Transformations
Dr. Hugh Blanton
ENTC 3331
Dr. Blanton - ENTC 3331 - Coordinate Transformations 2 / 29
• 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.
Dr. Blanton - ENTC 3331 - Coordinate Transformations 3 / 29
• 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.
Dr. Blanton - ENTC 3331 - Coordinate Transformations 4 / 29
• 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.
Dr. Blanton - ENTC 3331 - Coordinate Transformations 5 / 29
• 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
Dr. Blanton - ENTC 3331 - Coordinate Transformations 6 / 29
Cylindrical-to-CartesianCylindrical-to-Cartesian
z
y
x
r
(x,y,z) = (r,,z)
cosrx
sinry
zz
Dr. Blanton - ENTC 3331 - Coordinate Transformations 7 / 29
Cartesian-to-CylindricalCartesian-to-Cylindrical
z
y
x
r
(x,y,z) = (r,,z)
x
y
22222 yxryxr
x
y1tan
z = z
Dr. Blanton - ENTC 3331 - Coordinate Transformations 8 / 29
• Find the cylindrical coordinates of the point whose Cartesian coordinates are
(1, 2, 3)
Dr. Blanton - ENTC 3331 - Coordinate Transformations 9 / 29
Cylindrical Coordinates -- Answer 1 Cylindrical Coordinates -- Answer 1
Dr. Blanton - ENTC 3331 - Coordinate Transformations 10 / 29
• Find the Cartesian coordinates of the point whose cylindrical coordinates are
(2, /4, 3)
Dr. Blanton - ENTC 3331 - Coordinate Transformations 11 / 29
Cylindrical Coordinates -- Answer 2Cylindrical Coordinates -- Answer 2
3
2
z
yx
Dr. Blanton - ENTC 3331 - Coordinate Transformations 12 / 29
• Spherical coordinates consist of the three quantities (R
Dr. Blanton - ENTC 3331 - Coordinate Transformations 13 / 29
• First there is R. • This is the distance from the origin to
the point. • Note that R 0.
Dr. Blanton - ENTC 3331 - Coordinate Transformations 14 / 29
• 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
Dr. Blanton - ENTC 3331 - Coordinate Transformations 15 / 29
• 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 ≤ ≤.
Dr. Blanton - ENTC 3331 - Coordinate Transformations 16 / 29
• 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.
Dr. Blanton - ENTC 3331 - Coordinate Transformations 17 / 29
• 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.
Dr. Blanton - ENTC 3331 - Coordinate Transformations 18 / 29
Spherical-to-CylindricalSpherical-to-Cylindrical
z
y
x
r
(R) = (r,,z)
x
y
R
cosRz sinRr
=
Dr. Blanton - ENTC 3331 - Coordinate Transformations 19 / 29
Cylindrical-to-SphericalCylindrical-to-Spherical
z
y
x
r
(R) = (r,,z)
x
y
R
cosRz sinRr
=
22222 rzRrzR
z
r1tan
=
Dr. Blanton - ENTC 3331 - Coordinate Transformations 20 / 29
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
Dr. Blanton - ENTC 3331 - Coordinate Transformations 21 / 29
Cartesian-to-SphericalCartesian-to-Spherical
z
y
x
r
(R) = (r,,z)
x
y
R
cosRz sinRr
z
yx 221tan
222 zyxR
x
y1tan
Dr. Blanton - ENTC 3331 - Coordinate Transformations 22 / 29
Spherical-to-CartesianSpherical-to-Cartesian
z
y
x
r
(R) = (r,,z)
x
y
R
cosRz sinRr
cosRz
cosrx
sinry
cossinRx
sinsinRy
Dr. Blanton - ENTC 3331 - Coordinate Transformations 23 / 29
• 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.
Dr. Blanton - ENTC 3331 - Coordinate Transformations 24 / 29
• 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
Dr. Blanton - ENTC 3331 - Coordinate Transformations 25 / 29
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
Dr. Blanton - ENTC 3331 - Coordinate Transformations 26 / 29
• Next, let’s find R.
22222 62 RrzR
22862 R
Dr. Blanton - ENTC 3331 - Coordinate Transformations 27 / 29
• 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
Dr. Blanton - ENTC 3331 - Coordinate Transformations 28 / 29
• So, the spherical coordinates of this point will are
3,
4,22
Dr. Blanton - ENTC 3331 - Coordinate Transformations 29 / 29
4
3,
4
3,2