analyze. The equations can often be expressed in more simple terms using cylindrical coordinates. For example, the cylinder described by equation

x^{2} + y^{2} = 25

Spherical Coordinates

In the Cartesian coordinate system, the location of a point in space is described using an ordered triple in which each coordinate represents a distance. In the cylindrical coordinate system, location of a point in space is described using two distances

(r and z)

In the spherical coordinate system, we again use an ordered triple to describe the location of a point in space. In this case, the triple describes one distance and two angles. Spherical coordinates make it simple to describe a sphere, just as cylindrical coordinates make it easy to describe a cylinder. Grid lines for spherical coordinates are based on angle measures, like those for polar coordinates.

The formulas to convert from spherical coordinates to rectangular coordinates may seem complex, but they are straightforward applications of trigonometry. Looking at [link], it is easy to see that

r = ρ sin φ .

is similar. [link] also shows that

ρ^{2} = r^{2} + z^{2} = x^{2} + y^{2} + z^{2}

(from the first equation) yields

φ = arccos (\frac{z}{\sqrt{r^{2} + z^{2}}}) .

Also, note that, as before, we must be careful when using the formula

tan θ = \frac{y}{x}

As we did with cylindrical coordinates, let’s consider the surfaces that are generated when each of the coordinates is held constant. Let

c

Points on these surfaces are at a fixed distance from the origin and form a sphere. The coordinate

θ

in the spherical coordinate system is the same as in the cylindrical coordinate system, so surfaces of the form

θ = c

The points on these surfaces are at a fixed angle from the z-axis and form a half-cone ([link]).

Identifying Surfaces in the Spherical Coordinate System

Describe the surfaces with the given spherical equations.

$θ = \frac{π}{3}$
$φ = \frac{5 π}{6}$
$ρ = 6$
$ρ = sin θ sin φ$

The variable $θ$
represents the measure of the same angle in both the cylindrical and spherical coordinate systems. Points with coordinates
$(ρ, \frac{π}{3}, φ)$
lie on the plane that forms angle
$θ = \frac{π}{3}$
with the positive x-axis. Because
$ρ > 0,$
the surface described by equation
$θ = \frac{π}{3}$
is the half-plane shown in [link].
Equation $φ = \frac{5 π}{6}$
describes all points in the spherical coordinate system that lie on a line from the origin forming an angle measuring
$\frac{5 π}{6}$
rad with the positive z-axis. These points form a half-cone ([link]). Because there is only one value for
$φ$
that is measured from the positive z-axis, we do not get the full cone (with two pieces).

To find the equation in rectangular coordinates, use equation
$φ = arccos (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}) .$

$\begin{matrix} \frac{5 π}{6} & = & arccos (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}) \\ cos \frac{5 π}{6} & = & \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}} \\ - \frac{\sqrt{3}}{2} & = & \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}} \\ \frac{3}{4} & = & \frac{z^{2}}{x^{2} + y^{2} + z^{2}} \\ \frac{3 x^{2}}{4} + \frac{3 y^{2}}{4} + \frac{3 z^{2}}{4} & = & z^{2} \\ \frac{3 x^{2}}{4} + \frac{3 y^{2}}{4} - \frac{z^{2}}{4} & = & 0. \end{matrix}$

This is the equation of a cone centered on the z-axis.
Equation $ρ = 6$
describes the set of all points
$6$
units away from the origin—a sphere with radius
$6$
([link]).
To identify this surface, convert the equation from spherical to rectangular coordinates, using equations $y = ρ sin φ sin θ$
and
$ρ^{2} = x^{2} + y^{2} + z^{2} :$

$\begin{matrix} ρ & = & sin θ sin φ \\ ρ^{2} & = & ρ sin θ sin φ & Multiply both sides of the equation by ρ . \\ x^{2} + y^{2} + z^{2} & = & y & Substitute rectangular variables using the equations above. \\ x^{2} + y^{2} - y + z^{2} & = & 0 & Subtract y from both sides of the equation. \\ x^{2} + y^{2} - y + \frac{1}{4} + z^{2} & = & \frac{1}{4} & Complete the square. \\ x^{2} + {(y - \frac{1}{2})}^{2} + z^{2} & = & \frac{1}{4} . & Rewrite the middle terms as a perfect square. \end{matrix}$

