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In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a ''torus of revolution''. Real-world examples of (approximately) toroidal objects include inner tubes, swim rings, and the surface of a doughnut or bagel. A torus should not be confused with a solid torus, which is formed by rotating a disk, rather than a circle, around an axis. A solid torus is a torus plus the volume inside the torus. Real-world approximations include doughnuts, vadai or vada, many lifebuoys, and O-rings. In topology, a ring torus is homeomorphic to the Cartesian product of two circles: ''S''1 × ''S''1, and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1. The ring torus is one way to embed this space into three-dimensional Euclidean space, but another way to do this is the Cartesian product of the embedding of ''S''1 in the plane. This produces a geometric object called the Clifford torus, a surface in 4-space. In the field of topology, a torus is any topological space that is topologically equivalent to a torus. ==Geometry== A torus can be defined parametrically by: : where :''θ'', ''φ'' are angles which make a full circle, so that their values start and end at the same point, :''R'' is the distance from the center of the tube to the center of the torus, :''r'' is the radius of the tube. ''R'' is known as the "major radius" and ''r'' is known as the "minor radius". The ratio ''R'' divided by ''r'' is known as the "aspect ratio". A doughnut has an aspect ratio of about 2 to 3. An implicit equation in Cartesian coordinates for a torus radially symmetric about the ''z''-axis is or the solution of , where Algebraically eliminating the square root gives a quartic equation, The three different classes of standard tori correspond to the three possible aspect ratios between ''R'' and ''r'': * When , the surface will be the familiar ring torus. * corresponds to the horn torus, which in effect is a torus with no "hole". * describes the self-intersecting spindle torus. * When ''R'' = 0, the torus degenerates to the sphere. When , the interior of this torus is diffeomorphic (and, hence, homeomorphic) to a product of an Euclidean open disc and a circle. The surface area and interior volume of this torus are easily computed using Pappus's centroid theorem giving : These formulas are the same as for a cylinder of length 2π''R'' and radius ''r'', created by cutting the tube and unrolling it by straightening out the line running around the center of the tube. The losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side. As a torus is the product of two circles, a modified version of the spherical coordinate system is sometimes used. In traditional spherical coordinates there are three measures, ''R'', the distance from the center of the coordinate system, and θ and φ, angles measured from the center point. As a torus has, effectively, two center points, the centerpoints of the angles are moved; φ measures the same angle as it does in the spherical system, but is known as the "toroidal" direction. The center point of θ is moved to the center of ''r'', and is known as the "poloidal" direction. These terms were first used in a discussion of the Earth's magnetic field, where "poloidal" was used to denote "the direction toward the poles". In modern use these terms are more commonly used to discuss magnetic confinement fusion devices. 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「torus」の詳細全文を読む スポンサード リンク
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