constantdiameter

Constantdiameter is a term used in geometry to describe a property of sets or figures whose diameter remains fixed under a given consideration, or more loosely, to describe a family of shapes that share the same diameter. In Euclidean space, the diameter of a nonempty bounded set S is the greatest distance between any two points of S.

A precise formulation often appears in two forms. First, a family F of sets has constant diameter D if every member S in F satisfies diam(S) = D. Second, a single set S can be said to have constant diameter under a group G of transformations if diam(g(S)) = diam(S) = D for all g in G. Because the diameter is invariant under translations and rotations (isometries), the second form is most meaningful when G includes non-isometric maps or when one compares diameters across a family of placements.

Relation to constant width: Constantdiameter is closely related to the classical notion of shapes of constant width. A planar figure of constant width w has the same width in every direction, and for such figures the diameter equals w. Circles are the simplest examples, while noncircular shapes like Reuleaux polygons provide nontrivial instances of constant width and, hence, constant diameter.

Examples and applications: Circles and Reuleaux polygons are standard examples. In design and engineering, shapes with constant diameter can provide predictable clearance and rotational behavior, useful in mechanical parts, bearings, and coins. In computational geometry, constant-diameter families aid in shape matching and clustering where a fixed maximum pairwise distance is desirable.

See also: diameter (mathematics), width (geometry), shapes of constant width, Reuleaux polygon.