A series of reflections on the physical and philosophical nature of power electronics, where the transformation of energy reveals the fundamental rules of the world.
Scatter a handful of points on a plane and ask a simple question: which pairs should be connected? Not by nearness alone — nearness is cheap and often wrong — but by a rule that produces exactly one well-behaved triangulation out of the combinatorial chaos of every possible way to draw lines between them.
The Delaunay triangulation answers with a geometric guarantee: for any three points to form a triangle in the triangulation, the circle passing through all three must contain no other point of the set. An empty circumcircle is the whole law. Enforce it everywhere and the triangulation that falls out maximizes the minimum angle of every triangle in the mesh — no dangerously thin slivers, the enemy of every downstream simulation that will walk across this mesh solving differential equations.
Its dual lives on the other side of the same coin. Draw the set of points closer to one seed than any other and the plane partitions itself into convex cells — the Voronoi diagram, one region per point, boundaries falling exactly where two seeds tie for nearest. Connect each pair of seeds whose cells share a border and the connecting lines are, edge for edge, the Delaunay triangulation turned inside out. Build one structure and the other exists for free, a shadow cast by the same underlying geometry.
Neither structure needs a slow method to exist. Chan's algorithm builds a related convex hull in output-sensitive time — cost scaling with the number of hull vertices that survive, not the number of candidate points that don't — and spatial trees (k-d trees slicing space by alternating axis, R-trees grouping by bounding rectangle) answer nearest-neighbor and range queries without ever touching most of the data. Geometry, done right, refuses to look at what it doesn't need to.
Seed: Computational Geometry & Spatial Data Structures (Delaunay Triangulation, Voronoi Diagrams, Convex Hulls, k-d/R-Trees)