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In geometry, Villarceau circles 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 (1813–1883). Mannheim (1903) showed that the Villarceau circles meet all of the parallel circular cross-sections of the torus at the same angle, a result that he said a Colonel Schoelcher had presented at a congress in 1891. ==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, ''x''2 + ''y''2 = 22 and ''x''2 + ''y''2 = 82. Slicing with the ''x'' = 0 plane produces two side-by-side circles, (''y'' − 5)2 + ''z''2 = 32 and (''y'' + 5)2 + ''z''2 = 32. Two example Villarceau circles can be produced by slicing with the plane 3''x'' = 4''z''. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Villarceau circles」の詳細全文を読む スポンサード リンク
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