Fill space between two BezierCurves with random points

Question

I asked a question about filling the space between two curves (Sin and Cos) with random points and the answer I received does not work for InterpolatingFunctions. How can I fill the space between two BezierCurves or InterpolatingFunctions?

For example, I have the BezierCurves c1 and c2:

c1 = {{0, 0}, {2, 0}, {2, 1}};
c2 = {{0, 0.25}, {1.75, 0.25}, {1.75, 1}};

Graphics[{BezierCurve@c1,BezierCurve@c2}]

I can use RandomPoint by turning these curves into a Polygon:

f[c_] := Quiet@
   Interpolation[BezierFunction[c][#] & /@ Range[0, 1, 0.01]];
g[c_] := {#, f[c][#]} & /@ Range[0, c[[-1, 1]], 0.01];

h1 = Join[{c2[[1]]}, g@c1]; h2 = Join[g@c2, {c1[[-1]]}];

Graphics[{
  Point@RandomPoint[Polygon@Join[h1, h2], 500],
  Thick, Line@h1, Line@h2
  }]

My question is, is there a better/more efficient way of doing this?

Answer 1

• Only add a Line and set CurveClosed -> True to draw a closed contour.

c1 = {{0, 0}, {2, 0}, {2, 1}};
c2 = {{0, 0.25}, {1.75, 0.25}, {1.75, 1}};

reg = JoinedCurve[{BezierCurve@c1, Line[{Last@c1, Last@c2}], 
     BezierCurve@Reverse@c2}, CurveClosed -> True] // 
   BoundaryDiscretizeGraphics;
Graphics[{{EdgeForm[Cyan], FaceForm[LightGreen], reg}, Red, 
  Point@RandomPoint[reg, 200]}]

Answer 2

you can do it with sorcery...

c1 = {{2, 1}, {2, 0}, {0, 0}};
c2 = {{0, 0.25}, {1.75, 0.25}, {1.75, 1}};
gk = JoinedCurve[{Line[{{0, 0}, {0, 0.25}}], BezierCurve@c2, 
Line[{{1.75, 1}, {2, 1}}], BezierCurve@c1}];
Graphics[{Black, 
FilledCurve[{BezierCurve@c1, Line[{{0, 0}, {0, 0.25}}], 
BezierCurve@c2, Line[{{1.75, 1}, {2, 1}}]}], White, 
Point[RandomPoint[Disk[{1, 0.45}, {1.3, 0.9}], 5000]]}]

Answer 3

(Not an answer -- a comment/follow-up of the answer by @cvgmt.)

I expected NIntegerate to work also for this.

c1 = {{0, 0}, {2, 0}, {2, 1}};
c2 = {{0, 0.25}, {1.75, 0.25}, {1.75, 1}};

reg = JoinedCurve[{BezierCurve@c1, Line[{Last@c1, Last@c2}], 
     BezierCurve@Reverse@c2}, CurveClosed -> True] // 
   BoundaryDiscretizeGraphics;

res = Reap@
  NIntegrate[x, x \[Element] reg, Method -> "MonteCarlo", 
   PrecisionGoal -> 1.2, EvaluationMonitor :> Sow[{x}]]

But it gives the monitored point unevaluated:

{{0.596312, 0.204349}, {{{x}, {x}, {x}, {x},...

This work around can be used, but that is suboptimal:

res = Reap@
   NIntegrate[x*Boole[{x, y} \[Element] reg], {x, -2, 2}, {y, -2, 2}, 
    Method -> "MonteCarlo", PrecisionGoal -> 1.2, 
    EvaluationMonitor :> Sow[{x, y}]];

Graphics[Point@Pick[res[[2, 1]], # \[Element] reg & /@ res[[2, 1]]], 
 AspectRatio -> Automatic]