Cylindrical and Spherical Cylindrical and Spherical CoordinatesCoordinates

Cylindrical and Spherical Cylindrical and Spherical CoordinatesCoordinates

Written by Dr. Julia ArnoldWritten by Dr. Julia Arnold

Associate Professor of MathematicsAssociate Professor of Mathematics

Tidewater Community College, Norfolk Campus, Norfolk, VATidewater Community College, Norfolk Campus, Norfolk, VA

With Assistance from a VCCS LearningWare GrantWith 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

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

sin

cos

ry

rx

Polar Coordinates

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

In terms of x and y,

x

y

yxr

1

22

tan

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

A point P is represented by an ordered triple of . zr ,,

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

zzx

y

yxr

1

22

tan

To change from rectangular to cylindrical:

zzry

rx

,sin

cos

To change from

cylindrical to rectangular:

 

Common UsesThe 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.3

Solution:

zz

n

r

31

3tan

231

1

Work it out before you go to the next slide.

Express the point (x,y,z) = (1, ,2) in cylindrical coordinates.3

Solution:

zz

n

r

31

3tan

231

1

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

2,3,1

2,3

2,22,3,1

2,3

,22,3,1

or

See picture on next slide.

x

y

z

(1,-sqr3,2)(2,-pi/3,2)

(-2,-2pi/3,2)

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

The animation at the link below shows the points represented by constant values of the first coordinate as it varies from zero to one.

http://www.math.montana.edu/frankw/ccp/multiworld/multipleIVP/cylindrical/body.htm

             

            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  

10022 zr

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  

10022 zr

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

100222 zyx

10

10

10

Example 2  Identify the surface for each of the following equations.(a) r = 5 (b) (c) z = r  

10022 zr

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

x

y

z

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

The ordered triple is: ),,(,, phithetarho

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

0

0r

is the angle between the positive z-axis and the line segment 0,OP

P

zO

The figure at right shows the Rectangular coordinates (x,y,z) andThe spherical coordinates ,,

(x,y,z)

Conversion Formulas:

Spherical to Rectangular:

cos,sinsin,cossin zyx

Rectangular to Spherical:

222

222 arccos,tan,zyx

z

x

yzyx

Spherical to cylindrical ( ):

cos,,sin222 zr

0r

Cylindrical to spherical ( ):0r

22

22 arccos,,zr

zzr

Example 3A. Find a rectangular equation for the graph represented by the cylindrical equation

12cos 22 zr

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

1. 222 zyx

2. 04222 zzyx

Answers follow

Example 3A. Find a rectangular equation for the graph represented by the cylindrical equation

12cos 22 zr

1

1sincos

1)sin(cos

12cos

222

22222

2222

22

zyx

zrr

zr

zr

x

y

z

xy

z

Example 3

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

1. 222 zyx cos,sinsin,cossin zyx

4

3,

4

1tan

1tan

1cos

sin

cossin

cossin

cos)sin(cossin

cossinsincossin

2

22

22

2222

2222

222222

22222222

222

zyx

x

y

z

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. 04222 zzyx

cos,sinsin,cossin zyx

cos4

cos4

0cos4

0cos4)cos(sin

0cos4cossin

0cos4cos)sin(cossin

0cos4cossinsincossin

04

2

2

222

2222

222222

22222222

222

zzyx

A sphere

x

y

z

Spherical to Rectangular: cos,sinsin,cossin zyx

Rectangular to Spherical:

222

222 arccos,tan,zyx

z

x

yzyx

Spherical to cylindrical ( ): cos,,sin222 zr0r

Cylindrical to spherical ( ):0r

22

22 arccos,,zr

zzr

x

y

yxr

1

22

tan

Rectangular to cylindrical:

sin

cos

ry

rx

Cylindrical to

rectangular:

For comments on this presentation you may email the author Dr. Julia Arnold at jarnold@tcc.edu.