Creating a Domain of Regions Defined by Random, Non-Intersecting Curves

Question

I am trying to create a rectangular domain containing many curves that start and stop on either the left or right edges of the domain. These curves should not intersect. Additionally, I need to know the areas of the regions created by these curves as shown in this picture.

I found a couple of other questions that helped me, but I am still working on the problem.

Random Curves in 3D

Diving a Domain into Regions by Straight Lines

Here are the tasks I am still trying to complete:

1. Define regions with curves rather than straight lines. The second link divides the domain into regions defined by straight lines and I have taken that example and gotten the area of each of those regions, but I can't seem to make the same work for curves.
2. Create random curves that never intersect and start and stop on the left and right edges of the boundary. I have been trying to tackle this issue from a differential geometry perspective. I have tried to make some mapping that preserves the curve and where it starts and ends. This mapping would go from a square domain on which an initial random curve is defined to a domain that is entirely within other previously placed curves as seen in the picture below. One could then just randomly choose any existing region, map a curve into it, subset that region into two new regions, and repeat.

If anyone can offer any help with either of these tasks, I would appreciate it. Also please let me know if the question is poorly posed. I have some Mathematica experience, but not a lot.

Answer 1

Update: If we extract Polygons (instead of FilledCurves) we can use many Mesh* and Region* functions on extracted polygons:

SeedRandom[1];
cp = ContourPlot[Evaluate[Sum[Sin[RandomReal[5, 2] . {x, 5 y}], {10}]], 
  {x, 0, 5}, {y, 0, 5}, Contours -> 20, ContourShading -> None];


polygons = Cases[Normal@cp, 
   Line[x_] /; x[[{1, -1}, 1]] == {0., 0.} || x[[{1, -1}, 1]] == {5., 5.} :> 
    Polygon @ x, All];

Graphics[{RandomColor[], #} & /@ polygons]

largest20 = TakeLargestBy[polygons, Area, 20];

Graphics[{RandomColor[], #} & /@ largest20]

Multicolumn[DiscretizeRegion /@ largest20, 5]

Original answer:

Contour lines of a surface do not intersect.

So, we can take contour lines of a random surface and select the lines that start and end on the left or the right end of the horizontal range:

SeedRandom[1];
cp = ContourPlot[Evaluate[Sum[Sin[RandomReal[5, 2] . {x, 5 y}], {10}]], 
  {x, 0, 5}, {y, 0, 5}, Contours -> 20, ContourShading -> None];

Graphics @ Cases[Normal @ cp, 
  l : Line[x_] /; x[[{1, -1}, 1]] == {0., 0.} || x[[{1, -1}, 1]] == {5., 5.} :> 
  {RandomColor[], FilledCurve@l}, All] 

We can also use ListContourPlot:

SeedRandom[1];
lcp = ListContourPlot[RandomReal[{-.25, .25}, {20, 15}], 
   InterpolationOrder -> 3, Contours -> 20, ContourShading -> None];


Graphics @ Cases[Normal@lcp, 
  l : Line[x_] /; 
    x[[{1, -1}, 1]] == {1., 1.} || x[[{1, -1}, 1]] == {15., 15.} :>
    {l, RandomColor[], FilledCurve @ l}, All]