Superegg¶

A superegg is the solid of revolution formed by rotating an elongated superellipse of exponent greater than two about its long axis.

Version: v1 · 2026-09-28 · History
Domain-specific #: 7771
Origin domain: Geometry

Core Idea¶

A superegg is the solid of revolution produced by rotating an elongated superellipse with exponent \(p>2\) about its longest axis. With horizontal radius \(R\) and half-height \(h\), the solid is described by [ \left|\frac{\sqrt{x^2+y2}}{R}\right|^p+ \left|\frac{z}{h}\right|^p\leq1. ] Its horizontal sections are circles, and the exponent controls the flattening at the equator and tips. Using equality rather than inequality selects only the boundary surface; the inequality includes the three-dimensional interior.

Scope of Application¶

Superegg operates within geometry and exact shape realization wherever an elongated superellipse with exponent p > 2 is revolved about its long axis, producing an axially symmetric solid with circular horizontal sections. The map requires that construction and parameter regime: a merely egg-like body, an ellipsoid at p = 2, or a general superellipsoid lacking the revolution structure lies outside it.

Analytic solid definition — the inequality in radius and height coordinates specifies the filled three-dimensional body through R, h, and p.
Surface-of-revolution modeling — replacing the inequality by equality selects the boundary surface while preserving the same generating profile and axis.
Cross-section analysis — horizontal slices are derived as circles whose radii vary with height according to the defining relation.
Parameter-family comparison — R/h controls elongation and p controls flattening, allowing instances to vary without leaving the family while p > 2 remains satisfied.

Clarity¶

Naming a superegg distinguishes a particular solid of revolution from an object that merely looks egg-shaped. The generating profile must be an elongated superellipse, the rotation must be about its long axis, and the exponent must exceed two. This also locates the boundaries cleanly: \(p=2\) gives an ellipsoid, a general superellipsoid need not have circular horizontal sections, and replacing the defining inequality with equality denotes only the boundary surface rather than the filled solid.

Manages Complexity¶

Rotationally symmetric egg-like solids could be described point by point or by a large mesh, but the superegg equation reduces the entire shape to three geometric parameters and one construction. The horizontal radius R and half-height h set scale and elongation; the exponent p controls how sharply the profile transitions between its equator and tips; revolution about the long axis guarantees circular horizontal sections.

Abstract Reasoning¶

A classification move runs from a rotational solid's generating profile and axis to its geometric type. If the profile is an elongated superellipse, rotation is about its long axis, and the exponent satisfies p > 2, the result is a superegg. Circular horizontal sections support that diagnosis; an egg-like silhouette by itself does not, because an unrelated solid can have a similar outline without the required construction.

Knowledge Transfer¶

Within geometry, the superegg transfers literally across analytic descriptions, cross-section calculations, solid modeling, and physical realizations. The same construction carries from an ideal equation to a manufactured object: choose an elongated superellipse with p > 2, revolve it about its long axis, and preserve circular horizontal sections. The parameters R, h, and p, the equality-versus-inequality distinction, and the diagnostic boundary at p = 2 let geometers compare scale, elongation, flattening, surface, and solid without confusing a visual resemblance with membership in the family.

Relationships to Other Abstractions¶

Current abstraction Superegg Domain-specific

Parents (1) — more general patterns this builds on

Superegg is a kind of Superquadrics Domain-specific

A superegg satisfies the superquadric family's signed-power, positive-scale, positive-exponent geometry: its horizontal axes share radius R, its vertical scale is h, and its equal exponent p > 2 produces the stipulated elongated solid or boundary.

Hierarchy path (1) — routes to 1 parentless root

Superegg → Superquadrics → Scale

Neighborhood in Abstraction Space¶

Superegg sits in a sparse region of the domain-specific corpus (78^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08