# How to find out if three points on Earth are on the same great circle arc?

I am trying to figure out whether 3 particular points on Earth's surface are on the same arc (of a great circle) or not. Using Google Earth's (also ArcGIS Earth's) line tool, I see that the points are approximately on the same arc as the one in the middle is only 3 km off where the distance between the neighboring points are more than 1000 km each.

Considering the oblate spheroid shape of Earth, can I confidently conclude from Google (and ArcGIS) Earth measurements that these points are not exactly on the same arc? How probable is it that approximations or some other factors cause 3 km deviation in 1000 km? Is there any method or tool that can help reaching to exact results?

• Through any three points on the surface of an ellipsoid, you can draw a circle. Therefore, these three points are on the same arc.
– KVO
Jul 5, 2023 at 23:34
• Perhaps you meant the arc of a great circle, which lies in a plane passing through the center of the Earth.
– KVO
Jul 5, 2023 at 23:38
• @KVO yes I mean the arc of a great circle, thanks for pointing it out. Jul 6, 2023 at 7:10
• I was not aware that there is a SE community of GIS. I have asked the question there too. gis.stackexchange.com/questions/462900/… Jul 6, 2023 at 7:27
• What you could do is to define the two great circles passing through two pairs of point, i.e., the one passing through A–B, and the one passing through B–C. Then you could compute the intersection points of these two great circles. Matlab has a function for that: mathworks.com/help/map/ref/gcxgc.html, you could write something similar in Python. The function will find the two intersection points between the two great circles (if so, then A–B–C are not on the same one) or will tell you that the pair of great circles are identical, meaning that the three points are on the same one. Jul 6, 2023 at 7:36

I would take the same approach as Jean-Marie Prival, but using a different toolset, because I'm not familiar with GIS in general and have never been in a room that has ever seen Mathematica. But I do have, and have used a few times, a collection of aviation-relevant algorithms, which includes a tool that will take two Lat-Long pairs and give you the great circle course between them. For the (putative arc A-B-C, if the direction from A->B is the same as from A->C, then you're on the same great circle. (I'd double-check by comparing C->A and C->B, because I know I'm at the limits of my geometry.)

In practical surveying - steering oil wells to hit each other, or to avoid hitting each other, treating the Earth as a simple oblate spheroid is sufficient ; you don't need to go around adding in the second and higher orders of difference from a sphere. Whether your survey tool is sitting exactly axially in your drilling string (or is centralised in the borehole if surveying on wireline) is more significant, and has cause more blowouts (that I've heard of).

• Thanks. Can you give a reference to such a "tool that will take two Lat-Long pairs and give you the great circle course between them"? Jul 13, 2023 at 10:44
• I got it from a collection on Compuserve (now deceased or buried in AOL ; on their takeover, I left CI\$. I don't know if the forum archives still exist, or where.) It was called the "Aviation Formulary" then. No idea if it's still around. Unfortunately, it didn't help me with my main problem of converting "local" coordinates (m N/E from a fixed point) into or from UTM or "geographical" coordinates. But I kept the text file in my "Work notes" folder. Jul 17, 2023 at 16:01