Cranioid

Cranioid

A curve whose name means skull-like. It is given by the polar equation

r=asint+bsqrt(1-pcos^2t)+csqrt(1-qcos^2t),

where a,b,c>0, a<b+c, 0<p<1, 0<q<1, and p!=q. The top of the curve corresponds to t in [0,pi], while the bottom corresponds to t in [pi,2pi].

It has area given by

A=1/2pi[a^2-b^2(p-2)-c^2(q-2)+4bcF_1(1/2;-1/2,-1/2;1;p,q)],

where F_1(a;b_1,b_2;c;x,y) is an Appell hypergeometric function.

Portions of this entry contributed by Margherita Barile

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Shikin, E. V. Handbook and Atlas of Curves. Boca Raton, FL: CRC Press, pp. 140-142, 1995.