Spiric Section

A curve with Cartesian equation

$$(r^2 - a^2 + c^2 + x^2 + y^2) = 4r^2(x^2 + c^2).$$

Around 150 BC, Menaechmus constructed Conic Sections by cutting a Cone by a Plane. Two hundred years later, the Greek mathematician Perseus investigated the curves obtained by cutting a Torus by a Plane which is Parallel to the line through the center of the Hole of the Torus (MacTutor).

In the Formula of the curve given above, the Torus is formed from a Circle of Radius $a$ whose center is rotated along a Circle of Radius $r$. The value of $c$ gives the distance of the cutting Plane from the center of the Torus.

When $c = 0$, the curve consists of two Circles of Radius $a$ whose centers are at $(r,0)$ and $(-r,0)$. If $c = r+a$, the curve consists of one point (the origin), while if $c > r+a$, no point lies on the curve. The above curves have $(a,b,r)=(3,4,2)$, (3, 1, 2) (3, 0.8, 2), (3, 1, 4), (3, 1, 4.5), and (3, 0, 4.5).

References

MacTutor History of Mathematics Archive. ``Spiric Sections.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Spiric.html.