Re: implcit plot with undefined expression

*To*: mathgroup at smc.vnet.net*Subject*: [mg18863] Re: implcit plot with undefined expression*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Sun, 25 Jul 1999 03:30:04 -0400*Organization*: University of Western Australia*References*: <7n13kb$bf2@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

zoominzero at hotmail.com wrote: > Hi, > I've spent 4 days to try to use ImplicitPlot on an equation, without > success: > The Equation I try to plot has 2 nested square roots, ans 2 variables : > something like Sqrt[2*Sqrt[2X^2 -Y^2] - 2*Y +X^2]==0 > (it's not exactly this one: it's just to give you an idea ) > > Of course this equation is not defined everywhere, and the domain where > it's defined is complex. When i try to "ImplicitPlot" this equation, > i've an answer : > "The contour is attempting to traverse a cell in which some of the > points have not evaluated to numbers, and it will be dropped" > followed by > "Further output of ContourGraphics::"ctpnt" will be suppressed during > this calculation." > > So i don't have any answers ( and there are answers of course :) > > Is there some options or another solution to draw this implicit equation? The following Notebook highlights the problem and indicates one approach to solving it. Notebook[{ Cell["\<\ Modify the function so that if the result is complex, a specific \ numerical value (here 1) is returned.\ \>", "Text"], Cell[BoxData[ \(TraditionalForm\`f[x_?NumericQ, y_?NumericQ] := With[{t = N[\ at \(x\^2 - 2\ y + 2\ \ at \(2\ x\^2 - y\^2\)\)]}, If[Head[t] === Complex, 1, t]]\)], "Input"], Cell[TextData[{ "To get an idea of the behavior, we make a plot for fixed ", Cell[BoxData[ \(TraditionalForm\`y\)]], "." }], "Text"], Cell[BoxData[ \(TraditionalForm\`\(Plot[f[x, 1.5], {x, \(-2\), 2}, PlotPoints -> 100];\)\)], "Input"], Cell[TextData[{ "We see that the zero contour will be exceedingly sharp\[LongDash]and \ difficult for a non-adaptive plotting routine such as ", Cell[BoxData[ FormBox[ StyleBox[\(\(ContourPlot\)\(\ \)\), "Input"], TraditionalForm]]], " to find. Here we plot the 0.3 contour." }], "Text"], Cell[BoxData[ \(TraditionalForm\`\(cp = ContourPlot[f[x, y], \ {x, \(-2\), 2}, {y, \(-1\), 3}, PlotPoints -> 140, Contours -> {0.3}];\)\)], "Input"], Cell["In this example, we can attempt direct solution", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm\`Solve[\ at \(x\^2 - 2\ y + 2\ \ at \(2\ x\^2 - y\^2\)\) == 0, y]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{{y -> 1\/4\ \((x\^2 - \ at \(16\ x\^2 - x\^4\))\)}, {y -> 1\/4\ \((x\^2 + \ at \(16\ x\^2 - x\^4\))\)}}\)], "Output"] }, Open ]], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"pl", "=", RowBox[{"Plot", "[", RowBox[{ RowBox[{"Evaluate", "[", FormBox[\(y /. %\), "TraditionalForm"], "]"}], ",", \({x, \(-2\), 2}\), ",", \(PlotStyle -> Hue[1]\)}], "]"}]}], ";"}], TraditionalForm]], "Input"], Cell[BoxData[ \(TraditionalForm\`\(Show[cp, pl];\)\)], "Input"], Cell[TextData[{ "The agreement is quite good and you can show that the curves with ", Cell[BoxData[ \(TraditionalForm\`y < 0\)]], " are spurious solutions." }], "Text"] } ] Cheers, Paul ____________________________________________________________________ Paul Abbott Phone: +61-8-9380-2734 Department of Physics Fax: +61-8-9380-1014 The University of Western Australia Nedlands WA 6907 mailto:paul at physics.uwa.edu.au AUSTRALIA http://physics.uwa.edu.au/~paul God IS a weakly left-handed dice player ____________________________________________________________________