Is a straight line actually a circle?

Question

Hi there,

 

A straight line wrapped around the world would eventually come back to join the beginning of the line. We often describe this when describing parallel lines for example. However, though the line is straight, if you were to look down onto the earth you would see the line become a circle. So the question I ask is a straight line straight? 

 

P.s I am from the UK, 23 and in University, I do not wish to apply for US tests or exams etc.

 

Thank you, 

Marcus

Answer 1

This is a really good question, and the honest answer is that both things are true at the same time, because the word straight is doing two different jobs depending on where you are standing when you ask.

There is straight as measured from inside the surface, and straight as seen from outside it. If you are walking on the earth and you never turn left or right, you are going as straight as it is possible to go. That path is called a geodesic, and it is the real meaning of a straight line on a curved surface, since it is both the shortest route between two points and the one that involves no turning. From where you are standing, nothing about it feels curved at all.

Someone floating out in space, though, watches that same path bend, because the surface underneath you is bending. What they see is a great circle, a circle whose center is the center of the earth, like the equator or any line of longitude. So the line is straight intrinsically and curved extrinsically, and neither description is the wrong one.

A quick way to feel the difference is to draw a straight line on a flat sheet of paper and then roll the paper into a tube. The ink never moved, and as far as the paper is concerned the line is exactly as straight as it was, but from outside it now spirals.

The parallel lines part of your question is where this gets genuinely interesting. On a flat plane, two straight lines that start out parallel stay the same distance apart forever. On a sphere that just fails. Take two lines of longitude crossing the equator. They both head due north, perfectly parallel, neither one ever turning, and they still collide at the north pole. So a sphere has no parallel lines at all, and that is precisely where Euclid's parallel postulate stops holding. Spherical geometry is one of the non-Euclidean geometries, and it is not a loophole or a technicality, it is a completely consistent geometry that simply is not the flat one you learned first.

One more thing worth knowing, because it catches people out. Not every circle you can draw on a globe counts as straight in this sense. Only the great circles do. A line of latitude like the 50th parallel is a perfectly good circle, but if you walk along it you are steering slightly left the whole way to stay on it, so it is not a geodesic. That is why a flight from the UK to North America arcs up over Greenland instead of following the latitude line. On a flat map that arc looks like a detour, but it is actually the straight path, and the latitude line is the curved one.

So, is a straight line straight? Yes, if straight means never turning within the surface you are on. And yes, it is also a circle, if you step outside the surface and look at the shape it traces in three dimensional space. Those two answers are not fighting each other, they are answering two different questions.