Direct Approach to the Villarceau Circles of a Torus

摘要

Let T and T' be two tori which are linked like the two consecutive elements of a chain. Moreover the author assumes that T and T' have central circles of equal radii. By central circle of a torus he means the locus of the center of the sphere of constant radius which envelopes the torus. It is shown that the linked tori can be so placed that they are tangent to each other along simple closed curve gamma which is not a circle. In this mutually tangent position there is no gap between the two tori T and T'. It is shown that the above property is equivalent to the (slanting) circles of a torus discovered by Yvon Villarceau in 1848.