As with two dimensional space the standard coordinate system is called the Cartesian
coordinate system. In the last two
sections of this chapter we’ll be looking at some alternate coordinate systems
for three dimensional space.
We’ll start off with the cylindrical coordinate system. This one is fairly simple as it is nothing
more than an extension of polar coordinates
into three dimensions. Not only is it an
extension of polar coordinates, but we extend it into the third dimension just
as we extend Cartesian coordinates into the third dimension. All that we do is add a z on as the third coordinate.
The r and θ are the same as with polar coordinates.
Here is a sketch of a point in .
The conversions for x
and y are the same conversions that
we used back when we were looking at polar coordinates. So, if we have a point in cylindrical
coordinates the Cartesian coordinates can be found by using the following
conversions.
The third equation is just an acknowledgement that the zcoordinate of a point in Cartesian and
polar coordinates is the same.
Likewise, if we have a point in Cartesian coordinates the
cylindrical coordinates can be found by using the following conversions.
Let’s take a quick look at some surfaces in cylindrical
coordinates.
Example 1 Identify
the surface for each of the following equations.
(a)
(b)
(c)
Solution
(a) In two dimensions we know that this is a
circle of radius 5. Since we are now
in three dimensions and there is no z
in equation this means it is allowed to vary freely. So, for any given z we will have a circle of radius 5 centered on the zaxis.
In other words, we will have a cylinder of radius 5
centered on the zaxis.
(b) This
equation will be easy to identify once we convert back to Cartesian
coordinates.
So, this is a sphere centered at the origin with radius
10.
(c) Again, this
one won’t be too bad if we convert back to Cartesian. For reasons that will be apparent
eventually, we’ll first square both sides, then convert.
From the section on quadric
surfaces we know that this is the equation of a cone.
