Has anyone noted this rather obvious astroid = trisectrix before? On Fri, May 17, 2013 at 6:27 AM, Bill Gosper <billgosper@gmail.com> wrote:
[Somebody should export this as an mgif or something.]
gosper.org/tripenta.gif I just noticed that, viewed from the center of the moving circles, the trisection is of the negative of the angle of the short rotor wrt the positive x axis. I.e. the tri-blade counter-rotates. --rwg
and a rather handy one. Wikipedia mentions this in neither its astroid article nor its list of trisectrices. It is obvious from a diagram. Will Wikipedia, as usual, reject this as "original research"? --rwg Here is Julian's animation, with the comment
Using the quadricuspid cycloids to trisect and pentasect an arbitrary angle: The angle to be tri/pentasected is between the short green segment and horizontal. Note the hypocycloid gives the three correct angles (w.r.t. horizontal, cyan [rigid asterisk]) and five [magenta] bogons, while the epicycloid gives the five correct angles (dark blue [rigid asterisk]) and three [red] bogons. We've never understood the geometric significance of the bogons. --Bill and Julian
Block[{three = ParametricPlot[{3/4*Cos[t] + 1/4*Cos[-3*t], 3/4*Sin[t] + 1/4*Sin[-3*t]}, {t, -\[Pi], \[Pi]}, Axes -> None], five = ParametricPlot[{5/4*Cos[t] - 1/4*Cos[5*t], 5/4*Sin[t] - 1/4*Sin[5*t]}, {t, -\[Pi], \[Pi]}, Axes -> None]}, ListAnimate[ Table[Show[ Graphics[{AbsolutePointSize[10], AbsoluteThickness[2], Circle[{Cos[t], Sin[t]}/4, 5/4], Black, Point[{0, 0}], Red, Point[{Cos[t], Sin[t]}/4], Green, Line[{{0, 0}, {Cos[t], Sin[t]}}/4], Red, Line[Transpose[{Table[{Cos[t], Sin[t]}/4, {Length[#]}], #}] &@ Select[{x, y} /. (NSolve[{(x - Cos[t]/4)^2 + (y - Sin[t]/4)^2 == 25/16, 5/4*Cos[\[Theta]] - 1/4*Cos[5*\[Theta]] == x, 5/4*Sin[\[Theta]] - 1/4*Sin[5*\[Theta]] == y}, {x, y, \[Theta]}, WorkingPrecision -> MachinePrecision] /. {C[1] -> 0, Or -> List, And -> List, Equal -> Rule}), Re[#] == # &]], Blue, Line[ Transpose[{Table[{Cos[t], Sin[t]}/4, {5}], Table[{Cos[t], Sin[t]}/4 + 5/4*{Cos[t/5 + 2*k*\[Pi]/5 + \[Pi]], Sin[t/5 + 2*k*\[Pi]/5 + \[Pi]]}, {k, 0, 4}]}]]}, PlotRange -> {{-3/2, 3/2}, {-3/2, 3/2}}, ImageSize -> 1000], Graphics[{AbsolutePointSize[10], AbsoluteThickness[2], Circle[{Cos[t], Sin[t]}/4, 3/4], Black, Point[{0, 0}], Red, Point[{Cos[t], Sin[t]}/4], Green, Line[{{0, 0}, {Cos[t], Sin[t]}}/4], Magenta, Line[Transpose[{Table[{Cos[t], Sin[t]}/4, {Length[#]}], #}] &@ Select[{x, y} /. (NSolve[{(x - Cos[t]/4)^2 + (y - Sin[t]/4)^2 == 9/16, 3/4*Cos[\[Theta]] + 1/4*Cos[-3*\[Theta]] == x, 3/4*Sin[\[Theta]] + 1/4*Sin[-3*\[Theta]] == y}, {x, y, \[Theta]}, WorkingPrecision -> MachinePrecision] /. {C[1] -> 0, Or -> List, And -> List, Equal -> Rule}), Re[#] == # &]], Cyan, Line[ Transpose[{Table[{Cos[t], Sin[t]}/4, {3}], Table[{Cos[t], Sin[t]}/4 + 3/4*{Cos[-t/3 + 2*k*\[Pi]/3], Sin[-t/3 + 2*k*\[Pi]/3]}, {k, 0, 2}]}]]}, PlotRange -> {{-3/2, 3/2}, {-3/2, 3/2}}, ImageSize -> 900], three, five], {t, 0, 2 \[Pi], \[Pi]/60.}], 30, AnimationRunning -> False]]