Longshapes

Longshapes are geometric figures whose overall form is markedly elongated along a principal axis. In practical terms, they have a length L that greatly exceeds their width W, with an aspect ratio commonly greater than 2 or 3 depending on the context. They may be straight or curved and can be defined by the locus of a central spine with a width function w(s) along the length.

Canonical examples include rectangles with high aspect ratio, capsules (stadium shapes) produced by rounding the ends of a rectangle, elongated ellipses, and tubular or polyomino-like shapes composed of aligned units. Variants allow tapered ends or varying width; some longshapes are generated by sweeping a profile along a curve or by extruding a 2D cross-section along a path. Their geometry often features a relatively uniform cross-section over much of the length, with changes near the ends.

Applications of longshapes appear in tiling and packing problems where high anisotropy influences density and stability, in computer graphics and procedural generation for roads, rivers, or corridors, and in data visualization as elongated bars or streamer-like elements. They also serve in shape analysis to model elongated features in images or spatial data, where directionality and continuity along the length are of interest.

Notes: The term longshape is used in some design and modeling discussions but is not a formal mathematical category; it overlaps with concepts such as slender shapes, capsules, and elongated rectangles. Related terms include aspect ratio, slender body, and capsule.