Spiric Sections

Cartesian equation:
(r2 - a2 + c2 + x2 + y2)2 = 4r2(x2 + c2)



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 a whose centre is rotated along a circle of radius r. The value of c gives the distance of the cutting plane from the centre of the torus.

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

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


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JOC/EFR/BS January 1997

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