The Dupin Cyclide family of surfaces is a member of a larger group of surfaces referred to as Canal surfaces or Swept surfaces, all of which are envelopes of sweeping objects.
This formula is the inversion of a torus in the x-z plane, with the origin as inversion center.
(r1^2 - dy^2 - (dx + r0)^2)*(r1^2 - dy^2 - (dx - r0)^2)* (x^4+y^4+z^4)+ 2*((r1^2 - dy^2 - (dx + r0)^2 )* (r1^2 - dy^2 - (dx - r0)^2)* (x^2*y^2+x^2*z^2+y^2*z^2))+ 2*ri^2*((-dy^2-dx^2+r1^2+r0^2)* (2*x*dx+2*y*dy-ri^2)-4*dy*r0^2*y)* (x^2+y^2+z^2)+ 4*ri^4(dx*x+dy*y)*(-ri^2+dy*y+dx*x)+ 4*ri^4*r0^2*y^2+ri^8=0Where
r0=4.9,r1=5,dx=2,dy=0,ri=3 (double crescent) r0=3,r1=5,dx=3,dy=0,ri=9 (degenerate w. arch r0=6,r1=0.5,dx=3,dy=0,ri=12 (plain)
Here is the PoV3-source I used
I have also made an inc to generate PoV "poly" code. You can get it here.
Note on program. To find "interesting" shapes, it is helpful to have minor radius of the generating Torus larger than the major. Also, place the inversion center inside the Torus.