cylindrical coordinates: a point in three-dimensional space is
represented by the ordered triple 
, where r and 
are
polar coordinates of the projection of the point into the xy-plane, and z
is the distance of the point from the xy-plane.
To convert from cylindrical to rectangular coordinates, use
To convert from rectangular to cylindrical coordinates, use
spherical coordinates: a point P in three-dimensional space is
represented by the ordered triple 
, where
-
is the distance of the point from the origin; -
is the angle of the projection in the xy-plane (as in cylindrical
coordinates); and
-
is the angle between the positive z-axis and the line OP through P
and the origin.
To convert from spherical to rectangular coordinates, use
It's uglier to translate in the other direction (although of course, it can -- and will! -- be done...).
Cylindrical coordinates are great for problems with cylindrical symmetry (that is, with symmetry about a line). Spherical coordinates are great for problems with spherical symmetric (with symmetry about a point).
When you consider each coordinate system, ask yourself what constant equations
look like. Generally, these generate surfaces which are ``orthogonal'' to each other. For example, in Cartesian coordinates (x,y), the equations x=c and y=d give lines that are orthogonal to each other.
- For rectangular coordinates, x=c is a plane (and similarly for the other coordinates, producing a set of mutually perpendicular planes).
-
For cylindrical coordinates, z=c is a plane; r=c is a cylinder; and
is a plane orthogonal to the z-planes, and the cylinder (in the
sense that when they meet, they meet at right angles). -
For spherical coordinates,
is a spherical shell; 
is a cone
orthogonal to the shell; and is a plane orthogonal to the spherical
shell, and the cone.