Quadric error metrics (QEM)
The error metric that drives mesh simplification: each vertex carries a quadric measuring how far a collapse would move the surface from the original.
Michael Garland and Paul Heckbert published the method in 1997. Each vertex gets a 4×4 matrix — the sum of squared distances to the planes of its incident faces. The cost of collapsing an edge is that quadric evaluated at the merged position, and because the sum of two quadrics is again a quadric, the position that minimises the error can be solved directly.
Collapses run out of a priority queue, cheapest first. The result is fast, preserves sharp features surprisingly well, and has become the default for automatic decimation in everything from game engines to CAD.