Vectors in Space

# 14 Cylindrical and Spherical Coordinates

### Learning Objectives

• Convert from cylindrical to rectangular coordinates.
• Convert from rectangular to cylindrical coordinates.
• Convert from spherical to rectangular coordinates.
• Convert from rectangular to 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.

Definition

In the cylindrical coordinate system, a point in space ((Figure)) is represented by the ordered triple where

• are the polar coordinates of the point’s projection in the xy-plane
• is the usual in the Cartesian coordinate system
The right triangle lies in the xy-plane. The length of the hypotenuse is and is the measure of the angle formed by the positive x-axis and the hypotenuse. The
z-coordinate describes the location of the point above or below the xy-plane.

In the xy-plane, the right triangle shown in (Figure) provides the key to transformation between cylindrical and Cartesian, or rectangular, coordinates.

Conversion between Cylindrical and Cartesian Coordinates

The rectangular coordinates and the cylindrical coordinates of a point are related as follows:

As when we discussed conversion from rectangular coordinates to polar coordinates in two dimensions, it should be noted that the equation has an infinite number of solutions. However, if we restrict to values between and then we can find a unique solution based on the quadrant of the xy-plane in which original point is located. Note that if then the value of is either or depending on the value of

Notice that these equations are derived from properties of right triangles. To make this easy to see, consider point in the xy-plane with rectangular coordinates and with cylindrical coordinates as shown in the following figure.

The Pythagorean theorem provides equation Right-triangle relationships tell us that and

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 is a constant, then in rectangular coordinates, surfaces of the form or 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 are planes parallel to the xy-plane. Now, let’s think about surfaces of the form 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 The points on a surface of the form are at a fixed angle from the x-axis, which gives us a half-plane that starts at the z-axis ((Figure) and (Figure)).

In rectangular coordinates, (a) surfaces of the form are planes parallel to the yz-plane, (b) surfaces of the form are planes parallel to the xz-plane, and (c) surfaces of the form are planes parallel to the xy-plane.

In cylindrical coordinates, (a) surfaces of the form are vertical cylinders of radius (b) surfaces of the form are half-planes at angle from the x-axis, and (c) surfaces of the form are planes parallel to the xy-plane.

Converting from Cylindrical to Rectangular Coordinates

Plot the point with cylindrical coordinates and express its location in rectangular coordinates.

Conversion from cylindrical to rectangular coordinates requires a simple application of the equations listed in (Figure):

The point with cylindrical coordinates has rectangular coordinates (see the following figure).

The projection of the point in the xy-plane is 4 units from the origin. The line from the origin to the point’s projection forms an angle of with the positive x-axis. The point lies units below the xy-plane.

Point has cylindrical coordinates . Plot and describe its location in space using rectangular, or Cartesian, coordinates.

The rectangular coordinates of the point are

Hint

The first two components match the polar coordinates of the point in the xy-plane.

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.

Converting from Rectangular to Cylindrical Coordinates

Convert the rectangular coordinates to cylindrical coordinates.

Use the second set of equations from (Figure) to translate from rectangular to cylindrical coordinates:

We choose the positive square root, so Now, we apply the formula to find In this case, is negative and is positive, which means we must select the value of between and

In this case, the z-coordinates are the same in both rectangular and cylindrical coordinates:

The point with rectangular coordinates has cylindrical coordinates approximately equal to

Convert point from Cartesian coordinates to cylindrical coordinates.

Hint

and

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 in the Cartesian system can be represented by cylindrical equation

Identifying Surfaces in the Cylindrical Coordinate System

Describe the surfaces with the given cylindrical equations.

1. When the angle is held constant while and are allowed to vary, the result is a half-plane (see the following figure).
In polar coordinates, the equation describes the ray extending diagonally through the first quadrant. In three dimensions, this same equation describes a half-plane.

