Filling between several parametric plots

Question

I have several (four) parametric curves defining a closed loop in the plane, and I wish to fill it with some color. Now, I have seen several nice answers for the case of two curves, where the simplest solution is to shift between them as $t \:\gamma_1+(1-t)\gamma_2$, but this is easy only for two curves.

xs[u_, v_, λ_] := (-(1/4)+I/4) Sqrt[π]  λ(Erf[(1/2+I/2) u]-Erf[(1/2+I/2) (u-I v)]+Erfi[(1/2+I/2) u]-Erfi[(1/2+I/2) (u+I v)])+1/λ Sqrt[π]  FresnelC[u/Sqrt[π]]
ys[u_, v_, λ_] := (-(1/4)-I/4) Sqrt[π] λ(Erf[(1/2+I/2) u]-Erf[(1/2+I/2) (u-I v)]-Erfi[(1/2+I/2) u]+Erfi[(1/2+I/2) (u+I v)])+1/λ Sqrt[π]  FresnelS[u/Sqrt[π]]
    
loop[λ_] := {{xs[-1, t, λ], ys[-1, t, λ]}, {xs[2.2, t, λ], ys[2.2, t, λ]}, {xs[t, -1, λ], ys[t, -1, λ]}, {xs[t, 2.2, λ], ys[t, 2.2, λ]}}
      
ParametricPlot[{loop[0.6]}, {t, -1, 2.2} ]

Answer 1

• We can mapping the rectangle -1<=s<=2.2, -1<=t<=2.2 to such region.

xs[u_, v_, λ_] := (-(1/4) + 
     I/4) Sqrt[π] λ (Erf[(1/2 + I/2) u] - 
     Erf[(1/2 + I/2) (u - I v)] + Erfi[(1/2 + I/2) u] - 
     Erfi[(1/2 + I/2) (u + I v)]) + 
  1/λ Sqrt[π] FresnelC[u/Sqrt[π]]
ys[u_, v_, λ_] := (-(1/4) - 
     I/4) Sqrt[π] λ (Erf[(1/2 + I/2) u] - 
     Erf[(1/2 + I/2) (u - I v)] - Erfi[(1/2 + I/2) u] + 
     Erfi[(1/2 + I/2) (u + I v)]) + 
  1/λ Sqrt[π] FresnelS[u/Sqrt[π]]
ParametricPlot[{xs[s, t, .6], ys[s, t, .6]}, {s, -1, 2.2}, {t, -1, 
  2.2}, PlotStyle -> Cyan]

• We can compare with the mapping domain and the mapping range as below.

{ParametricPlot[{s, t}, {s, -1, 2.2}, {t, -1, 2.2}, PlotStyle -> Cyan,
    Axes -> None, Mesh -> {10, 10}, 
   MeshShading -> {{Red, Blue}, {Yellow, Green}}], 
  ParametricPlot[{xs[s, t, .6], ys[s, t, .6]}, {s, -1, 2.2}, {t, -1, 
    2.2}, PlotStyle -> Cyan, Axes -> None, Mesh -> {10, 10}, 
   MeshShading -> {{Red, Blue}, {Yellow, Green}}]} // GraphicsRow

• Another way is take care of the orientation of paths.

xs[u_, v_, λ_] = (-(1/4) + 
      I/4) Sqrt[π] λ (Erf[(1/2 + I/2) u] - 
      Erf[(1/2 + I/2) (u - I v)] + Erfi[(1/2 + I/2) u] - 
      Erfi[(1/2 + I/2) (u + I v)]) + 
   1/λ Sqrt[π] FresnelC[u/Sqrt[π]];
ys[u_, v_, λ_] = (-(1/4) - 
      I/4) Sqrt[π] λ (Erf[(1/2 + I/2) u] - 
      Erf[(1/2 + I/2) (u - I v)] - Erfi[(1/2 + I/2) u] + 
      Erfi[(1/2 + I/2) (u + I v)]) + 
   1/λ Sqrt[π] FresnelS[u/Sqrt[π]];
loop[λ_] = {{xs[-1, t, λ], 
    ys[-1, t, λ]}, {xs[2.2, t, λ], 
    ys[2.2, t, λ]}, {xs[t, -1, λ], 
    ys[t, -1, λ]}, {xs[t, 2.2, λ], 
    ys[t, 2.2, λ]}};
{l1, l2, l3, l4} = ParametricPlot[#, {t, -1, 2.2}] & /@ loop[.6];
{pts1, pts2, pts3, pts4} = 
  Cases[#, Line[a_] :> a, Infinity] & /@ {l1, l2, l3, l4};
pts = Join[pts3[[1]], pts2[[1]], Reverse@pts4[[1]], Reverse@pts1[[1]]];
Graphics[{Green, FilledCurve[Line[pts]]}, Axes -> True]
(* Graphics[{Green, WindingPolygon[pts]}, Axes -> True] *) 

• BoundaryDiscretizeGraphics also work if we use JoinedCurve.

