Transformations

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Transformations

• Dr. Hugh Blanton
• ENTC 3331

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• 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.
  
• 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)
y
f
r
x
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Cartesian-to-Cylindrical
z
z z
(x,y,z) (r,f,z)
y
f
x
r
y
x
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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
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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 x-axis 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 z-axis 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. 
  
• Lets 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,q,f) (r,f,z)
R
q
y
f f
f
x
r
y
x
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Cylindrical-to-Spherical
z
f f
(R,q,f) (r,f,z)
R
q
y
f f
f
x
r
y
x
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Cartesian-to-Spherical
z
Recall from Cartesian-to-cylindrical
transformations
f f
(R,q,f) (r,f,z)
R
q
y
f f
f
x
r
y
x
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Cartesian-to-Spherical
z
(R,q,f) (r,f,z)
R
q
y
f
x
r
y
x
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Spherical-to-Cartesian
z
(R,q,f) (r,f,z)
R
q
y
f
x
r
y
x
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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 lets work an example of
  each.

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• Perform each of the following conversions.
• (a) Convert the point   from
  cylindrical to spherical coordinates.
•      
• (b) Convert the point   from
  Cartesian to spherical coordinates.

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• Solution
• (a) Convert the point   from
  cylindrical to spherical coordinates.
•  
• Well start by acknowledging that is the
  same in both coordinate systems.

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• Next, lets find R.

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• Finally, lets get q. 
• To do this we can use either the conversion for r
  or z.
• Well use the conversion for z.   

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• So, the spherical coordinates of this point will
  are

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