Calculate points on earth along an isochrone (from a satellite) - geometry

Looking to calculate a set of locations on the ground that have the same time delay to a satellite in orbit (GEO, MEO, or LEO). Any pointers to some geometry formulas that can help? I would like to factor in the oblateness of the earth. I suppose I can do this trial and error, but hoping for a more elegant solution.

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How to know if vectors in a KDTree are consecutive?

I am currently working on a hole detection problem in 3D point cloud data. I am referring to this paper "Detecting Holes in Point Set Surfaces" by Gerhard H Bendels, Ruwen Schnabel and Reinhard Klein. One of the criterions mentioned is an Angle Criterion in which we need to determine angles between consecutive points in a KD Tree(Radially Nearest Neighbors to a given point).
See Image:
Angle between points
I am using Open3D to extract a KD Tree but I believe it is giving me an unsorted list of points rather than a list of consecutive points.
See Image:
List of Nearest neighbors
The point below '______' is the point of interest and rest are it's neighbors. Now my question is,
How do I know which point is next to which point?
And if that's not possible to know, How can I find the angles as shown in the first image.
I just need the angles to find the boundary probability for each point, so an answer would really help me progress.
Thanks
What I've Tried so far..
I have tried generating vectors out of all the points and calculated the angles using dot product. But that seems wrong because I believe I may be calculating dot products between first and third point.

Distance between point to polygon

I've read this article and try to use it with my project whitch is a point to several polygons shortest distance.
https://medium.com/analytics-vidhya/calculating-distances-from-points-to-polygon-borders-in-python-a-paris-example-3b597e1ea291
enter image description here
but it says"In order to keep our algorithm lean, let’s not account for these specific cases and always calculate the distance to the middle point of our lines. Especially in accurately defined polygons (on a small space), the differences are negligible."
So now it won't work if I want to know the shortest distance,and it'll sometimes got the wrong vertex(I'll figure out later) is there any solutions to know the polygon and the points shortest distance?
btw I'm not in English so sorry for the grammer...

Deviation analysis color to vertex color using Meshlab

I would to know if is possible to do deviation analysis with Meshlab and transfer the result to vertex color in a mesh. So expand those 2 ideas...
1st. Is it possible to do deviation analysis with MeshLab? I have a scanned mesh and I will compare with a "ideal model". The difference between these 2 will generate a (grey or color) scale information that represent the distance I have from the points of the scanned model to the "ideal" one.
2nd. I want to get this information (color/grey grading that shows how distant the points are) and transfer to a vertex color information.
I don't know it was clear, but if you know what deviation analysis means I think you got it. The difference is that I would like the generate a 3d mesh with the vertex color provided by this deviation analysis.
Seems that mesh lab can compare two models and can deal with vertex colorizing, but I don't Know if is possible to deal with real measurements, transfer this information to vertex color and export a mesh that show it.
If its possible and If you know how just point me some direction. I'm not familiar with Meshlab and click here and there trying a impossible task can be very frustrating, so it will be good if someone can give me some tips.
Thanks.
Yes, MeshLab can compute deviation analysis between two similar surfaces (and the required alignment preprocessing too).
Estimating the deviation between two meshes means computing the hausdorff distance.
There is a small tutorial on how to compute and visualize it in MeshLab here:
http://meshlabstuff.blogspot.com/2010/01/measuring-difference-between-two-meshes.html

Orthographic projection of point to line in WGS84

I need to find points (from a rather small dataset) which are close enough to a polyline. All coordinates are WGS84.
I think of some r-tree thing to reduce the data to just a few candidates which then have to be checked in more detail.
While i managed to do this using "great circle" arithmetic, i am sure this is too pedantic for the following reasons:
The segmentation of those polylines is quite high. A single segment of a polyline can be considered to be no longer than 10 km.
The points in question are not more than a few hundred meters away from segments.
The area in question is Europe, so the algorithm does not need to be valid for extreme (near pole?) conditions. Again: points don't need to be checked agains the whole polyline (which could be hundrets of kilometers). Only the "nearby" segments need to be considured.
Do i need to transform the WGS84 coordinates to
some local cartesian reference system
to a mercator system
Or can i even just calculate with "angle differences"? I know that this is just a matter of accuracy: I can accept an error which is below ~50 meters.
I highly appreciate your suggestions!
On how to measure distance from point to polyline:
you have to measure distances from all your points to all segments of a polyline.
See Distance from a point to a polygon
You can do without converting coordinates to cartesian (especially if the area is rather small, you don't mind 50 meters error and you don't need exact distances, just relative) See https://en.wikipedia.org/wiki/Decimal_degrees.

Transition from spherical coordinate system (local) to cartesian

I have got this problem, so there is a set of data as points in the spherical coordinate system - local (not a faithful arrangement of geographic or mathematical)and I'm trying to convert it to a Cartesian system to preview it in any program to draw the shape which should rise from these points .
Points are collected by the meter with a rotating laser head (thus slightly noisy). The head rotates in two axes, called phi, theta and the distance r.
Where
phi - is left-right rotation (-90 to 90)
theta - is up-down rotation (-90 to 90)
r - the distance
This can be seen in the figure below:
I tried to convert the data to Cartesian (xyz) according to the following formulas:
Unfortunately, every time something happens to me run them down and the picture that I get is incorrect.
For sample collection:
sample
I get such a picture (seen from top):
The expected result should be a rectangular tub (with bare upper part). This first arc (at the point where data has not yet been ran over) is called. lens effect, resulting from the fact that the meter was close to the wall, and second end of graph puzzles me end where the data are arranged in a completely unexpected.
With this number of points is hard for me to figure out what causes failure or bad for the conversion of data, or simply meter so measured. I would be grateful for verification my way of thinking and any advices.
Thank you in advance.
I think i am late to answer this question.
I can't see the images, anyway you can go through enter link description here .
It will give you clear idea how to convert spherical cordinate data into cartesian cordinate system.

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