Matpic
mathematics, in pictures
Concept

Lifting map

Sending a planar point to the point above it on the paraboloid, so that its height is the square of its distance from the origin. It turns every circle in the plane into a plane in space, and every inside-the-circle question into an inequality.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The triangulation as the underside of a hull. Eight points on a floor, lifted onto a paraboloid above them. The faces of the lifted set's lower convex hull are shaded, and their shadows on the floor are the Delaunay triangulation of the original points.

One dimension up, and the circles disappear

The Delaunay triangulation is defined by a condition about circles, which is awkward to compute and awkward to reason about. Lift every point onto a paraboloid and the circles turn into planes, the condition turns into convexity, and a two-dimensional problem is solved by looking at a three-dimensional shape from underneath.

geometry · Voronoi The diagram when the sites are not the same size. 7 weighted sites drawn as circles of different radius, with the power diagram over them. The boundaries are straight, as in the unweighted diagram, but each one is pushed towards the smaller of the two circles it separates.

When the sites are not the same size

Give every site a weight and the boundaries slide. The cells stay convex and the edges stay straight, which is surprising, and one thing happens that the unweighted diagram never allows — a site can end up owning nothing at all.

geometry · Voronoi

Named alongside it

The objects these essays reach for when they reach for this one.

Convex hullConvexityDualityVoronoi diagramCircleCircumcircleDelaunay triangulationParaboloidPower diagramRadical axis

All concepts