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 is 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.
Definitions Formally, a planar support function can be defined as a continuously differentiable function h {\displaystyle h} from the unit circle in the plane to real numbers, or equivalently as a function f ( θ ) = h ( ( cos θ , sin θ ) ) {\displaystyle f(\theta )=h{\bigl (}(\cos \theta ,\sin \theta ){\bigr )}} from angles to real numbers. For each point q {\displaystyle q} on the unit circle, it defines a line, the set of points p {\displaystyle p} for which p ⋅ q = h ( q ) {\displaystyle p\cdot q=h(q)} . This line is perpendicular to vector q {\displaystyle q} , passes through the point q h ( q ) {\displaystyle qh(q)} , and is at distance | h ( q ) | {\displaystyle |h(q)|} from the origin. A support function is anti-symmetric when, for all q {\displaystyle q} , h ( − q ) = − h ( q ) {\displaystyle h(-q)=-h(q)} , or equivalently in terms of angles f ( θ ) = − f ( θ + π ) {\displaystyle f(\theta )=-f(\theta +\pi )} , so that q {\displaystyle q} and − q {\displaystyle -q} define the same line as each other. Given any support function h {\displaystyle h} , its hedgehog is denoted H h {\displaystyle {\mathcal {H}}_{h}} . In terms of the function f {\displaystyle f} and the angle θ {\displaystyle \theta } it has the parametric equations
x = f ( θ ) cos θ − f ′ ( θ ) sin θ y = f ( θ ) sin θ + f ′ ( θ ) cos θ {\displaystyle {\begin{aligned}x&=f(\theta )\cos \theta -f'(\theta )\sin \theta \\y&=f(\theta )\sin \theta +f'(\theta )\cos \theta \end{aligned}}}
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. Hedgehogs can also be defined in the same way in higher dimensions, as envelopes of hyperplanes defined by support functions.
Examples The support function describing the supporting lines for a convex set K {\displaystyle K} is defined by h ( q ) = max { p ⋅ q ∣ p ∈ K } {\displaystyle h(q)=\max\{p\cdot q\mid p\in K\}} . The hedgehog of the support function of any strictly convex set is its boundary, parameterized by the angle of its supporting lines. 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. The astroid provides an example of a non-convex hedgehog.
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.
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