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Cylindrical and Spherical Coordinates

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Cylindrical and Spherical Coordinates

Written by Dr. Julia Arnold

Associate Professor of Mathematics

Tidewater Community College, Norfolk Campus, Norfolk, VA

With Assistance from a VCCS LearningWare Grant

- In this lesson you will learn
- about cylindrical and spherical coordinates
- how to change from rectangular coordinates to cylindrical coordinates or spherical coordinates
- how to change from spherical coordinates to rectangular coordinates or cylindrical coordinates
- how to change from cylindrical coordinates to rectangular coordinates or spherical coordinates

Polar Coordinates

The polar coordinates r (the radial coordinate) and (the angular coordinate, often called the polar angle) are defined in terms of Cartesian Coordinates by

where r is the radial distance from the origin, and is the counterclockwise angle from the x-axis.

In terms of x and y,

A point P is represented by an ordered triple of .

To change from rectangular to cylindrical:

To change from cylindrical to rectangular:

Cylindrical coordinates are a generalization of two-dimensional polar coordinates to three dimensions by superposing a height (z) axis.

As you can see, this coordinate system lends itself well to cylindrical figures.

Common Uses

The most common use of cylindrical coordinates is to give the equation of a surface of revolution. If the z-axis is taken as the axis of revolution, then the equation will not involve theta at all.

Examples:

A paraboloid of revolution might have equation

z = r2. This is the surface you would get by rotating the parabola z = x2 in the xz-plane about the z-axis. The Cartesian coordinate equation of the paraboloid of revolution would be z = x2 + y2.

A right circular cylinder of radius a whose axis is the z-axis has equation

r = R.

A a sphere with center at the origin and radius R will have equation

r + z2 = R2.

A right circular cone with vertex at the origin and axis the z-axis has equation

z = m r.

As another kind of example, a helix has the following equations:

r = R, z = a theta.

http://mathforum.org/dr.math/faq/formulas/faq.cylindrical.html

Express the point (x,y,z) = (1, ,2) in cylindrical coordinates.

Solution:

Work it out before you go to the next slide.

Express the point (x,y,z) = (1, ,2) in cylindrical coordinates.

Solution:

You have two choices for r and infinitely many choices for theta.

Thus the point can be represented by non unique cylindrical coordinates. For example

See picture on next slide.

This graph was done using Win Plot in the two different coordinate systems.

The animation below the one above shows the points represented by constant values of the second coordinate as it varies from zero to 2 pi.

You can also view at this link:

http://www.tcc.edu/faculty/webpages/JArnold/movies.htm

Example 2 Identify the surface for each of the following equations.

(a) r = 5

(b)

(c) z = r

Solution:

a. In polar coordinates we know that r = 5 would be a circle of radius 5 units. By adding the z dimension and allowing z to vary we create a cylinder of radius 5.

5

Example 2 Identify the surface for each of the following equations.

(a) r = 5

(b)

(c) z = r

Solution:

b. This is equivalent to which we know to be a sphere centered at the origin with a radius of 10.

10

10

10

Example 2 Identify the surface for each of the following equations.

(a) r = 5

(b)

(c) z = r

Solution:

c. Since the radius equals the height and the angle is any angle we get a cone.

Spherical coordinates are a system of curvilinear coordinates that are natural for describing positions on a sphere or spheroid.

The ordered triple is:

For a given point P in spherical coordinates

is the distance between P and the origin

is the same angle theta used in cylindrical coordinates for

is the angle between the positive z-axis and the line segment

(x,y,z)

P

z

O

The figure at right shows the

Rectangular coordinates (x,y,z) and

The spherical coordinates

Conversion Formulas:

Spherical to Rectangular:

Rectangular to Spherical:

Spherical to cylindrical ( ):

Cylindrical to spherical ( ):

Example 3

A. Find a rectangular equation for the graph represented by the cylindrical equation

B. Find an equation in spherical coordinates for the surface represented by each of the rectangular equations and identify the graph.

1.

2.

Answers follow

Example 3

A. Find a rectangular equation for the graph represented by the cylindrical equation

Example 3

B. Find an equation in spherical coordinates for the surface represented by each of the rectangular equations and identify the graph.

1.

A double cone.

Example 3

B. Find an equation in spherical coordinates for the surface represented by each of the rectangular equations and identify the graph.

2.

A sphere

Rectangular to cylindrical:

Cylindrical to rectangular:

Review

Spherical to Rectangular:

Rectangular to Spherical:

Spherical to cylindrical ( ):

Cylindrical to spherical ( ):

For comments on this presentation you may email the author

Dr. Julia Arnold at [email protected]