The equation describes a sphere centered at point
$(0, \frac{1}{2}, 0)$
with radius
$\frac{1}{2} .$

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

x^{2} + y^{2} + z^{2} = c^{2}

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

4000

mi. We express angle measures in degrees rather than radians because latitude and longitude are measured in degrees.

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.

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.

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 $z = k r,$
where
$k$
is a constant. In spherical coordinates, we have seen that surfaces of the form
$φ = c$
are half-cones. Last, in rectangular coordinates, elliptic cones are quadric surfaces and can be represented by equations of the form
$z^{2} = \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} .$
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.

Key Concepts

Use the following figure as an aid in identifying the relationship between the rectangular, cylindrical, and spherical coordinate systems.

This figure is the first octant of the 3-dimensional coordinate system. There is a line segment from the origin upwards. It is labeled “rho.” The angle between this line segment and the z-axis is labeled “phi.” There is also a broken line from the origin to the shadow of the point. This line segment is in the x y-plane and is labeled “r.” The angle between r and the x-axis is labeled “theta.” For the following exercises, the cylindrical coordinates $(r, θ, z)$

of a point are given. Find the rectangular coordinates $(x, y, z)$

of the point.

(4, \frac{π}{6}, 3)

(2 \sqrt{3}, 2, 3)

(3, \frac{π}{3}, 5)

(4, \frac{7 π}{6}, 3)

(−2 \sqrt{3}, −2, 3)

(2, π, −4)

For the following exercises, the rectangular coordinates $(x, y, z)$

of a point are given. Find the cylindrical coordinates $(r, θ, z)$

of the point.

(1, \sqrt{3}, 2)

(2, \frac{π}{3}, 2)

(1, 1, 5)

(3, −3, 7)

(3 \sqrt{2}, - \frac{π}{4}, 7)

(−2 \sqrt{2}, 2 \sqrt{2}, 4)

For the following exercises, the equation of a surface in cylindrical coordinates is given.

Find the equation of the surface in rectangular coordinates. Identify and graph the surface.

[T] $r = 4$

A cylinder of equation $x^{2} + y^{2} = 16,$

with its center at the origin and rulings parallel to the z-axis,* * *

This figure is a right circular cylinder, vertical. It is inside of a box. The edges of the box represent the x, y, and z axes.

[T] $z = r^{2} {cos}^{2} θ$

[T] $r^{2} cos (2 θ) + z^{2} + 1 = 0$

Hyperboloid of two sheets of equation $− x^{2} + y^{2} - z^{2} = 1,$

with the y-axis as the axis of symmetry,* * *

This figure is a elliptic cone surface that is horizontal. It is inside of a box. The edges of the box represent the x, y, and z axes.

[T] $r = 3 sin θ$

[T] $r = 2 cos θ$

Cylinder of equation $x^{2} - 2 x + y^{2} = 0,$

with a center at $(1, 0, 0)$

and radius $1,$

with rulings parallel to the z-axis,* * *

This figure is a right circular cylinder, vertical. It is inside of a box. The edges of the box represent the x, y, and z axes.

[T] $r^{2} + z^{2} = 5$

[T] $r = 2 sec θ$

Plane of equation $x = 2,$

This figure is a vertical parallelogram where x = 2 and parallel to the y z-plane. It is inside of a box. The edges of the box represent the x, y, and z axes.

[T] $r = 3 csc θ$

For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in cylindrical coordinates.

z = 3

z = 3

x = 6

x^{2} + y^{2} + z^{2} = 9

r^{2} + z^{2} = 9

y = 2 x^{2}

x^{2} + y^{2} - 16 x = 0

r = 16 cos θ, r = 0

x^{2} + y^{2} - 3 \sqrt{x^{2} + y^{2}} + 2 = 0

For the following exercises, the spherical coordinates $(ρ, θ, φ)$

of a point are given. Find the rectangular coordinates $(x, y, z)$

of the point.

