We in mathematics say (so if you'll have to use, what will sound natural to you "on/in"?):
Two objects are coplanar if they both lie in the same plane.
Or
Two objects are coplanar if they both lie on the same plane.
So what sounds natural: "on" or "in" or both of them sound equally natural?
Also, is there any AmE-BrE difference?
Best Answer
Both are completely natural, but imply different ways of looking at the spatial orientation of the elements.
"In the same plane" is probably the more accurate from the mathematical perspective, since a plane has no depth. A point cannot rest "on" the plane the way a cup would rest on a table, since that would mean it is above the plane and therefore not part of the plane.
However, "on the same plane" is a common expression. Even though they should know it represents a mathematical fallacy, mathematicians persist in using the same language to describe these spatial relationships that they would use for real-world objects.
So, use whichever sounds better to you.