12.4
Cylindrical & Spherical Coordinates
Contemporary Calculus
2
CYLINDRICAL COORDINATES
Cylindrical coordinates are basically "polar coordinates with
altitude." The cylindrical coordinates (r,
θ
, z) specify the
point P that is z units above the point on the xy–plane whose
polar coordinates are r and
θ
(Fig. 3).
Example 1
:
Plot the points given by the
cylindrical
coordinates A(3,
π
/3, 1),
B(1, 0, –2), and C(2, 180
o
, –1).
Solution: The points are plotted in Fig. 4.
Practice 1
:
On Fig. 4, plot the points given by the cylindrical coordinates
P(3,
π
/6, –1), Q(3,
π
/2, 2)), and R(0,
π
, 3).
We can start to develop an understanding of the effect of each variable in the
ordered triple by holding two of the variables fixed and letting the other one vary.
Fig. 5 shows the results in the rectangular coordinate system of fixing x and y
(x = 1, y = 2) and letting z vary: we get a vertical line parallel to the z–axis. Similarly, in the rectangular
coordiante system, when we fix x and z (x = 1, z = 3) and let y vary we get a line parallel to the y–axis.
We can try the same process in the cylindrical coordinate system.
Fixing two of the variables and letting the other one vary:
In the cylindrical coordinate system, if we fix r and
θ
(r = 2,
θ
=
π
/3 = 60
o
) and let z vary
(Fig. 6a) we get a line parallel to the z–axis.
If we fix r and z (r = 2, z = 3) and let
θ
vary (Fig. 6b) we get a circle of radius 1 centered around
the z–axis at a height of 3 units above the xy plane.
Fig. 3
x
y
z
P(
r,
!
, z
)
z
r
3D Cylindrical
2D Polar
!
x
y
P(
r,
!
)
r
x
y
z
A
B
C
Fig. 4
Fig. 5: Rectangular coordinates: two variables fixed
line
(1, 2, z)
x
y
z
1
2
x = 1
y = 2
z varies
(1, y, 3)
1
x
y
x = 1
z = 3
y varies
line
(x, 2, 3)
y = 2
z = 3
x varies
1
3
x
y
z