(3, 0, π)

(0, 0, −3)

(1, \frac{π}{6}, \frac{π}{6})

(12, - \frac{π}{4}, \frac{π}{4})

(6, −6, \sqrt{2})

(3, \frac{π}{4}, \frac{π}{6})

For the following exercises, the rectangular coordinates $(x, y, z)$

of a point are given. Find the spherical coordinates $(ρ, θ, φ)$

of the point. Express the measure of the angles in degrees rounded to the nearest integer.

(4, 0, 0)

(4, 0, 90 °)

(−1, 2, 1)

(0, 3, 0)

(3, 90 °, 90 °)

(−2, 2 \sqrt{3}, 4)

For the following exercises, the equation of a surface in spherical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface.

[T] $ρ = 3$

Sphere of equation $x^{2} + y^{2} + z^{2} = 9$

centered at the origin with radius $3,$

This figure is a sphere. It is inside of a box. The edges of the box represent the x, y, and z axes.

[T] $φ = \frac{π}{3}$

[T] $ρ = 2 cos φ$

Sphere of equation $x^{2} + y^{2} + {(z - 1)}^{2} = 1$

centered at $(0, 0, 1)$

with radius $1,$

This figure is a sphere of radius 1 centered in a box. The center of the sphere is the point (0, 0, 1).

[T] $ρ = 4 csc φ$

[T] $φ = \frac{π}{2}$

The xy-plane of equation $z = 0,$

This figure is a parallelogram representing a plane. It is parallel to the x y-plane at z = 0. It is inside of a box. The edges of the box represent the x, y, and z axes.

[T] $ρ = 6 csc φ sec θ$

For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in spherical coordinates. Identify the surface.

x^{2} + y^{2} - 3 z^{2} = 0,

z \neq 0

φ = \frac{π}{3}

or $φ = \frac{2 π}{3};$

Elliptic cone

x^{2} + y^{2} + z^{2} - 4 z = 0

z = 6

ρ cos φ = 6;

Plane at $z = 6$

x^{2} + y^{2} = 9

For the following exercises, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle $φ$

in radians rounded to four decimal places.

[T] $(1, \frac{π}{4}, 3)$

(\sqrt{10}, \frac{π}{4}, 0.3218)

[T] $(5, π, 12)$

(3, \frac{π}{2}, 3)

(3 \sqrt{2}, \frac{π}{2}, \frac{π}{4})

(3, - \frac{π}{6}, 3)

For the following exercises, the spherical coordinates of a point are given. Find its associated cylindrical coordinates.

(2, - \frac{π}{4}, \frac{π}{2})

(2, - \frac{π}{4}, 0)

(4, \frac{π}{4}, \frac{π}{6})

(8, \frac{π}{3}, \frac{π}{2})

(8, \frac{π}{3}, 0)

(9, - \frac{π}{6}, \frac{π}{3})

For the following exercises, find the most suitable system of coordinates to describe the solids.

The solid situated in the first octant with a vertex at the origin and enclosed by a cube of edge length $a,$

where $a > 0$

Cartesian system, ${(x, y, z) \| 0 \leq x \leq a, 0 \leq y \leq a, 0 \leq z \leq a}$

A spherical shell determined by the region between two concentric spheres centered at the origin, of radii of $a$

and $b,$

respectively, where $b > a > 0$

A solid inside sphere $x^{2} + y^{2} + z^{2} = 9$

and outside cylinder ${(x - \frac{3}{2})}^{2} + y^{2} = \frac{9}{4}$

Cylindrical system, ${(r, θ, z) \| r^{2} + z^{2} \leq 9, r \geq 3 cos θ, 0 \leq θ \leq 2 π}$

A cylindrical shell of height $10$

determined by the region between two cylinders with the same center, parallel rulings, and radii of $2$

and $5,$

respectively

[T] Use a CAS to graph in cylindrical coordinates the region between elliptic paraboloid $z = x^{2} + y^{2}$

and cone $x^{2} + y^{2} - z^{2} = 0 .$

The region is described by the set of points ${(r, θ, z) \| 0 \leq r \leq 1, 0 \leq θ \leq 2 π, r^{2} \leq z \leq r} .$

This figure is a paraboloid, vertical. It is inside of a box. The edges of the box represent the x, y, and z axes.

