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.
Read more about this topic: Villarceau Circles
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“Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
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