Cylindrical Coordinates - Three Dimensional Space

Since with two dimensional space the standard (x, y, z) coordinate system is known as the Cartesian coordinate system.  In the previous two sections we will be looking at some alternate coordinate systems for three dimensional (3D) spaces.

We'll start off along with the cylindrical coordinate system.  This one is quite simple as it is nothing much more as compared to an extension of polar coordinates into three dimensions. Not just only is it an extension of polar coordinates, however we extend it into the third dimension just like we extend Cartesian coordinates into the third dimension.  All we do is add a z on as the third coordinate. The r and Θ are similar as with polar coordinates.

Here is a diagram of a point in R³.

The conversions for x and y are similar conversions that we used back while we were looking at polar coordinates.  Thus, if we have a point in cylindrical coordinates the Cartesian coordinates can be found by using the subsequent conversions.

x = r cos Θ

y = r sin Θ

z =z

The above third equation is just an acknowledgement that the z-coordinate of a point in Cartesian and polar coordinates is similar.