Plot of 2 equation and 3 variables

Question

I have two equation for Ex(t1,t2,t) and Ey(t1,t2,t). Please,could you tell me, how I can make a plot Ex(Ey)?

Ex[t1_, t2_, t_] = ((Sin[t2] - Sin[t1])/(t1 - t2) - Cos[t])^2

Ey[t1_, t2_, t_] = (2 (Cos[t1] - Cos[t2])/(t1 - t2) - Sin[t])^2

Answer 1

What you really have is a family of parametric curves. I would go about doing this in the following way (and you can switch out t for t1 or t2 if you want; just follow the general construction).

As an example, consider:

With[{t1 = 0.5, t2 = 0},
  ParametricPlot[{Ey[t1, t2, t], Ex[t1, t2, t]}, {t, 0, 2 π}, AspectRatio -> 1]
 ]

If you want to see some general behavior, we can make a table over one of the other variables. For instance,

Multicolumn[
 Table[
   With[{t2 = 0}, ParametricPlot[{Ey[t1, t2, t], Ex[t1, t2, t]}, {t, 0, 2 π}, AspectRatio -> 1]],
   {t1, {0.01, 0.1, 0.5, 1, 2, 5}}],
 3]

Answer 2

If you are not after a continuous solution, then this is straightforward:

min = 0.;
max = 2.;
step = 0.1;

ListPlot[
  Transpose@{Flatten@
     Quiet@Table[
       Ey[x, y, z], {x, min, max, step}, {y, min, max, step}, {z, min,
         max, step}], 
    Flatten@Quiet@
      Table[Ex[x, y, z], {x, min, max, step}, {y, min, max, step}, {z,
         min, max, step}]}, Frame -> True, AspectRatio -> 1, 
  FrameLabel -> {"Ey", "Ex"}]

Explanation: for a set {x,y,z} the functions Ex[x,y,z] and Ey[x,y,z] take some values, say, E1 and E2, respectively. So for sure when the function Ex takes the value E1, then function Ey takes the value E2. So one needs to make a list of such pairs {E2, E1}. Quiet is because when the denominator (t1-t2 in the definitions) is zero, then both functions are Indeterminate, and I didn't want to play around with discarding such values as they don't influence the plot.