2. Substitute into equation to express the rectangular form of the equation: This equation describes a sphere centered at the origin with radius (see the following figure).
The sphere centered at the origin with radius can be described by the cylindrical equation

3. To describe the surface defined by equation is it useful to examine traces parallel to the xy-plane. For example, the trace in plane is circle the trace in plane is circle and so on. Each trace is a circle. As the value of increases, the radius of the circle also increases. The resulting surface is a cone (see the following figure).
The traces in planes parallel to the xy-plane are circles. The radius of the circles increases as increases.

Describe the surface with cylindrical equation

This surface is a cylinder with radius

Hint

The and components of points on the surface can take any value.

### 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 and an angle measure 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.

Definition

In the spherical coordinate system, a point in space ((Figure)) is represented by the ordered triple where

• (the Greek letter rho) is the distance between and the origin
• is the same angle used to describe the location in cylindrical coordinates;
• (the Greek letter phi) is the angle formed by the positive z-axis and line segment where is the origin and
The relationship among spherical, rectangular, and cylindrical coordinates.

By convention, the origin is represented as in spherical coordinates.

Converting among Spherical, Cylindrical, and Rectangular Coordinates

Rectangular coordinates and spherical coordinates of a point are related as follows:

If a point has cylindrical coordinates then these equations define the relationship between cylindrical and spherical coordinates.

The formulas to convert from spherical coordinates to rectangular coordinates may seem complex, but they are straightforward applications of trigonometry. Looking at (Figure), it is easy to see that Then, looking at the triangle in the xy-plane with as its hypotenuse, we have The derivation of the formula for is similar. (Figure) also shows that and Solving this last equation for and then substituting (from the first equation) yields Also, note that, as before, we must be careful when using the formula to choose the correct value of

The equations that convert from one system to another are derived from right-triangle relationships.

As we did with cylindrical coordinates, let’s consider the surfaces that are generated when each of the coordinates is held constant. Let be a constant, and consider surfaces of the form Points on these surfaces are at a fixed distance from the origin and form a sphere. The coordinate in the spherical coordinate system is the same as in the cylindrical coordinate system, so surfaces of the form are half-planes, as before. Last, consider surfaces of the form The points on these surfaces are at a fixed angle from the z-axis and form a half-cone ((Figure)).

In spherical coordinates, surfaces of the form are spheres of radius (a), surfaces of the form are half-planes at an angle from the x-axis (b), and surfaces of the form are half-cones at an angle from the z-axis (c).

Converting from Spherical Coordinates

Plot the point with spherical coordinates and express its location in both rectangular and cylindrical coordinates.

Use the equations in (Figure) to translate between spherical and cylindrical coordinates ((Figure)):

The projection of the point in the xy-plane is units from the origin. The line from the origin to the point’s projection forms an angle of with the positive x-axis. The point lies units above the xy-plane.

The point with spherical coordinates has rectangular coordinates

Finding the values in cylindrical coordinates is equally straightforward:

Thus, cylindrical coordinates for the point are

Plot the point with spherical coordinates and describe its location in both rectangular and cylindrical coordinates.

Cartesian: cylindrical:

Hint

Converting the coordinates first may help to find the location of the point in space more easily.

Converting from Rectangular Coordinates

Convert the rectangular coordinates to both spherical and cylindrical coordinates.

Start by converting from rectangular to spherical coordinates:

Because then the correct choice for is

There are actually two ways to identify We can use the equation A more simple approach, however, is to use equation We know that and so

and therefore The spherical coordinates of the point are

To find the cylindrical coordinates for the point, we need only find

The cylindrical coordinates for the point are

Identifying Surfaces in the Spherical Coordinate System

Describe the surfaces with the given spherical equations.

1. The variable represents the measure of the same angle in both the cylindrical and spherical coordinate systems. Points with coordinates lie on the plane that forms angle with the positive x-axis. Because the surface described by equation is the half-plane shown in (Figure).
The surface described by equation is a half-plane.

