How can I fill the intersection of multiple ellipses?

Questions
,
1

Related to a previous question I had, I am interested in filling the intersection of multiple ellipses.

Specifically, I need to figure out how to draw partial arcs of ellipses, using only two points, the radius, and the center.

Here is an example of the kind of diagram I want, which I created using a straight line approximation:

image
image681×328 26.7 KB

#import "@preview/cetz:0.4.2": *

#canvas({
  import draw: *
  
  circle((0,0), radius: (8, 4), name: "G")
  circle((rel: (-6, 0.4), to: "G.center"), radius: 0.25, name: "A", fill:black)
  

  circle("A", radius: 3, name: "inner", stroke: gray)
  circle("A", radius: 4, name: "outer", stroke: gray)

  
  intersections("i", "G", "inner")
  intersections("o", "G", "outer")
  
  merge-path(
    stroke: color.purple + 2pt,
    fill: color.purple.transparentize(80%),
    close: true,
    {
      arc-through("i.0", "inner.east", "i.1")
      line("i.1", "o.1")
      arc-through("o.1", "outer.east", "o.0")
      line("o.0", "i.0")
    }
  )
})

This does not work for larger segments:

image
image936×535 54.1 KB

#canvas({
  import draw: *
  
  circle((0,0), radius: (8, 4), name: "G")
  circle((rel: (-6, 0.4), to: "G.center"), radius: 0.25, fill: black, name: "A")
  

  circle("A", radius: 3, name: "inner", stroke: gray)
  circle("A", radius: 10, name: "outer", stroke: gray)

  
  intersections("i", "G", "inner")
  intersections("o", "G", "outer")
  
  merge-path(
    stroke: purple + 2pt,
    fill: purple.transparentize(80%),
    close: false,
    {
      arc-through("i.0", "inner.east", "i.1")
      line("i.1", "o.0")
      arc-through("o.0", "outer.east", "o.1")
      line("o.1", "i.0")
    }
  )
  // If I had the angle of these lines, could I draw the partial arc?
  line("G", "i.0")
  line("G", "o.0")
  line("G", "i.1")
  line("G", "o.1")
})

It really seems like with the center, radius and two points, I should have all of the information necessary to draw these partial arcs. I think my best idea was to compute the angle of the lines in the figure I added, but I thus far have not been able to put anything together that works.

Thank you in advance for any ideas.

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