peratshape

Peratshape is a term used in discussions of parametric shape representations to denote a class of shapes produced by blending base primitives through parameterized coefficients. A peratshape is typically described by a function S that maps a surface parameter pair and a morph parameter to three-dimensional coordinates, i.e., S(u, v, t) describes how the shape changes as t varies.

The name is a portmanteau of “per” and “shape,” and it appears mainly in informal writing and tutorials as a way to illustrate continuous interpolation between shapes. It is not part of formal geometric taxonomy and does not have a standardized mathematical definition in peer-reviewed literature as of now.

A common construction represents S(u, v, t) as a blend of basis shapes B_i(u, v) with coefficients c_i(t), such that S(u, v, t) = sum_i c_i(t) B_i(u, v). The bases can be simple primitives or learned implicit shapes, and the coefficients are chosen to ensure smooth transitions, normalization, or other desirable constraints. Depending on the application, the parameter t may control morphing, detail levels, or other shape attributes.

Applications of peratshape ideas include controlled morphing in computer graphics, procedural generation, and interactive modeling workflows. The approach emphasizes intuitive control over global form through coefficient schedules but relies on the designer to select appropriate bases and morphing schemes. Its lack of formal standard means practitioners must consider compatibility and performance on a case-by-case basis.

See also: Parametric surface, Bezier surface, NURBS, Morphable model, Implicit surface.