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Toric Sections: Spiric Sections, Villarceau Circles, Lemniscate of Bernoulli, Cassini Oval, Hippopede, Toric Section

Chapters: Spiric Sections, Villarceau Circles, Lemniscate of Bernoulli, Cassini Oval, Hippopede, Toric Section. Source: Wikipedia. Pages: 28. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In geometry, Villarceau circles (pronounced ) are a pair of circles produced by cutting a torus diagonally through the center at the correct angle. Given an arbitrary point on a torus, four circles can be drawn through it. One is in the plane (containing the point) parallel to the equatorial plane of the torus. Another is perpendicular to it. The other two are Villarceau circles. They are named after the French astronomer and mathematician Yvon Villarceau (18131883). 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, x + y = 2 and x + y = 8. Slicing with the x = 0 plane produces two side-by-side circles, (y 5) + z = 3 and (y + 5) + z = 3. 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 (5, 0, 5) and at (5, 0, 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. A proof of the circles existence can be constructed from the fact that the slicing plane is tangent to the torus at two points. One characterization of a torus is that it is a surface of revolution. Without loss of generality, choose a coordinate system so that the axis of revolution is the z axis. Be...More: