A curve whose name means skull-like. It is given by the polar equation
 |
where 
, 
, 
, 
, and 
. The top of the curve corresponds to ![t in [0,pi]](/images/equations/Cranioid/Inline6.svg)
, while the bottom corresponds to ![t in [pi,2pi]](/images/equations/Cranioid/Inline7.svg)
.
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)],](/images/equations/Cranioid/NumberedEquation2.svg) |
where 
is an Appell
hypergeometric function.
Cranioid
Portions of this entry contributed by Margherita Barile
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References
Shikin, E. V. Handbook and Atlas of Curves. Boca Raton, FL: CRC Press, pp. 140-142, 1995.Cite this as:
Barile, Margherita and Weisstein, Eric W. "Cranioid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Cranioid.html