FOLIOID

 Curve studied by P. van Geer in 1918. Loria 3D p. 240

 Polar equation: with n real > 0, i.e. with . Algebraic curve iff n is rational: if p and q are the numerator and denominator of n, the degree is 2(p +q) if p and q are odd, and 4(p + q) otherwise. For n =1, it is the circle with centre (b, 0) and radius a. For a = b (e = 1), it is the rose .

Case e < 1: the curve is rather close to a double conchoid of a rose ): In blue, the two conchoids of a rose, and in red the folioid, here with n = 4.

Case e > 1: the curve is composed of p closed curves if p and q are odd, and 2p curves otherwise. n = 1: circle n = 2 n = 3 n = 4 n = 5 n = 1/2 n = 3/2 n = 5/2 n = 7/2 n = 9/2 n = 1/3 n = 2/3 n = 4/3 n = 5/3 n = 7/3 n = 1/4 n = 3/4 n = 5/4 n = 7/4 n = 9/4 n = 1/5 n = 2/5 n = 3/5 n = 4/5 n = 6/5

If it is wrapped around the plane of the circle into a cone with vertex O and half-angle , with axis Oz, the projection on xOy of this wrapped around circle is the folioid . This provides a construction of folioids based on a simple circle in the case n < 1.