Mesh reduction is performed to accelerate render and save memory when lowering the level of detail in a model is considered acceptable. For example, the model may be far from the observer, making rendering at its highest possible level of detail wasteful. For 3D printing, mesh reduction can reduce the amount of material that's required to produce the object and eliminate details that a particular printer may have no capacity to produce.
A "progressive mesh" reduces a model's complexity by repeated use of an edge collapse algorithm, this paper by Hoppe originally presented the concept.
Melax's method finds and collapses the edge that when eliminated affects the model's topology the least. Melax defines a criterion that eliminates the edge with highest coplanarity in the model.
A tutorial video showing how to install GLUI in Visual Studio 2010 can be seen here.
Code tested in Visual Studio 2010 Professional SP 1, Win 7, lightning and controls done with GLUI.
1. Select edge uv to collapse according to a criterion of lowest cost of loss of topology.
Melax chooses to assign lowest cost of loss of topology to the edge uv that is most highly coplanar.
Cost of collapsing an edge = length of edge * curvature term determined by dot products of face normals
To determine this "curvature term", we seek the triangle adjacent to vertex u that's furthest away from the other triangles that are along uv.
2. Remove triangles on the selected edge uv (u is going to be removed)
3. Update the remaining triangles that used vertex u, to use vertex v instead.
4. Remove vertex u.