vollecone

The vollecone is a generalized geometric shape that extends the classical cone by allowing the cross-sectional radius to vary along its axis. It is defined by an apex A, an axis direction d (a unit vector), a height H > 0, and a radius profile R: [0, H] → [0, ∞). The object consists of all points whose perpendicular distance to the axis at height z is at most R(z), with z measured from the apex along the axis. Intuitively, the radius can grow, shrink, or oscillate as height increases, producing a wide range of forms beyond the standard conical surface.

Formally, for each z ∈ [0, H], the cross-section perpendicular to d at the point A + z d is a circle of radius R(z). If R is linear in z, the vollecone reduces to a ordinary cone segment; if R is constant, the shape resembles a cylindrical segment. Nonlinear profiles yield tapered, flared, or more intricate boundary shapes, while smooth profiles produce smooth lateral surfaces. The boundary is a ruled surface when R is piecewise linear and a smooth surface when R is differentiable.

The volume of a vollecone is given by V = ∫_0^H π [R(z)]^2 dz, linking its two-dimensional cross-sections to its three-dimensional size. In computer graphics, architectural design, and CAD, vollecones are used as flexible primitives or in swept-surface constructions to model tapered or variable-radius components.

Terminology varies; the term vollecone is not universally standardized and is often described more generally as a generalized cone or tapered cone with a variable radius profile. It serves as a conceptual tool for exploring conic-like shapes with nonuniform cross-sections.