# Knots on a Torus

The applet below displays the curves that wind on the surface of a torus a prescribed number of times. The applet makes use of the JavaView library and is a slight modification (hopefully, an improvement) of one of their tutorial samples.

Torus is a two dimensional surface defined in a 3d space, which means that, to describe a torus, one needs three coordinates, say

x(*u*, *v*) = (R + r cos *v*) cos *u*,

y(*u*, *v*) = (R + r cos *v*) sin *u*,

z(*u*, *v*) = r sin *v*.

Torus is described by a circle of radius r centered at the points of circle *u* is responsible for the position of the variable circle relative to the xy-coordinate plane; parameter *v* describes the position of a point on that circle. Both parameters range over the interval

The curves that the applet displays are defined with a single parameter,

u(t) = Ut, v(t) = Zt, t ∈[0, 2π].

Parameter U tells us how many times the curve winds around z-axis. Parameter Z gives the number of turns around the basic circle,

What if applet does not run? |

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