On 9/12/2010 5:11 PM, Fred E.J. Linton wrote:
The knot? -- the familiar trefoil knot, aka (2,3)-torus knot. The offending Wikipedia page? -- http://en.wikipedia.org/wiki/Torus_knot . The problem? [...] if one uses PostScript to "draw" the curve with the parametrization given above (with p=2 and q=3, of course), the result is the "lumpy" figure I've put up, as .ps and .png files, respectively, here:
http://tlvp.net/~tlvp/Trefoil/Wiki-2-3-Torus.ps , http://tlvp.net/~tlvp/Trefoil/Wiki-2-3-Torus.png .
There are really two trefoil knots: (i) the (2,3)-torus knot, which is a creature of topology, being defined up to homeomorphism of the complementary space, for which lumpiness is not an invariant; (ii) The trefoil arising as a a common motif in iconography and the visual arts. Thare are a number of variants of these, each defined up to similarity. They are dealt with in the section "Trefoils in religion and culture" on the Wikipedia page http://en.wikipedia.org/wiki/Trefoil_knot You seem to be interested in the second. Conceivably you could try modifying the topology article, but you will have to deal somehow with the objection that "lumpiness" is not a topological invariant, so why spoil a perfectly good parametrization by replacing it with a more complicated one that's solving a non-problem for this article? A better approach might be to expand the art section with a little mathematics giving a suitable parameterization of the kind you want. The ellipse-based one you suggest seems to work fine, and translates back from your postscript if I'm not mistaken as x = sin(3t)cos(t) + 3 cos(3t)sin(t) y = 3 cos(3t)cos(t) - sin(3t)sin(t) Here's another I came up with just now that also works. r = 4 - cos(3t) x = r sin(2t) - 3 sin(t) y = r cos(2t) + 3 cos(t) The parametrization in the torus knot article is this one with "4 - " replaced by "2 + " and the second half deleted from the expressions for x and y. Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]