Ratioshape

Ratioshape is a term used in geometry and design to describe the proportional relationship that characterizes a shape, captured by a ratio of two geometric quantities. It serves as a compact descriptor that can classify shapes into families and guide the generation of new shapes by fixing a ratio parameter.

Formally, let S be a shape and let F1(S) and F2(S) denote two geometric quantities (such as a length, area, or radius). The ratioshape r of S with respect to the pair (F1, F2) is r = F1(S) / F2(S). Two shapes S and T share the same ratioshape for that pair if F1(S)/F2(S) = F1(T)/F2(T). Different pairs (F1, F2) yield different ratioshapes for the same shape, reflecting different proportional aspects.

Common cases include the aspect ratio, where F1 = width and F2 = height, and the circle-based ratio of area to a squared radius for certain analyses. For polygons, one may compare characteristic lengths to circumradius, or area to a product of characteristic lengths.

Properties emphasize that ratioshape is generally scale-sensitive unless the chosen features are themselves scale-invariant or combined to cancel scale. It is not a universal invariant; it depends on the selected feature pair and on the transformation applied to the shape.

Applications appear in computer graphics, responsive design, architectural patterning, and shape-based retrieval in image databases, where a fixed ratioshape serves as a compact descriptor to compare and generate shapes.

See also: aspect ratio, shape descriptor, similarity (geometry). Notes: Ratioshape is presented as a conceptual term for classification and design purposes; it is not a single standardized mathematical category across all disciplines.