Polar, Cylindrical, and Spherical Coordinates
1. (a) In polar coordinates, what shapes are described by r = k and θ = k, where k is a constant?
Solution. r = k describes a circle of radius k centered at the origin.
θ = k describes a ray from the origin which makes an angle of θ when measured counter-clockwise
from the x-axis.
Θ
(b) Draw r = 0, r = 2π
3 , r =
can’t we draw θ = 2π?)
4π
3 ,
r = 2π, θ = 0, θ =
2π
3 ,
and θ =
4π
3
on the following axes. (Why
Solution. Here is the picture, with r = constant curves drawn in blue and θ = constant drawn
in red. We only allow θ to be in [0, 2π), so we can’t actually have θ = 2π.
y
2Π
2Π
-2 Π
x
-2 Π
(c) On the axes in (b), sketch the curve with polar equation r = θ.
Solution. The curve is a spiral starting from (x, y) = (0, 0). Notice that it does not actually
contain the point (x, y) = (2π, 0), since θ cannot actually equal 2π.
1
y
2Π
x
2Π
-2 Π
-2 Π
2. In cylindrical coordinates, what shapes are described by r = k, θ = k, and z = k, where k is a constant?
Solution. r = k describes a cylinder of radius k centered around the z-axis.
θ = k describes a vertical half-plane whose intersection with the plane z = 0 is just the ray θ = k in
polar coordinates.
z = k describes a plane parallel to the xy-plane.
3. In spherical coordinates, what shapes are described by ρ = k, θ = k, and φ = k, where k is a constant?
Solution. ρ = k describes a sphere of radius k centered at the origin.
θ = k describes exactly the same thing as it does in cylindrical coordinates. (After all, θ means the
same thing in both coordinate systems.)
φ = k describes a half-cone. (Remember that the quadric surface that we call a cone actually opens
in two directions. A half-cone is what we typically think of as a cone; it opens only in one direction.)
For instance, here are pictures of φ = π6 and φ = 5π
6 .
z
z
y
x
φ=
y
x
π
6
φ=
5π
6
4. (a) In cylindrical coordinates, let’s look at the surface r = 5. What does z = k look like on this
surface? How about θ = k? (k is a constant.)
Solution. The surface r = 5 is a cylinder of radius 5 centered about the z-axis. On this surface,
each curve z = k is a circle (shown in red), and each curve θ = k is a vertical line (shown in blue):
2
z
y
x
(b) In spherical coordinates, let’s look at the surface ρ = 5. What does θ = k look like on this surface?
How about φ = k?
Solution. The surface ρ = 5 is a sphere of radius 5 centered about the origin. On this surface,
each curve θ = k is a half-circle running from the top point of the sphere to the bottom point. (If
you imagine the sphere as a globe, these are like lines of longitude.) These are shown in blue on
the picture. Each curve φ = k is a circle (these are like lines of latitude), and these are shown in
red.
z
y
x
√
√
5. Write the point (x, y, z) = ( 6, − 6, −2) in cylindrical and spherical coordinates.
√
√
Solution. To write the point in cylindrical coordinates, we essentially convert x = 6, y =
p− 6 from
Cartesian
coordinates to polar coordinates. Since x = r cos θ and y = r sin θ, we have r = x2 + y 2 =
√
7π
2 3 and tan θ = xy = −1. Now, just knowing tan θ = −1 tells us that θ is either 3π
4 or 4 . But
√
we also know that cos θ = xr = 22 , so θ must be 7π
4 . Thus, in cylindrical coordinates, the point is
√ 7π
(r, θ, z) = 2 3, 4 , −2 .
p
Next, we need to find the point in spherical coordinates. We know that ρ = x2 + y 2 + z 2 = 4 and
√
7π 2π
that tan φ = zr = − 3, so φ = 2π
.
3 . So, in spherical coordinates, the point is (ρ, θ, φ) = 4, 4 , 3
6. Consider the surface whose equation in cylindrical coordinates is z = r. How could you describe this
surface in Cartesian coordinates? Spherical? Can you sketch the surface?
p
Solution. To write in Cartesian coordinates, we can use the fact that r = x2 + y 2 , so z = r is the
3
same as z =
elliptic cone.
p
x2 + y 2 . This is the portion of z 2 = x2 + y 2 with z ≥ 0, so it is the top half of an
To write in spherical coordinates, remember that z = ρ cos φ and r = ρ sin φ, so this is the surface with
ρ cos φ = ρ sin φ. In order for this to be true, cos φ = sin φ, so tan φ = 1. Since 0 ≤ φ ≤ 1, φ = π4 .
Here is the surface:
z
y
x
7. Most of the time, a single equation like 2x + 3y + 4z = 5 in Cartesian coordinates or ρ = 1 in
spherical coordinates defines a surface. Can you find examples in Cartesian, cylindrical, and spherical
coordinates where this is not the case?
Solution. In Cartesian coordinates, xyz = 0 is just the three coordinate axes. x2 + y 2 + z 2 = 0 is a
point.
In cylindrical coordinates, r = 0 is just the z-axis.
In spherical coordinates, ρ = 0 is a single point, the origin. Also, φ = 0 is the non-negative portion of
the z-axis. If you have trouble visualizing this, you might want to change to Cartesian coordinates:
x
= ρ sin φ cos θ
y
= ρ sin φ sin θ
z
=
ρ cos φ
When φ = 0, sin φ = 0 and cos φ = 1, so we have x = 0, y = 0, and z = ρ.
Of course, these are not the only examples.
4