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单词 Torus
释义

torus


Visually, the torus looks like a doughnut. Informally, we take a rectangleMathworldPlanetmathPlanetmath, identify two edges to form a cylinderPlanetmathPlanetmath, and then identify the two ends of the cylinder to form the torus. Doing this gives us a surface of genus one. It can also be described as the Cartesian product of two circles, that is, S1×S1. The torus can be parameterized in Cartesian coordinatesMathworldPlanetmath by:

x=cos(s)(R+rcos(t))
y=sin(s)(R+rcos(t))
z=rsin(t)

with R the major radius and r the minor radius are constant, and s,t[0,2π).


Figure 1: A torus generated with Mathematica 4.1

To create the torus mathematically, we start with the closed subset X=[0,1]×[0,1]2. Let X* be the set with elements:

{x×0,x×10<x<1}
{0×y,1×y0<y<1}

and also the four-point set

{0×0,1×0,0×1,1×1}.

This can be schematically represented in the following diagram.

Diagram 1: The identifications made on I2 to make a torus.

Opposite sides are identified with equal orientations, and the four corners

are identified to one point.

Note that X* is a partitionMathworldPlanetmathPlanetmath of X, where we have identified opposite sides of the square together, and all four corners together. We can then form the quotient topology induced by the quotient map p:XX* by sending each element xX to the corresponding element of X* containing x.

Titletorus
Canonical nameTorus
Date of creation2013-03-22 12:55:17
Last modified on2013-03-22 12:55:17
OwnerDaume (40)
Last modified byDaume (40)
Numerical id15
AuthorDaume (40)
Entry typeDefinition
Classificationmsc 54B15
Classificationmsc 51H05
Related topicMobiusStrip
Related topicNTorus
Related topicSurfaceOfRevolution2
Definesmajor radius
Definesminor radius
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更新时间:2025/5/4 14:39:11