Clear[xs, ys, l1, l2, l3, l4]; 
xs[u_, v_, λ_] = (-(1/4) + 
      I/4) Sqrt[π] λ (Erf[(1/2 + I/2) u] - 
      Erf[(1/2 + I/2) (u - I v)] + Erfi[(1/2 + I/2) u] - 
      Erfi[(1/2 + I/2) (u + I v)]) + 
   1/λ Sqrt[π] FresnelC[u/Sqrt[π]] // Re;
ys[u_, v_, λ_] = (-(1/4) - 
       I/4) Sqrt[π] λ (Erf[(1/2 + I/2) u] - 
       Erf[(1/2 + I/2) (u - I v)] - Erfi[(1/2 + I/2) u] + 
       Erfi[(1/2 + I/2) (u + I v)]) + 
    1/λ Sqrt[π] FresnelS[u/Sqrt[π]] // Re;
l1 = Cases[
    ParametricPlot[{xs[s, -1, .6], ys[s, -1, .6]}, {s, -1, 2.2}], 
    Line[a_] :> a, Infinity][[1]];
l2 = Cases[
    ParametricPlot[{xs[2.2, t, .6], ys[2.2, t, .6]}, {t, -1, 2.2}], 
    Line[a_] :> a, Infinity][[1]];
l3 = Cases[
     ParametricPlot[{xs[s, 2.2, .6], ys[s, 2.2, .6]}, {s, 2.2, -1}], 
     Line[a_] :> a, Infinity][[1]] // Reverse;
l4 = Cases[
     ParametricPlot[{xs[-1, t, .6], ys[-1, t, .6]}, {t, 2.2, -1}], 
     Line[a_] :> a, Infinity][[1]] // Reverse;
Graphics[JoinedCurve[
   Line[Join[l1, l2, l3, l4]]]] // BoundaryDiscretizeGraphics

• Some related.

https://mathematica.stackexchange.com/a/252935/72111

https://mathematica.stackexchange.com/a/262992/72111

Answer 2

An alternative approach: post-process one-parameter ParametricPlot output to order line coordinates and add a polygon.

ClearAll[addFilling]
addFilling[style_ : ColorData[97]@1] := Graphics[{Opacity[.3], style, 
         Polygon @ #[[Last@FindShortestTour[#]]] & @
           MeshCoordinates @ DiscretizeGraphics @ #}] &;

Use addFilling[] with the option DisplayFunction:

ppA1 = ParametricPlot[loop[0.6], {t, -1, 2.2}, 
  ImageSize -> 700, PlotStyle -> AbsoluteThickness[.5], Axes -> False, Frame -> True,
  DisplayFunction -> (Show[#, addFilling[] @ #] &)]

Compare with the two-parameter ParametricPlot method from cvgmt's answer:

ppA2 = ParametricPlot[{xs[s, t, .6], ys[s, t, .6]}, {s, -1, 2.2}, {t, -1, 2.2}, 
  Axes -> False, ImageSize -> 700]

Zoom in to see the artifacts in the second picture:

GraphicsColumn[{Show[ppA1, PlotRange -> {{.5, 2}, {.5, 1}}],
   Show[ppA2, PlotRange -> {{.5, 2}, {.5, 1}}]}]

Using λ = .3:

ppB1 = ParametricPlot[loop[0.3], {t, -1, 2.2}, 
   Axes -> False, PlotStyle -> AbsoluteThickness[.5], Frame -> True, ImageSize -> 700, 
   DisplayFunction -> (Show[#, addFilling[] @ #] &)]

ppB2 = ParametricPlot[{xs[s, t, .3], ys[s, t, .3]}, {s, -1, 2.2}, {t, -1,  2.2}, 
   Axes -> False, ImageSize -> 700]

GraphicsColumn[{Show[ppB1, PlotRange -> {{-2, 2}, {-1, 1}}],
  Show[ppB2, PlotRange -> {{-2, 2}, {-1, 1}}]}]