Torus is a doughnut shapedrotateA surface is generated by a circle revolving around an axis coplanar with the circle.Topologically, a torus is a closed surface defined as the product of two circles.
The homeomorphic surface is called torus, which isGenusA orientable closed surface that is 1.In general, the torus can be seen as the result of a box overlapping the left and right sides and the upper and lower sides in a counterclockwise direction.
Figure 1
Definition 2: If a linear algebraic group G is isomorphic to a D (n, k), then G is said to be a torus.The connected diagonalizable algebraic group must be a torus.[1]
Geometric meaning
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Geometrically, atorusIt is a doughnut shaped rotating surface generated by a circle revolving around an axis coplanar with the circle.The ball can be regarded as a special case of the torus, that is, when the axis of rotation is the diameter of the circle.If the shaft does not intersect the circle, there is a hole in the middle of the circle, like a doughnut, a hula hoop, or a inflated tire surface.On the other hand, when the axis is a chord of a circle, a flattened spherical surface will be generated, just like a round cushion.
The torus can be parametrically defined as:
amongu,v∈ [0, 2π],RIs the distance from the center of the pipe to the center of the picture,rIs the radius of the tube.
The torus equation of z-axis azimuth symmetry in rectangular coordinate system is:
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The surface area and internal volume of the torus are as follows:
According to a more general definition, the generator of torus need not be a circle, but an ellipse or any conic curve.
topology
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Topologically, atorusIs a closed surface defined as the product of two circles.
N-dimensional torus
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Algebraic groupAn important part ofsubgroup。Refers to an algebraic group isomorphic to the group D (n, K) formed by all invertible diagonal matrices of order n.The rational representations of torus are completely reducible, and the irreducible representations are one-dimensional.Therefore, the representation theory of torus is completely characterized by the characteristic group.The maximal torus subgroup of an algebraic group plays an important role in the structure and representation theory of this algebraic group.Different maximal torus are mutual in algebraic groupconjugateOf.
nature
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The torus is Abelian Lie group.
Connected Abelian Lie groups are isomorphic to the product of linear space and torus.[2]
Toroidal lens is a hat shaped lens that is equivalent to cutting at the edge of the torus.The shape of the lens surface is like a piece cut off from the edge of a doughnut.The highest and lowest curvatures are arc curvatures.Therefore, unlike a very popular view, toroidal lens is not an ellipsoid.Mainly used forcontact lenses,KeratoscopeAnd intraocular lens to correct high corneal astigmatism.
