Villarceau Circles - Example

Example

For example, let the torus be given implicitly as the set of points on circles of radius three around points on a circle of radius five in the xy plane

Slicing with the z = 0 plane produces two concentric circles, x2 + y2 = 22 and x2 + y2 = 82. Slicing with the x = 0 plane produces two side-by-side circles, (y − 5)2 + z2 = 32 and (y + 5)2 + z2 = 32.

Two example Villarceau circles can be produced by slicing with the plane 3x = 4z. One is centered at (0, +3, 0) and the other at (0, −3, 0); both have radius five. They can be written in parametric form as

and

The slicing plane is chosen to be tangent to the torus while passing through its center. Here it is tangent at (16⁄5, 0, 12⁄5) and at (−16⁄5, 0, −12⁄5). The angle of slicing is uniquely determined by the dimensions of the chosen torus, and rotating any one such plane around the vertical gives all of them for that torus.

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