Spherical Coordinates
In this section we will introduce spherical coordinates. Spherical coordinates can take
a little getting used to. It’s probably easiest to start things off with a sketch.
Spherical coordinates consist of the following three quantities.
First there is
. This is the distance from the origin to the point and we will
require
.
Next there is
. This is the same angle that we saw in polar/cylindrical
coordinates. It is the angle between the positive
x
axis and the line above denoted
by
r
(which is also the same
r
as in polar/cylindrical coordinates). There are no
restrictions on
.
Finally there is
. This is the angle between the positive
z
axis and the line from
the origin to the point. We will require
.
In summary,
is the distance from the origin of 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.
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. So, we
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and what to find
. Of course
we really only need to find
r
and
z
since
is the same in both coordinate systems.
We will be able to do all of our work by looking at the right triangle shown above in
our sketch. With a little geometry we see that the angle between
z
and
is
and so we can see that,
and these are exactly the formulas that we were looking for. So, given a point in
spherical coordinates the cylindrical coordinates of the point will be,
Note as well that,
Or,
Next, let’s find the Cartesian coordinates of the same point. To do this we’ll start with
the cylindrical conversion formulas from the
previous section
.
Now all that we need to do is use the formulas from above for
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 Fall '08
 prellis
 Polar coordinate system, coordinates, polar/cylindrical

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