Extrapoloida

Extrapoloida is a term used in geometry to describe a class of solids generated by extrapolating a base polygonal cross-section along a fixed axis according to a prescribed rule. It can be defined more formally: let C0 be a base polygon in a plane z = 0. For t in [0, 1], let C(t) be obtained from C0 by applying a similarity transformation S(t) with scale factor s(t) and optional rotation R(t). The extrapoloid E is the union of all cross-sections C(t) as t ranges from 0 to 1.

If the scale factor s(t) is constant, E is a prism. If s(t) decreases linearly to zero, E resembles a generalized cone with an apex at t = 1. When the parameterization yields linear cross-sections connected by planar faces, the surface resembles a frustum-like solid; with nonlinear or nonuniform s(t) the side surfaces may be curved in the sense defined by the trajectory of C(t). The volume can be computed by V = ∫0^1 Area(C(t)) dt, and the surface area depends on the chosen path and cross-section transformations.

Variants of extrapoloida include linear, radial, and affine forms, each producing different degrees of shape complexity. Applications occur in computer-aided design, architectural modeling, and procedural generation, where a consistent cross-section is extended along a trajectory to create a family of related shapes. The concept serves as a generalization of extrusion and lofting, offering a flexible framework for generating solids from a base polygon.

See also: extrusion, lofting, frustum, polyhedral solids.