`HARMONY` starts with a molecular surface computed from available programs.
For example, this image
shows the crambin molecular surface computed by the program `MSMS` by
Michel Sanner.

Before computing harmonics, this surface must be topologically mapped to a sphere because the spherical harmonic functions are defined on a unit sphere. A restriction of this techniques is that the input surfaces must be topologically equivalent to a sphere. Surfaces like donuts won't work. After the spherical mapping, a unique spherical coordinate (theta, phi), can be assigned to each point of the molecular surface.

We can understand the effect of the mapping by color-coding surfaces by
the X Cartesian coordinate of the **original** molecular surface.
Note that the number of vertices, edges, and triangles for the
original surface and its spherical map is the same.

The first step in computing the topological map is to project the molecular surface to a unit sphere. This produces a surface that has overlaps.

The second step is a stochastic optimization that removes back-facing triangles and evens out the triangle mesh over the sphere surface. In this step, the surface is modified by translating the vertices to remove the overhanging regions.

After these steps, there is a one-to-one mapping from the molecular surface to the unit sphere. This new surface is isomorphic to a unit sphere and each ray from the center of the sphere intersects the spherical surface in only one place.

Spherical harmonic expansion coefficients can now be computed
using the coordinate data from the **original**
molecular surface, and the parameter ((theta,phi) spherical coordinates)
data from the **spherical** surface.

After the `tmap` procedure computes
the spherical map, the `fitsurf`
program evaluates the surface integrals for the X, Y, and
Z Cartesian coordinates to obtain three sets of expansion coefficients,
one for each Cartesian coordinate.

After coefficients are computed, the `mksurf`
program computes spherical harmonic surfaces using
the desired number of coefficients.

The following examples show results of this procedure.
In each instance, the spherical harmonic approximation is
on the left and the original `MSMS` surface is on the right.

Tue Feb 21 15:12:21 PST 1995