Concept

Support function

For a convex shape, the distance from a chosen origin to the supporting line with a given outward normal. It describes the shape by the lines that touch it rather than by its boundary, and it turns adding shapes into adding functions.

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

A Reuleaux triangle. A curve of constant width on 3 vertices, with 6 pairs of parallel supporting lines drawn across it. Every pair is 180.1 apart.

Round is not the only way to be the same width

A shape that measures the same in every direction sounds like a description of a circle. It is not — there are infinitely many others, one of them is on a coin in most people's pockets, and a drill built from one cuts a nearly square hole.

geometry · Constant width

The two numbers that make up a width. A Reuleaux polygon with 3 sides, its centre marked as the origin, and the two supporting lines with normals 33° and 213°. The perpendicular distances from the origin to the two lines are marked; they add to the width.

The shape described from outside

A convex shape can be given by its boundary or by the family of lines that touch it, and the second description turns the constant-width condition into one line of arithmetic — after which the perimeter falls out, and curves with no corners at all can simply be written down.

geometry · Constant width

Named alongside it

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

Constant width Convexity Area Barbier's theorem Convex hull Curvature Duality Fourier series Integral geometry Perimeter Pi Reuleaux triangle

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