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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:
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