SPIRIC OF PERSEUS

 Perseus, 2nd century BC: Greek scholar.

 Reduced Cartesian equation:  with A > B, i.e. . Bicircular quartic, rational if C = 0, having real points only if . Polar equation: .

The spirics of Perseus are the bicircular quartics with a symmetry centre. Therefore, they are the cyclic curves with respect to a centred conic, the deferent (corresponding to the "initial curve" in the link) of which is this conic. In other words, they are the envelopes of the circles the centres of which describe a centred conic, and such that the centre of the conic has a constant power with respect to these circles.

 Case where 0 < B < A are fixed, and C is variable. the curves inside the two circles, in the left and right parts, are not planar sections of a real torus. Case where B < 0 < A are fixed, and C is variable.

Historically, these curves were defined as the sections of a torus by a plane parallel to its axis; but to obtain all the real curves given above, one has to consider complex tori.
For a torus with centre O, axis Oz, with major and minor radii a and b, cut by the plane parallel to Oz located at distance d from O, we get, in a frame with origin the projection of O on this plane, the Cartesian equation above with: .
This comes from the equation:  of these curves.

Note that the torus above is real only if .

 Spirics of Perseus of a ring torus Spirics of Perseus of a spindle torus

When , i.e. db (distance of the plane to the axis equal to the minor radius), we get the Cassinian ovals, which reduce to the lemniscate of Bernoulli when C = 0, i.e. a  = 2 b.
When C = 0, i.e.  (plane tangent inside the torus), we get the Booth curves (or Hippopedes of Proclus), which also reduce to the lemniscate of Bernoulli when a  = 2 b.

The limit case A = B (case where the torus is reduced to a sphere) gives circles.

The spirics of Perseus are also the isoptic curves of the centred conics.

 Picture by Mr Konieczka and Ms Gautherin, Laboratory of Mechanics of the Department of Mechanical Engineering of École normale supérieure (ENS) de Cachan Disk composed of a birefringent diametrically charged material observed under polarised monochromatic light. The black lines - the isochromatic lines - are the loci of the points for which the difference of the 2 principal constraints is constant.