Spiric Sections

Cartesian equation:
(r2a2+c2+x2+y2)2=4r2(x2+c2)(r^{2} - a^{2} + c^{2} + x^{2} + y^{2})^{2} = 4r^{2}(x^{2} + c^{2})


After Menaechmus constructed conic sections by cutting a cone by a plane, around 150 BC which was 200 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 centre of the hole of the torus.

In the formula of the curve given above the torus is formed from a circle of radius aa whose centre is rotated along a circle of radius rr. The value of cc gives the distance of the cutting plane from the centre of the torus.

Whenc=0c = 0 the curve consists of two circles of radius aa whose centres are at (r,0)(r, 0) and (r,0)(-r, 0).

If c=r+ac = r + a the curve consists of one point, namely the origin, while if c>r+ac > r + a no point lies on the curve.