geometrides

Geometrides are a class of abstract planar figures defined as connected shapes whose boundary is a closed chain composed of straight line segments and circular arcs with a fixed radius of curvature. Adjacent edges meet at a common endpoint and at a constant turning angle, so the boundary is locally regular. The construction allows a continuum between polygons (all edges straight) and curvilinear polygons (all edges curved). Geometrides are studied in recreational geometry, tiling theory, and computer graphics as a framework for exploring how the mix of straight and curved edges affects area, perimeter, and symmetry.

Geometrides are categorized by three parameters: n, the total number of boundary edges; s, the number of straight edges; and θ, the uniform turn angle at each vertex. Each curved edge has constant curvature along its length. Shapes that do not cross themselves are called simple geometrides; those that self-intersect are self-intersecting. The symmetry group of a geometride is determined by the arrangement of edge types and the angle θ.

Examples include a square with rounded corners (four straight edges and four equal circular corners) and a hexagon with alternating straight and arc edges. In applications, geometrides appear in tiling puzzles, architectural design, and computer graphics where soft-edged polygonal forms are desirable.

See also: curvilinear polygon, rounded polygon, Reuleaux polygon, tiling.