2. Equation describes all points in the spherical coordinate system that lie on a line from the origin forming an angle measuring rad with the positive z-axis. These points form a half-cone ((Figure)). Because there is only one value for that is measured from the positive z-axis, we do not get the full cone (with two pieces).
The equation describes a cone.

To find the equation in rectangular coordinates, use equation

This is the equation of a cone centered on the z-axis.

3. Equation describes the set of all points units away from the origin—a sphere with radius ((Figure)).
Equation describes a sphere with radius

4. To identify this surface, convert the equation from spherical to rectangular coordinates, using equations and

The equation describes a sphere centered at point with radius

Describe the surfaces defined by the following equations.

a. This is the set of all points units from the origin. This set forms a sphere with radius b. This set of points forms a half plane. The angle between the half plane and the positive x-axis is c. Let be a point on this surface. The position vector of this point forms an angle of with the positive z-axis, which means that points closer to the origin are closer to the axis. These points form a half-cone.

Hint

Think about what each component represents and what it means to hold that component constant.

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 (Figure). 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

Sydney, Australia is at and Express Sydney’s location in spherical coordinates.

Hint

Because Sydney lies south of the equator, we need to add to find the angle measured from the positive z-axis.

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 ((Figure)).

1. Find the center of gravity of a bowling ball.
2. Determine the velocity of a submarine subjected to an ocean current.
3. Calculate the pressure in a conical water tank.
4. Find the volume of oil flowing through a pipeline.
5. 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)

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.

Which coordinate system is most appropriate for creating a star map, as viewed from Earth (see the following figure)?

How should we orient the coordinate axes?

Spherical coordinates with the origin located at the center of the earth, the z-axis aligned with the North Pole, and the x-axis aligned with the prime meridian

Hint

What kinds of symmetry are present in this situation?

### Key Concepts

• In the cylindrical coordinate system, a point in space is represented by the ordered triple where represents the polar coordinates of the point’s projection in the xy-plane and represents the point’s projection onto the z-axis.
• To convert a point from cylindrical coordinates to Cartesian coordinates, use equations and
• To convert a point from Cartesian coordinates to cylindrical coordinates, use equations and
• In the spherical coordinate system, a point in space is represented by the ordered triple where is the distance between and the origin is the same angle used to describe the location in cylindrical coordinates, and is the angle formed by the positive z-axis and line segment where is the origin and
• To convert a point from spherical coordinates to Cartesian coordinates, use equations and
• To convert a point from Cartesian coordinates to spherical coordinates, use equations and
• To convert a point from spherical coordinates to cylindrical coordinates, use equations and
• To convert a point from cylindrical coordinates to spherical coordinates, use equations and

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

For the following exercises, the cylindrical coordinates of a point are given. Find the rectangular coordinates of the point.

For the following exercises, the rectangular coordinates of a point are given. Find the cylindrical coordinates of the point.

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]

A cylinder of equation with its center at the origin and rulings parallel to the z-axis,

[T]

[T]

Hyperboloid of two sheets of equation with the y-axis as the axis of symmetry,

[T]

[T]

Cylinder of equation with a center at and radius with rulings parallel to the z-axis,

[T]

[T]

Plane of equation

[T]

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

For the following exercises, the spherical coordinates of a point are given. Find the rectangular coordinates of the point.

For the following exercises, the rectangular coordinates of a point are given. Find the spherical coordinates of the point. Express the measure of the angles in degrees rounded to the nearest integer.

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]

Sphere of equation centered at the origin with radius

[T]

[T]

Sphere of equation centered at with radius

[T]

[T]

The xy-plane of equation

[T]

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.

or Elliptic cone

Plane at

For the following exercises, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle in radians rounded to four decimal places.

