How to plot a solution set? [duplicate]

Question

I have two functions, let's call them $F1$ and $F2$. Both take the same three arguments: $x1, x2$ and $\epsilon$. With every $\epsilon^* \in [0,1]$, there should be either zero or one unique point $(x1^*,\,x2^*)$ that is a solution to

$\qquad F1(x1^*,\,x2^*,\,\epsilon^*) = 0$

$\qquad F2(x1^*,\,x2^*,\,\epsilon^*)= 0.$

What I want to do is make a plot with $\epsilon$ on the x-axis and the two values of $(x1,\,x2)$ on the y-axis so that for every $\epsilon \in [0,1]$ I display the $(x1,x2)$ pair that solves the above equation (if it exists). I know I could do this numerically by gathering points by solving the system for different $\epsilon$ values, but is there a more elegant solution. If not, I would also appreciate help on the numerical solution, since I'm not very comfortable with the Wolfram Language.

Answer 1

I think it is actually possible to do this purely with plotting function.

We can abuse MeshFunctions and ContourPlot3D to do it.

f1[x1_, x2_, e_] := x1^2 + x2^2 - e
f2[x1_, x2_, e_] := x1 - x2 - e

ContourPlot3D[f1[x1, x2, e] == 0, {x1, -3, 3}, {x2, -3, 3}, {e, 0, 3},
  MeshFunctions -> {Function[{x1, x2, e}, f2[x1, x2, e]]}, 
 Mesh -> {{0}},
 PlotPoints -> 40,
 ContourStyle -> None,
 BoundaryStyle -> None, AxesLabel -> {"x1", "x2", "e"}]