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Geometrical Torus 3-D Formula

The word torus comes from the Latin word meaning cushion. In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional spaces about an axis coplanar with the circle. In most cases the axis does not touch the circle and in this case the surface has a ring shape.

This then becomes the ring torus or simply torus if the ring shape is implicit. Other types are the horn torus which is generated when the axis is tangent to the circle. The spindle torus is generated when the axis is a chord of the circle. Another kind is when the axis is a diameter of the circle and surface is a sphere. The ring torus bounds a solid known as a toroid.

The formulas for three dimensional torus measurements are of length 2πR and radius r, created by cutting the tube and unrolling it by straightening out the line running around the centre of the tube. The losses in surface area and volume on the inner side of the tube happen to exactly cancel out the gains on the outer side.

A torus can be defined parametrically by:

x(u, v) =  (R + r \cos{v}) \cos{u} \,

y(u, v) =  (R + r \cos{v}) \sin{u} \,

z(u, v) =  r \sin{v} \,

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, r is the radius of the tube.



Figure 8 Torus Gumdrop Torus

2 Torus

 
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