I've been looking at some various ways to calculate geodesics. One particularly good site collecting load of helpful information, is [this one](http://www.movable-type.co.uk/scripts/latlong-vincenty.html). There you can also find a link to [a map](https://geographiclib.sourceforge.io/scripts/geod-google.html) that shows a circle whose geodesic radius is accurately calculated by the WGS84 ellipsoidal approximation. In the resulting map, there are some additional info presented (as red arcs), called `envelope`. I'm struggling to understand what these are and what exactly they represent. The only partially useful resource I can find is from [this](https://www.maths.ox.ac.uk/about-us/departmental-art/quadric-surfaces/envelopes-geodesics-ellipsoids) website, but looking at the faint lines drawn on 2D image of a stone, does not enlighten anyone, especially as you cannot rotate the images, to see where those lines goes, and intersects. 

[![enter image description here][1]][1]

OR from another calculation: 

[![enter image description here][2]][2]

From the website, the [description](https://geographiclib.sourceforge.io/scripts/geod-google-instructions.html) only say:

> The geodesic envelopes as red curves; all the geodesics emanating from
> lat1, lon1 are tangent to the envelopes (providing they are extended
> far enough). The number of solutions to the inverse problem changes
> depending on whether lat2, lon2 lies inside the envelopes. For
> example, there are four (resp. two) approximately hemispheroidal
> geodesics if this point lies inside (resp. outside) the inner envelope
> (only one of which is a shortest path).


  
**Q: How can I better understand what those red arcs represent?**

 1. Why are they useful on this picture?
 2. How are they generated?

I'm more of a visual person, seeking a visual explanation, rather than a purely verbal description. 

  [1]: https://i.sstatic.net/PhNVC.png
  [2]: https://i.sstatic.net/dJzGa.png