For any conformal map *X* from C to IR^3
,
where

We now assume that and satisfy the differential equations:

If we assume and are of the form:

(1) and (2) can be rewritten in terms of *h*_{1} and *h*_{2} as follows:

Using (1)-(6), we obtain a surface of revolution as follows:

We have the differential equations (5) and (6) satisfied by *h*_{1} and *h*_{2}:

So:

As a result, the parametrization is a surface of revolution.

Thus, given the mean curvature half density of a surface, U, we can
solve the differential equations and obtain a parametrization for a surface
of revolution.