Vector Equation of a Toroidal Spiral

A toroidal spiral is defined by a vector function r(t)=((a+cos bt)cos t,(a+cos bt)sin t,sin bt) for some a,b.

In order to change the limits on t (currently 0 to about 4pi), use the sliders t_min and t_max near the center-top of the
screen. This also allows you to see the curve being "traced out" over time. If you want to rotate the figure, use the
sliders at the top left (theta and phi).

The equation may be adjusted by changing the radius and rate parameters on the top right. What impact do these
have on the curve? Why do the curves for integer values look so different from those with half-integer values?

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[For geogebra users: The view can be rotated by adjusting theta and phi... the perspective can be similarly adjusted by changing A_x and chi. The parametric curve is described as a locus, requiring the creation of a custom slider (at the top of the page).]

Elisha Peterson, 2-Apr-07, Created with GeoGebra