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MathGroup Archive 2005

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Re: Attempt to generalize a constant

  • To: mathgroup at smc.vnet.net
  • Subject: [mg57906] Re: Attempt to generalize a constant
  • From: "Narasimham" <mathma18 at hotmail.com>
  • Date: Sun, 12 Jun 2005 04:34:30 -0400 (EDT)
  • References: <d7rk50$bl7$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Thanks  Bob, Pratik and Wouter Meeussen. Wouter sent me a private email
and opines (hope it is ok to state it here,) that as z increases from
.5 to .999, the ImplicitPlot finds a string of small islands along the
crest of the ridges. The
numerics approach z=1 but rounding errors will obfuscate the nice
mathematical ( top ridge) line.

This has since been partially obtained in FindRoot/ListPlot.

<< Graphics`ParametricPlot3D`;
<< Graphics`ImplicitPlot`;
F[t_, mu_] := mu*JacobiSN[t, mu^2];
Jaco = ParametricPlot3D[{mu, t, F[t, mu]}, {mu, 1, 3, .1}, {t, Pi/6,
2Pi, Pi/12}, AspectRatio -> Automatic] ;
Plane = Plot3D[.95, {mu, 1, 3}, {t, Pi/6, 2Pi}] ;
Show[Jaco, Plane] ;

data = Table[{mu /. FindRoot[F[t, mu] == 1, {mu, 1, 3}, MaxIterations
-> 100], t}, {t, Pi/6, 2Pi, Pi/96}];
ListPlot[data, Frame -> True, Axes -> False, PlotJoined -> True,
PlotRange -> {{0.95, 3.15}, Automatic}, Epilog ->
{AbsolutePointSize[4], Red, Point /@ Select[data, IntegerQ[6*#[[2]]/Pi]
&]}, PlotStyle -> Blue, ImageSize -> 360];

" (*  In a simpler example, y is a variable, x1 was a known constant at
the outset/beginning in parabolic relation, before being variablized or
generalized to x, x1-> x ; It generalizes parabola to an ellipse while
including a second parameter *)"

G[x1_, y_] = x1^2 + x1* y + y^2 ;
Plot3D[G[x1, y], {y, -1.2, 1.2}, {x1, -1.2, 1.2}];
Impl = ImplicitPlot[G[x, y] == 1 , {x, -1.2, 1.2}, {y, -1.2,
1.2},AspectRatio -> Automatic];
dat = Table[{y /.
FindRoot[G[x1, y] == 1, {y, -1.2, 1.2}, MaxIterations -> 100],x1}, {x1,
-1.2, 1.2, .1}];
LP = ListPlot[dat, Frame -> True, Axes -> False, PlotJoined -> True,
PlotRange -> {{-1.2, 1.2}, Automatic}, Epilog -> {AbsolutePointSize[4],
Red,Point /@ Select[data, IntegerQ[6*#[[2]]/Pi] &]}, PlotStyle -> Blue,
ImageSize -> 360]; Show[Impl, LP];

(* Intersection of above Jacobi function and flat plane z =
 1 intersection curves look like  hyperbolae mu * t = constant. By
assembling ListPlot roots obtained from FindRoot either in the given
known ellipse case or in JacobiSN case we do not capture all of the
possible function points.  Is the problem due to convergence?
branching? seed value location? -- Implicit Plot / Contour Plot was
expected to capture ALL the roots! *)


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