[T] Use a CAS to graph in spherical coordinates the “ice cream-cone region” situated above the xy-plane between sphere $x^{2} + y^{2} + z^{2} = 4$

and elliptical cone $x^{2} + y^{2} - z^{2} = 0 .$

Washington, DC, is located at $39 °$

N and $77 °$

W (see the following figure). Assume the radius of Earth is $4000$

mi. Express the location of Washington, DC, in spherical coordinates.

This figure is an image of a globe. On the globe there is a point labeled where Washington, DC is located. It is labeled with 39 degrees north latitude and 77 degrees west longitude.

(4000, − 77 °, 51 °)

San Francisco is located at $37.78 ° N$

and $122.42 ° W .$

Assume the radius of Earth is $4000$

mi. Express the location of San Francisco in spherical coordinates.

Find the latitude and longitude of Rio de Janeiro if its spherical coordinates are $(4000, − 43.17 °, 102.91 °) .$

43.17 ° W,

22.91 ° S

Find the latitude and longitude of Berlin if its spherical coordinates are $(4000, 13.38 °, 37.48 °) .$

[T] Consider the torus of equation ${(x^{2} + y^{2} + z^{2} + R^{2} - r^{2})}^{2} = 4 R^{2} (x^{2} + y^{2}),$

where $R \geq r > 0 .$

Write the equation of the torus in spherical coordinates.
If $R = r,$
the surface is called a horn torus. Show that the equation of a horn torus in spherical coordinates is
$ρ = 2 R sin φ .$
Use a CAS to graph the horn torus with $R = r = 2$
in spherical coordinates.

a. $ρ = 0,$

ρ + R^{2} - r^{2} - 2 R sin φ = 0;

c.* * *

This figure is a torus. It is inside of a box. The edges of the box represent the x, y, and z axes.

[T] The “bumpy sphere” with an equation in spherical coordinates is $ρ = a + b cos (m θ) sin (n φ),$

with $θ \in [0, 2 π]$

and $φ \in [0, π],$

where $a$

and $b$

are positive numbers and $m$

and $n$

are positive integers, may be used in applied mathematics to model tumor growth.

Show that the “bumpy sphere” is contained inside a sphere of equation $ρ = a + b .$
Find the values of
$θ$
and
$φ$
at which the two surfaces intersect.
Use a CAS to graph the surface for $a = 14,$ $b = 2,$ $m = 4,$
and
$n = 6$
along with sphere
$ρ = a + b .$
Find the equation of the intersection curve of the surface at b. with the cone $φ = \frac{π}{12} .$
Graph the intersection curve in the plane of intersection.

Chapter Review Exercises

For the following exercises, determine whether the statement is true or false. Justify the answer with a proof or a counterexample.

For the following exercises, find the vector and parametric equations of the line with the given properties.

For the following exercises, find the equation of the plane with the given properties.

For the following exercises, find the traces for the surfaces in planes

x = k, y = k, and z = k .

For the following exercises, write the given equation in cylindrical coordinates and spherical coordinates.

For the following exercises, convert the given equations from cylindrical or spherical coordinates to rectangular coordinates. Identify the given surface.

The following problems consider your unsuccessful attempt to take the tire off your car using a wrench to loosen the bolts. Assume the wrench is

0.3

Cylindrical and Spherical Coordinates

Cylindrical Coordinates

Spherical Coordinates

Key Concepts

Chapter Review Exercises

Glossary