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, , z), two of

the coordinates – r and z -- measure length and, thus, are in units of length but

• the coordinate 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.

cosrx

sinry zz

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Cylindrical-to-CartesianCylindrical-to-Cartesian

z

y

x

r

(x,y,z) = (r,,z)

cosrx

sinry

zz

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Cartesian-to-CylindricalCartesian-to-Cylindrical

z

y

x

r

(x,y,z) = (r,,z)

x

y

22222 yxryxr

x

y1tan

z = z

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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 Cylindrical Coordinates -- Answer 1

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• Find the Cartesian coordinates of the point whose cylindrical coordinates are

(2, /4, 3)

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Cylindrical Coordinates -- Answer 2Cylindrical Coordinates -- Answer 2

3

2

z

yx

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• Spherical coordinates consist of the three quantities (R 

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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 .  • 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

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• Finally there is .  • This is the angle between the positive z-

axis and the line from the origin to the point. 

• We will require 0 ≤ ≤.

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• In summary, • R is the distance from the origin to the

point,  • is the angle that we need to rotate

down from the positive z-axis to get to the point and

• 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-CylindricalSpherical-to-Cylindrical

z

y

x

r

(R) = (r,,z)

x

y

R

cosRz sinRr

=

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Cylindrical-to-SphericalCylindrical-to-Spherical

z

y

x

r

(R) = (r,,z)

x

y

R

cosRz sinRr

=

22222 rzRrzR

z

r1tan

=

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Cartesian-to-SphericalCartesian-to-Spherical

z

y

x

r

(R) = (r,,z)

x

y

R

cosRz sinRr

=

222 rzR z

r 1tan

= Recall from Cartesian-to-cylindrical transformations:

222 yxr 2222 yxzR

222 zyxR

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Cartesian-to-SphericalCartesian-to-Spherical

z

y

x

r

(R) = (r,,z)

x

y

R

cosRz sinRr

z

yx 221tan

222 zyxR

x

y1tan

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Spherical-to-CartesianSpherical-to-Cartesian

z

y

x

r

(R) = (r,,z)

x

y

R

cosRz sinRr

cosRz

cosrx

sinry

cossinRx

sinsinRy

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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.• (a) Convert the point   from

cylindrical to spherical coordinates.

      • (b) Convert the point   from

Cartesian to spherical coordinates.

2,4

,6

2,1,1

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Solution(a) Convert the point   from cylindrical to spherical coordinates.

 • We’ll start by acknowledging that

is the same in both coordinate systems.

2,4

,6

4

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• Next, let’s find R.

22222 62 RrzR

22862 R

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• Finally, let’s get .  • To do this we can use either the

conversion for r or z.• We’ll use the conversion for z.   

cosRz

2

1

22

2cos 2

1cos 1

3

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• So, the spherical coordinates of this point will are

3,

4,22

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4

3,

4

3,2