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Standard torus

In mathematics, a standard torus is a circular torus of revolution, that is, any surface of revolution generated by rotating a circle in three dimensional space about an axis coplanar with the circle. When the axis passes through the center of the circle, the torus degenerates into a sphere. This case is normally excluded from the definition. With no further restrictions on the location of the axis in this plane, there are three classes of standard tori: the ring torus, where the axis is disjoint from the circle; the horn torus, where the axis is tangent to the circle; and the spindle torus, where the axis meets the circle in two distinct points. The three classes may also be characterized by the extent of their self-intersection.

Parametric equation and classes

After a change of coordinates so that the axis is the z axis, a standard torus can be defined parametrically by:

$x\left(u, v\right) = \left(R + rcos\left\{v\right\}\right) cos\left\{u\right\} ,$
$y\left(u, v\right) = \left(R + r cos\left\{v\right\}\right) sin\left\{u\right\} ,$
$z\left(u, v\right) = r sin\left\{v\right\} ,$
where
u, v are in the interval [0, 2π),
R is the distance from the center of the tube to the center of the torus, and
r is the radius of the tube.

The three different classes of standard tori correspond to the three possible relative sizes of r and R. When R > r, the surface will be the familiar ring torus. The case R = r corresponds to the horn torus, which in effect is a torus with no "hole". The case R < r describes a self-intersecting surface called a spindle torus. When used by itself, the word torus usually refers to the ring torus, but may also refer to any topological space homeomorphic to a ring torus.

Kepler called the external portion of the spindle torus the "apple", and defined the "lemon" as the interior portion of the surface. The two surfaces have the same equations for their cross-sections in the x-z plane

$z=pmsqrt\left\{r^2 - \left(x-R\right)^2\right\}$,
but for the apple, x is restricted to the to the interval $\left[-\left(r+R\right), \left(r+R\right)\right]$, and for the lemon, x is restricted to $\left[-\left(r-R\right), \left(r-R\right)\right]$.

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