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Let me give an example, 

probe1 = ContourPlot[10 == x + y, {x, 0, 10}, {y, 0, 10}, Frame -> True, FrameLabel -> {"x", "y"}]

This plots 10=f[x,y]=x+y, for a set of point (x,y) that solved the equality. Moreover, 

probe2 = ParametricPlot[{x y, x^2 + y^2}, {x, 0, 10}, {y, 0, 10}]

where g=g[x,y]=xy and h=h[x,y]=x^2+y^2, plots h=h[g[x,y]]. What I want is to plot only the case in which x+y=10. What should I do?

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  • 2
    $\begingroup$ How about a parametric region Region[ParametricRegion[{{x y, x^2 + y^2}, x + y == 10}, {{x, 0, 10}, {y, 0, 10}}], Frame -> True] $\endgroup$
    – Simon Woods
    Jul 5, 2019 at 21:02

2 Answers 2

Reset to default
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Update: Adding multiple mesh lines and legends:

mesh = {{9, Directive[Red, Opacity[1], Thick]}, 
        {10, Directive[Green, Opacity[1], Thick]}, 
        {12, Directive[Black, Opacity[1], Thick]}};

ParametricPlot[{u t, u^2 + t^2}, {u, 0, 10}, {t, 0, 10}, 
  MeshFunctions -> {Function[{x, y, u, t}, u + t]}, 
  Mesh -> {mesh}, 
  PlotLegends -> LineLegend[## & @@ Reverse[Transpose[mesh]], LegendLabel -> "u + t"]]

enter image description here

Original answer:

You can use the argument of ContourPlot as the MeshFunctions option value in ParametricPlot as follows:

ParametricPlot[{u t, u^2 + t^2}, {u, 0, 10}, {t, 0, 10}, 
 MeshFunctions -> {Function[{x, y, u, t}, u + t - 10]}, 
 Mesh -> {{0}}, 
 MeshStyle -> Directive[Red, Thick]]

enter image description here

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7
  • $\begingroup$ It worked. Thanks. Do you have a sugestion to include in the same ParametricPlot several cases of the condition, for example u+v-2 and u+v-5, etc? What should I change in your solution? $\endgroup$
    – Patrick El Pollo
    Jul 6, 2019 at 3:49
  • 1
    $\begingroup$ @PatrickElPollo, try ParametricPlot[{u t, u^2 + t^2}, {u, 0, 10}, {t, 0, 10}, MeshFunctions -> {Function[{x, y, u, t}, u + t]}, Mesh -> {{{9, Directive[Red, Opacity[1], Thick]}, {10, Directive[Green, Opacity[1], Thick]}, {12, Directive[Black, Opacity[1], Thick]}}}] $\endgroup$
    – kglr
    Jul 6, 2019 at 4:03
  • $\begingroup$ thanks, again. Only one more detail, sorry... is it possible to label each Mesh to have a legend of them? $\endgroup$
    – Patrick El Pollo
    Jul 6, 2019 at 4:27
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    $\begingroup$ Patrick, please see the update. $\endgroup$
    – kglr
    Jul 6, 2019 at 5:12
  • 1
    $\begingroup$ @PatrickElPollo, you can use PlotStyle -> White (but that also gets rid of the right boundary) $\endgroup$
    – kglr
    Jul 14, 2019 at 23:56
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This may be what you want

Clear["Global`*"]

eqn = 10 == x + y;

ysol[x_] = y /. Solve[eqn, y][[1]]

(* 10 - x *)

Show[
 ParametricPlot[
  {x y, x^2 + y^2},
  {x, 0, 10}, {y, 0, 10}],
 ParametricPlot[
  Evaluate[{x y, x^2 + y^2} /. y -> ysol[x]],
  {x, 0, 10},
  PlotStyle -> Red]]

enter image description here

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2
  • $\begingroup$ it works, but what I want is a way to extract the pairs from ContourPlot when Solve does not work to use them as parameters in ParametricPlot. $\endgroup$
    – Patrick El Pollo
    Jul 5, 2019 at 19:39
  • $\begingroup$ @PatrickElPollo - have you tried using a numeric solver such as NSolve or FindRoot? $\endgroup$
    – Bob Hanlon
    Jul 5, 2019 at 23:29

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