Torus

Geometry

In geometry, a torus (pl. tori) is a doughnut-shaped surface of revolution generated by revolving a circle about an axis coplanar with the circle. The sphere is a special case of the torus obtained when the axis of rotation is a diameter of the circle. If the axis of rotation does not intersect the circle, the torus has a hole in the middle and resembles a ring doughnut, a hula hoop or an inflated tire (U.K. tyre). The other case, when the axis of rotation is a chord of the circle, produces a sort of squashed sphere resembling a round cushion. Torus was the Latin word for a cushion of this shape.

Related Topics:
Geometry - Doughnut - Surface of revolution - Circle - Coplanar - Sphere - Diameter - Hula hoop - Tire - Chord - Latin

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A torus can be defined parametrically by

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:x(u, v) = (R + r cos v) cos u

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:y(u, v) = (R + r cos v) sin u

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:z(u, v) = r sin v

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where u, v ∈ , R is the distance from the center of the tube to the center of the torus, and r is the radius of the tube.

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The equation in Cartesian coordinates for a torus azimuthally symmetric about the z-axis is

Related Topics:
Cartesian coordinates - Axis

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:left(R - sqrt{x^2 + y^2} ight)^2 + z^2 = r^2

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The surface area and interior volume of this torus are given by

Related Topics:
Surface area - Volume

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:A = 4pi^2 Rr ,

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:V = 2pi^2R r^2. ,

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According to a broader definition, the generator of a torus need not be a circle but could also be an ellipse or any other conic section.

Related Topics:
Generator - Ellipse - Conic section

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