Lines in spherical geometry

Spherical geometry : A type of non-Euclidean geometry which forms a surface (2 dimensions) of a sphere (obviously 😉 )

A (straight) line have a different interpretation in non-Euclidean geometry from that in Euclidean geometry. Interpretation, not the definition. A Straight line is still defined as the shortest path between two points.

Geodesic : The generalisation of the notion of a “straight line” to “curved spaces”.

There is just one more definition to go before we can discuss the geodesic in spherical geometry. It is ‘Great Circle’ which represents the intersection of the sphere (which we are dealing with) and a plane that passes through the centre of the same sphere. (Similarly, there is a ‘small circle’ where the plane does not pass through the centre of the sphere). A trivial example: longitudes (vertical lines) on a globe are great circles and latitudes (horizontal lines) being small circles except for the equator.

terrestrial-globe-151897

If you have kept track of all the definitions above you could have guessed that the great circles are the geodesics in spherical geometry. Yes, you are almost correct.

In general, a geodesic (shortest path between two points) in any curved space can be found by writing the equation for the length¹ and then minimise it using some calculus. Elaborate treatment of this is difficult and lengthy to be mentioned here.

But I can give an intuitive but not completely satisfactory explanation of why the great circles are the geodesics of spherical geometry: Through the centre of a sphere and any two points on the surface only one plane can be drawn (except when the two points are the extremities of a diameter). Hence there can be only one great circle through any two points. (So, we chose a great circle over infinite possibilities of small circles that can be drawn through two points on the surface). And this great circle is unequally divided at the two points. The shorter arc among them is the arc of a great circle joining the two points, like a line segment (shortest path between two points).

And yes, in the case of points that are extremities of a diameter (called as antipodal points) there is no unique (straight) line joining them violating Euclid’s first postulate (isn’t this enough to call it non-Euclidean?).

Conclusion: The geodesics of spherical geometry are the great circles.

So what? 

  • The distance between any two points on earth is measured by drawing the arc of great circle joining them (Yes, that’s how they take care of curvature).
  • In general relativity, freely falling particles move on geodesics.

Extras:

  • It is easy to visualise that any two great circles on a sphere intersect at 2 points (which will also form extremities of a diameter also).
  • The above statement also implies that there is no such thing as parallel lines in spherical geometry. The surface of a sphere being finite does not have room for parallel lines in the first case. But there is another argument which says, for any two great circles we can draw a great circle such both form internal angles of 90º (like, equator forms an internal angle of 90º with all longitudes). This implies that all the great circles are parallel.
  • The sum of the angles of a triangle on a sphere is 180°(1 + 4f), where f is the fraction of the sphere’s surface that is enclosed by the triangle.
  • Equidistant curves on the sphere are called parallels of latitude analogous to the latitude lines (which are not geodesics except equator) on a globe.

[1] Such equations are functions from to manifold.
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