Hedgehog (geometry)

In differential geometry, a hedgehog or plane hedgehog is a type of plane curve, the envelope of a family of lines determined by a support function.

More intuitively, sufficiently well-behaved hedgehogs are plane curves with one tangent line in each oriented direction.

A projective hedgehog is a restricted type of hedgehog, defined from an anti-symmetric support function, and (again when sufficiently well-behaved) forms a curve with one tangent line in each direction, regardless of orientation.

Every closed strictly convex curve, the envelope of its supporting lines.

Hedgehogs can also be defined from support functions of hyperplanes in higher dimensions.

from the unit circle in the plane to real numbers, or equivalently as a function

on the unit circle, it defines a line, the set of points

it has the parametric equations[1] A hedgehog is non-singular when it has a tangent line at each of its points.

A projective hedgehog is defined by an anti-symmetric support function.

[4] When a convex set is not strictly convex (it has a line segment in its boundary), its support function is continuous but not continuously differentiable, and the parametric equations above jump discontinuously across the line segment instead of defining a continuous curve, so it is not defined as a hedgehog.

[5] An example of a projective hedgehog, defined from an anti-symmetric support function, is given by the deltoid curve.

The deltoid is a simple closed curve but other hedgehogs may self-intersect, or otherwise behave badly.

In particular, there exist anti-symmetric support functions based on the Weierstrass function whose corresponding projective hedgehogs are fractal curves that are continuous but nowhere differentiable and have infinite length.

[4] Every strictly convex body in the plane defines a projective hedgehog, its middle hedgehog, the envelope of lines halfway between each pair of parallel supporting lines.

The points of the middle hedgehog are the midpoints of line segments connecting the pairs of points where each pair of parallel supporting lines contact the body.

It has finite length, equal to half the perimeter of the given body.

Correspondingly, any sufficiently well-behaved projective hedgehog has a unique tangent line in each direction without respect to orientation.

Pairs of hedgehogs can be combined by the pointwise sum of their support functions.

When a projective hedgehog has finite length, a construction of Leonhard Euler shows that its involutes of sufficiently high radius are curves of constant width.

[8] More generally, hedgehogs are the natural geometrical objects that represent the formal differences of convex bodies: given (K,L) an ordered pair of convex bodies in the Euclidean vector space

Case of smooth convex bodies with positive Gauss curvature: Subtracting two convex hypersurfaces (with positive Gauss curvature) by subtracting the points corresponding to a same outer unit normal to obtain a (possibly singular and self-intersecting) hypersurface:

The idea of using Minkowski differences of convex bodies may be traced back to a couple of papers by A.D. Alexandrov and H. Geppert[9] in the 1930s.

[10] In a long series of papers, hedgehogs and their extensions were studied by Y. Martinez-Maure under various aspects.

[11] The most striking result of this hedgehog theory was the construction of counterexamples to an old conjectured characterization of the 2-sphere.