**Cartesian equation: **

(*r*^{2} - *a*^{2} + *c*^{2} + *x*^{2} + *y*^{2})^{2} = 4*r*^{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 *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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