Sketchup Sliceform – About the Torus

A torus is a shape like a ring doughnut (or donut in US English). It can be constructed geometrically by taking a circle and rotating about an axis.

You can find out more about a torus on Wikipedia or on Wolfram MathWorld

There are a number of ways to slice a torus to end up with circles. The first is to cut a perpendicular to the axis of rotation. The second is to cut with a plane through the axis; this yields the circles that are being rotated to generate the torus. I am going to slice the torus to show Villarceau circles details of which are on Wikipedia.

I will slice in the direction of a tangent plane.

(Image By Kieff  via Wikimedia Commons)

This video shows the slicing at the tangent I am going to use.

Villarceau’s original paper was published in 1848:

Théorème sur le tore de M. Villarceau (Yvon). Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale, Sér. 1, 7 (1848), p. 345-347

You can get this original paper by going here and searching for Villarceau.


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