[T]

[T]

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

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 where

Cartesian system,

A spherical shell determined by the region between two concentric spheres centered at the origin, of radii of and respectively, where

A solid inside sphere and outside cylinder

Cylindrical system,

A cylindrical shell of height determined by the region between two cylinders with the same center, parallel rulings, and radii of and respectively

[T] Use a CAS to graph in cylindrical coordinates the region between elliptic paraboloid and cone

The region is described by the set of points

[T] Use a CAS to graph in spherical coordinates the “ice cream-cone region” situated above the xy-plane between sphere and elliptical cone

Washington, DC, is located at N and W (see the following figure). Assume the radius of Earth is mi. Express the location of Washington, DC, in spherical coordinates.

San Francisco is located at and Assume the radius of Earth is mi. Express the location of San Francisco in spherical coordinates.

Find the latitude and longitude of Rio de Janeiro if its spherical coordinates are

Find the latitude and longitude of Berlin if its spherical coordinates are

[T] Consider the torus of equation where

1. Write the equation of the torus in spherical coordinates.
2. If the surface is called a horn torus. Show that the equation of a horn torus in spherical coordinates is
3. Use a CAS to graph the horn torus with in spherical coordinates.

a.
c.

[T] The “bumpy sphere” with an equation in spherical coordinates is with and where and are positive numbers and and are positive integers, may be used in applied mathematics to model tumor growth.

1. Show that the “bumpy sphere” is contained inside a sphere of equation Find the values of and at which the two surfaces intersect.
2. Use a CAS to graph the surface for and along with sphere
3. Find the equation of the intersection curve of the surface at b. with the cone 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 vectors and and any given scalar

True

For vectors and and any given scalar

The symmetric equation for the line of intersection between two planes and is given by

False

If then is perpendicular to

For the following exercises, use the given vectors to find the quantities.

a. b. c. Can’t dot a vector with a scalar; d.

Find the values of such that vectors and are orthogonal.

For the following exercises, find the unit vectors.

Find the unit vector that has the same direction as vector that begins at and ends at

Find the unit vector that has the same direction as vector that begins at and ends at

For the following exercises, find the area or volume of the given shapes.

The parallelogram spanned by vectors

The parallelepiped formed by and

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

The line that passes through point that is parallel to vector

The line that passes through points and

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

The plane that passes through point and has normal vector

The plane that passes through points

For the following exercises, find the traces for the surfaces in planes Then, describe and draw the surfaces.

trace: is a circle, trace: is a hyperbola (or a pair of lines if trace: is a hyperbola (or a pair of lines if The surface is a cone.

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

Cylindrical: spherical:

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

sphere

For the following exercises, consider a small boat crossing a river.

If the boat velocity is km/h due north in still water and the water has a current of km/h due west (see the following figure), what is the velocity of the boat relative to shore? What is the angle that the boat is actually traveling?

When the boat reaches the shore, two ropes are thrown to people to help pull the boat ashore. One rope is at an angle of and the other is at If the boat must be pulled straight and at a force of find the magnitude of force for each rope (see the following figure).

331 N, and 244 N

An airplane is flying in the direction of 52° east of north with a speed of 450 mph. A strong wind has a bearing 33° east of north with a speed of 50 mph. What is the resultant ground speed and bearing of the airplane?

Calculate the work done by moving a particle from position to along a straight line with a force

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 m long and you are able to apply a 200-N force.

Because your tire is flat, you are only able to apply your force at a angle. What is the torque at the center of the bolt? Assume this force is not enough to loosen the bolt.

Someone lends you a tire jack and you are now able to apply a 200-N force at an angle. Is your resulting torque going to be more or less? What is the new resulting torque at the center of the bolt? Assume this force is not enough to loosen the bolt.

More, J

### Glossary

cylindrical coordinate system
a way to describe a location in space with an ordered triple where represents the polar coordinates of the point’s projection in the xy-plane, and represents the point’s projection onto the z-axis
spherical coordinate system
a way to describe a location in space with an ordered triple where is the distance between and the origin is the same angle used to describe the location in cylindrical coordinates, and is the angle formed by the positive z-axis and line segment where is the origin and