Reuleaux Polygon

Reuleaux Polygon is named after Franz Reuleaux. Its feature is that it has a shape of constant diameter no matter the orientation.

This feature of constant diameter only works with polygons having odd-numbered sides. The perimeter is constructed using a series of circular arcs whose centres lie on the opposing edge.

Some countries use coins in the shape of Reuleaux Polygons. The Canadian CAD one-dollar coin is 11-sided while the UK GBP 50p and 20p have 7 sides. Besides being decorative one physical advantage is that such a shape requires a less complicated coin-operated machine mechanism to verify an exact expected diameter. The diameter is constant no matter the orientation of the coin so a simple set of parallel straight-edged feelers can pinch the coin from opposite sides.

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The simple Reuleaux Polygon can be constructed using a regular polygon as a reference. In this example it's a 3-sided polygon (triangle) shown in blue. The green arcs are parts of a circle whose centres are the vertices. Note that the vertices retain sharp corners. This is not always a desirable physical trait. However this resultant perimeter shape does satisfy the constant-diameter goal.

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If we introduce the concept of 'extension', as one site called it, the vertices can be rounded. Here the reference polygon's sides are extended (shown in red). The vertices are used as centres for smaller arcs, shown in orange, while also used to draw the larger arcs in opposition.

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Rounding the vertices in this way makes the Reuleaux Polygon approach a circle - I call "Circle Affinity". With an affinity of 0.5 (half-way towards a perfect circle) the ratio of arc radii is 3:1. The smaller arg being 1/3 the radius of the larger arc.

Here is a program that explores Reuleaux Polygons of various number of sides and degree of vertex rounding.

Revised 1 Jul 2020