Transformations
Dr. Hugh Blanton
ENTC 3331
• It is important to compare the units
that are used in Cartesian
coordinates with the units that are
used in cylindrical coordinates and
spherical coordinates.
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• In Cartesian coordinates, (x, y, z), all
three coordinates measure length
and, thus, are in units of length.
• In cylindrical coordinates, (r, f, z), two of
the coordinates – r and z -- measure
length and, thus, are in units of length
but
• the coordinate f measures angles and
is in "units" of radians.
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• The most important part of the
preceding slide is the quotation
marks around the word "units" –
• radians are a dimensionless quantity –
• That is, they do not have associated units.
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• The formulas below enable us to convert
from cylindrical coordinates to Cartesian
coordinates.
x  r cosf
y  r sin f
zz
• Notice the units work out correctly.
• The right side of each of the first two equations is a
product in which the first factor is measured in units
of length and the second factor is dimensionless.
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Cylindrical-to-Cartesian
z
(x,y,z) = (r,f,z)
zz
y
x  r cosf
x
f
r
y  r sin f
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Cartesian-to-Cylindrical
r 2  x2  y2  r  x2  y2
f  tan 1
z
y
x
z=z
(x,y,z) = (r,f,z)
y
x
x
f
r
y
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• Find the cylindrical coordinates of the
point whose Cartesian coordinates
are
(1, 2, 3)
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Cylindrical Coordinates -- Answer 1
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• Find the Cartesian coordinates of the
point whose cylindrical coordinates
are
(2, p/4, 3)
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Cylindrical Coordinates -- Answer 2
x 2 y
z 3
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• Spherical coordinates consist of the
three quantities (R,q,f).
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• First there is R.
• This is the distance from the origin to
the point.
• Note that R  0.
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• Next there is f.
• This is the same angle that we saw in
cylindrical coordinates.
• It is the angle between the positive xaxis and the line denoted by r (which is
also the same r as in cylindrical
coordinates).
• There are no restrictions on f.
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• Finally there is q.
• This is the angle between the positive zaxis and the line from the origin to the
point.
• We will require 0 ≤q ≤p.
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• In summary,
• R is the distance from the origin to the
point,
• q is the angle that we need to rotate
down from the positive z-axis to get to
the point and
• f is how much we need to rotate around
the z-axis to get to the point.
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• We should first derive some
conversion formulas.
• Let’s first start with a point in spherical
coordinates and ask what the cylindrical
coordinates of the point are.
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Spherical-to-Cylindrical
z
r  R sin q
z  R cosq
R
(R,q,f) = (r,f,z)
q
f=f
x
x
y
f
r
y
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Cylindrical-to-Spherical
R2  z 2  r 2  R  z 2  r 2
z
q  tan 1
r
z
r  R sin q
z  R cosq
f=f
R
(R,q,f) = (r,f,z)
q
f=f
x
x
y
f
r
y
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Cartesian-to-Spherical
R  z r
2
2
2
tan 1 q 
z
Recall from
Cartesian-tocylindrical
z  R cosq
transformations:
R
R2  z 2  x2  y 2
R  x2  y2  z 2
(R,q,f) = (r,f,z)
q
f=f
x
x
f=f
r  R sin q
r 2  x2  y2
r
z
y
f
r
y
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Cartesian-to-Spherical
R  x2  y2  z 2
f  tan 1
q  tan 1
y
x
z
r  R sin q
z  R cosq
x2  y2
z
R
(R,q,f) = (r,f,z)
q
y
x
x
f
r
y
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Spherical-to-Cartesian
z  R cosq
x  r cos f
x  R sin q cos f
y  r sin f
z
r  R sin q
z  R cosq
y  R sin q sin f
R
(R,q,f) = (r,f,z)
q
y
x
x
f
r
y
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• Converting points from Cartesian or
cylindrical coordinates into spherical
coordinates is usually done with the
same conversion formulas.
• To see how this is done let’s work an
example of each.
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• Perform each of the following
conversions.
p


6
,
,
2

 from
point
4


• (a) Convert the
cylindrical to spherical coordinates.
• (b) Convert the point  1,1, 2 ) from
Cartesian to spherical coordinates.
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Solution
p

6
,

(a) Convert the point  4 ,
cylindrical to spherical
coordinates.

2

from
p
• We’ll start by acknowledging that f 
4
is the same in both coordinate systems.
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• Next, let’s find R.
R  z r  R 
2
2
2
 2)   6)
2
2
R  26  8  2 2
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• Finally, let’s get q.
• To do this we can use either the
conversion for r or z.
• We’ll use the conversion for z.
z  R cosq
2 1
cos q 

2 2 2
1
q  cos
2
p
q
3
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1
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• So, the spherical coordinates of this
point will are
p p

 2 2, , 
4 3

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 3p 3p 
, 
 2,
 4 4 
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