Math Art
p-Norm Unit Sphere (superellipsoid, p knob)

p-Norm Unit Sphere (superellipsoid, p knob)

p-Norm Unit Sphere (superellipsoid, p knob) is an algebraic surface, defined by an implicit equation, immersed.

Open p-Norm Unit Sphere (superellipsoid, p knob) in the interactive viewer →

algebraic aperiodic closed closed-orientable def-implicit immersed implemented tradition-classical

Formula

abs⁡(x)4+abs⁡(y)4+abs⁡(z)4−1=0\operatorname{abs}\left(x\right)^{4} + \operatorname{abs}\left(y\right)^{4} + \operatorname{abs}\left(z\right)^{4} - 1 = 0

Properties

Family
algebraic
Given by
an implicit equation
Curvature
no condition imposed
Periodicity
not periodic
Ends
0
Embedding
immersed
Orientable
yes
Fidelity
exact
Exactness
elementary

Definition

The unit spheres of the p-norms on R3 (the equal-exponent Lame / superellipsoid family): |x|^p + |y|^p + |z|^p = 1, octahedron at p = 1, sphere at p = 2, cube as p -> infinity, convex for p >= 1. SHIPPED with p as a live knob (default 4); gated on two exact identities -- the p = 2 member IS the unit sphere (|F| = 0 on it) and every member is invariant under the full order-48 octahedral group.

Sources

Closed form reproduced numerically against the shipped implementation (math_art/surfaces/algebraic.py) over 240 sample points: matches oracle x 1 over 240 points (worst 0).