Difference between revisions of "Torus"
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[[Image:Ghdyt.png|right|thumb|300px]] | [[Image:Ghdyt.png|right|thumb|300px]] | ||
| − | A '''Torus''' is the mathematical term for a “[[doughnut]]-like” shape created by rotating a circle around | + | A '''Torus''' is the mathematical term for a “[[doughnut]]-like” shape created by rotating a [[circle]] around the [[x-axis]]. |
| − | If the “inner | + | If the “inner [[radius]]” (the width of the doughnut hole) is R, and the radius of the circular cross-section is r, then, the [[surface area]] of the torus is |
:S = 4 π<sup>2</sup> (R+r)r | :S = 4 π<sup>2</sup> (R+r)r | ||
| − | and the volume is | + | and the [[volume]] is |
:V = 2 π<sup>2</sup>(R+r)r<sup>2</sup> | :V = 2 π<sup>2</sup>(R+r)r<sup>2</sup> | ||
== Topology == | == Topology == | ||
| − | + | In [[topology]], a '''torus''' can be defined as the [[cartesian product]] of 2 [[circle]]s. It can be constructed from a [[rectangle]] by identifying its opposite [[edge]]s under the [[quotient topology]]. | |
This means that the torus surface is connected in the same way as the points on a rectangle, where you can “wrap-around” from one side of the rectangle to the other. This was a common topology used in the playing field of older video games. | This means that the torus surface is connected in the same way as the points on a rectangle, where you can “wrap-around” from one side of the rectangle to the other. This was a common topology used in the playing field of older video games. | ||
Revision as of 21:49, June 27, 2008
A Torus is the mathematical term for a “doughnut-like” shape created by rotating a circle around the x-axis.
If the “inner radius” (the width of the doughnut hole) is R, and the radius of the circular cross-section is r, then, the surface area of the torus is
- S = 4 π2 (R+r)r
and the volume is
- V = 2 π2(R+r)r2
Topology
In topology, a torus can be defined as the cartesian product of 2 circles. It can be constructed from a rectangle by identifying its opposite edges under the quotient topology.
This means that the torus surface is connected in the same way as the points on a rectangle, where you can “wrap-around” from one side of the rectangle to the other. This was a common topology used in the playing field of older video games.
The genus of a one-fold torus is 1.