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An envelope parameterized by its Gauss Map. The parametric equations for a hedgehog are

$\displaystyle x$ $\textstyle =$ $\displaystyle p(\theta)\cos\theta+p'(\theta)\sin\theta$  
$\displaystyle y$ $\textstyle =$ $\displaystyle p(\theta)\sin\theta+p'(\theta)\cos\theta.$  

A plane convex hedgehog has at least four Vertices where the Curvature has a stationary value. A plane convex hedgehog of constant width has at least six Vertices (Martinez-Maure 1996).


Langevin, R.; Levitt, G.; and Rosenberg, H. ``Hérissons et Multihérissons (Enveloppes paramétrées par leur application de Gauss.'' Warsaw: Singularities, 245-253, 1985. Banach Center Pub. 20, PWN Warsaw, 1988.

Martinez-Maure, Y. ``A Note on the Tennis Ball Theorem.'' Amer. Math. Monthly 103, 338-340, 1996.

© 1996-9 Eric W. Weisstein