Hedgehog (geometry)

Not to be confused with hedgehog (hypergraph) or hedgehog space.
The astroid, constructed as an envelope of lines. The corresponding support function ranges from a minimum of 0 for the points on the horizontal and vertical axes to a maximum at the four diagonal points on the unit circle.
A hypocycloid that forms a self-crossing hedgehog

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. The astroid forms a non-convex hedgehog, and the deltoid curve forms a projective hedgehog.

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