LOGO-L>Trefoil Knot Maths

["mike sandy" <MJSANDY@email.msn.com>]

"The trefoil knot is equivalent to the overhand knot.  One of the
projections
of the trefoil onto a plane is a hypotrochoid, which is the locus of
a point, P, on the radius vector of a circle (radius rs),which rolls inside
another circle (radius rl).
The parametric equations are:
 X = dr*cos(theta)+h*cos ((dr/rs)*theta)
 Y = dr*cos(theta)-h*sin ((dr/rs)*theta)
Where h = distance of P from the centre of the circle(rs)
      dr = rl-rs
If z =  sin(rl*theta/rs) and theta runs from 0 to 360*rs,
the trefoil knot is obtained if rs:rl = 2:3.
h ~ 0.15*rs, but will depend on the radius of the encircling surface
band......"

Dale, Thanks for the comment.
The above equations are correct in this sense: if a rope is tied in an
over-hand knot
and the ends are spliced together, it remains a knot however the rope is
deformed, so
the plotted surface is a particular model for the knot.  Regarding the
trefoil knot, I am not so sure.
The only definition I have found is "diagramatic" i.e. the diagram shows 3
cross-over points
and a particular symmetry.  The 3:2 hypotrochoid is the closest match I
could find (see
"A Catalog of Special Plane Curves. J D Lawrence. Dover '72".
Perhaps others with access to a good reference library could help.
The x and y equations are in: "A Book of Curves. E H Lockwood. CUP '63", but
not the deduction.
The z function I deduced as follows:
Theta for a rs:rl hypotrochoid goes thro' rs cycles and from
the symmetry, the cross-overpoints occur at:
 theta = 180 + 360*k/rl (k = 0 1,..(rl-1)).
I assumed at a cross-over point there would be a max and min, so in the rs
cycles there would
be rl max. For sin(rl*theta) there are rl max in 1 cycle and for sin
(rl*theta/rs) these are stretched
over rs cycles.
On comparison with x and y perhaps an additional factor, of dr, to give
dr*sin (rl*theta/rs)
is called for!
I don't know of any reference books for knots, though there have been
articles in Scientific American.
Incidentally, the hypotrochoid is a model for a braided mat, which opens
additional teaching projects.



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