nice. your 2d shadow is degree 4, while mine was degree 6. my construction went around a circle, with a fixed-length “elbow” tracing the knot out on the torus. C
On May 25, 2020, at 1:28 PM, Brad Klee <bradklee@gmail.com> wrote:
Comparing parametric definitions, Cris has cos(5t) when the minimal possible choice is cos(3t). The choice 5t makes the knot curvier than necessary, and makes Chebyshev map higher degree.
The algorithm I mentioned has an even / odd case split, but it did manage to calculate the following weak-invariant data:
{X,Y,Z}={Sin[t] + 2 Sin[2 t], Cos[t] - 2 Cos[2 t], Sin[3 t]} 0 = 27 - 27 X^2 + 4 X^4 - 12 X^2 Y - 27 Y^2 + 8 X^2 Y^2 + 4 Y^3 + 4 Y^4 0 = -X^3 + 3 X Y^2 - 9 Z + 4 X^2 Z + 4 Y^2 Z
So the Trefoil can be found using polynomials degree 4 in (X,Y) and degree 3 in (X,Y,Z). In particular, the height function Z(X,Y) is a ratio of polynomials in the (X,Y) variables.
I will see if I can fix the algorithm later, and print off more polynomials for the first few cases.
--Brad
On Mon, May 25, 2020 at 12:50 PM Cris Moore via math-fun < math-fun@mailman.xmission.com> wrote:
for the trefoil, I had in mind
r[t_] := 2 + Sin[3 t] z[t_] := Cos[3 t] x[t_] := r[t] Cos[2 t] y[t_] := r[t] Sin[2 t] ParametricPlot3D[ {x[t], y[t], z[t]}, {t, 0, 3 Pi}]
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun