How to determine which points are on an envelope of ellipses?

Question

I'm wondering how to determine the location of some of the points on the envelope of a family of ellipses? The picture I have is below:

For each of the individual ellipses, I know how to get the enclosed region using ImplicitRegion, and the code goes like either

ImplicitRegion[Evaluate[a*x^2+b*y^2+cx*y+dy*x+e<=0], {x, y}]

Or

ImplicitRegion[
 Sum[EuclideanDistance[{x, y}, p], {p, foci}] <= majoraxis, {x, y}]]

Supposedly, I'm looking for a function that (1. taking in a point in the region bounded by the envelope, (2. for some theta determine the closest colored point in the angle theta with regards to the input point.

I only need around 20-30 points so that I can run lm and decide whether the envelope is also an ellipse (and I already know how to do this part). Any suggestions on how I can locate points on the envelope?

I tried running RegionIntersection, RegionBoundary and then rasterize it to take random points from the boundary, but this takes more than 1hr on my MacBook for an envelope of 100 ellipses. Both RegionIntersection and RegionBoundary runs very slow, so I'm looking for a more time-wise efficient solution.

Answer 1

Update

n = 10;
centers = Table[RandomReal[{0, 1/2}, 2], n];
majoraxis = Table[RandomReal[{3, 6}, 2], n];
ellipsoids = MapThread[Ellipsoid, {centers, majoraxis}];
ellipses = MapThread[Circle, {centers, majoraxis}];
interior = RegionIntersection@ellipsoids;
pts = MeshPrimitives[
   RegionIntersection @@@ Subsets[DiscretizeRegion /@ ellipses, {2}] //RegionUnion, 0];
inpts = Select[pts, RegionWithin[interior, #] &];
outpts = Complement[pts, inpts];
Graphics[{{Yellow, DiscretizeRegion@interior}, 
  ellipses, {Cyan, outpts}, {Red, PointSize[Medium], inpts}}]

Or

n = 10;
centers = Table[RandomReal[{0, 1/2}, 2], n];
majoraxis = Table[RandomReal[{3, 6}, 2], n];
ellipsoids = MapThread[Ellipsoid, {centers, majoraxis}];
ellipses = MapThread[Circle, {centers, majoraxis}];
interior = RegionIntersection@ellipsoids;
(* pts=RegionIntersection@@@Subsets[ellipses,{2}] *)
fig = Graphics[ellipses];
pts = Graphics`Mesh`FindIntersections[fig, 
   Graphics`Mesh`AllPoints -> False];
inpts = Select[pts, RegionMember[interior, #] &];
outpts = Complement[pts, inpts];
Graphics[{{Yellow, DiscretizeRegion@interior}, 
  ellipses, {Cyan, Point@outpts}, {Red, PointSize[Medium], 
   Point@inpts}}]

Edit

Clear[ellipses, fig, pts];
ellipses = 
  Table[Circle[RandomReal[{0, 1/2}, 2], RandomReal[{3, 6}, 2]], 10];
fig = Graphics[ellipses];
pts = Graphics`Mesh`FindIntersections[fig, 
   Graphics`Mesh`AllPoints -> False];
Show[fig, Graphics[{PointSize[Medium], Red, Point[pts]}]]

Or

Clear[ellipses, pts];
ellipses = 
  Table[Circle[RandomReal[{0, 1/2}, 2], RandomReal[{3, 6}, 2]], 10];
pts = Cases[
   Table[RegionIntersection[ellipses[[i]], ellipses[[j]]], {i, 
     Length@ellipses}, {j, i - 1}], {x_Real, y_Real} :> {x, y}, 
   Infinity];
Graphics[{ellipses, PointSize[Medium], Red, Point[pts]}]

Original

Do you want such points?

reg = ImplicitRegion[4 x^2 + 3 y^2 - 3 x*y == 4, {x, y}];
pts = RandomPoint[reg, 5]
RegionMember[reg] /@ pts
RegionPlot[reg, Epilog -> {PointSize[Large], Red, Point[pts]}]

{True, True, True, True, True}