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, y, z), which depend on two parameters. Analytically,

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 x² + y² = R² and located in the plane through the z-axis (or by a rotation of the latter.) Parameter 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 [0, 2π].

torus by a rotating circle

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, x² + y² = R².

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

What if applet does not run?

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