Spherical Geometry

We will now see some basic aspects of spherical geometry. It is the study of geometry which is drawn on a spherical surface rather than a plane.

Let us take the case of Earth. Take 2 longitudes one at 0 (Greenwich, UK) and another 80 degrees west (Hyderabad, India) as 2 sides of a triangle drawn on the surface of the Earth. Take the part of the equator between these 2 longitudes as the base of the triangle.

At the equator, both the longitudes form right angles with the Equator. Due to the spherical shape of the Earth, when these longitudes are extended to the north, they meet at the North Pole! Now 2 theorems of Plane Geometry are proved wrong.

1. Though both the longitudes are at right angles to the equator, both the longitudes, when extended, meet at the North Pole. They start as parallel lines but meet at a point, when extended.

2. The triangle formed had 3 sides – the 2 longitudes and the third being the equator. The 2 base angles of this triangle is 2 Right Angles i.e a Straight Angle. Hence in this circle the sum of the angles of a triangle is greater than 2 Right angles!

Hence it was understood that Euclid’s Fifth Postulate is true only when the surface on which it is drawn is flat. These effects are not observable for small distances & areas on the Earth. But over large distances they need to be taken into account. Which means that the Fifth Postulate was a fundamental property of a plane surface!

In case of figures drawn on a spherical surface, the sum of the angles of a triangle would be greater than 2 right angles. The geometry of figures drawn on a spherical surface was named as Spherical Geometry. The surface is convex.

Straight Lines

What about the idea of a straight line on Earth? We have accepted the idea that a straight line is the shortest distance between 2 points.

Let us take the 2 points as the location of London & Chicago on Earth. What is the path taken by an airline, if it want to go by the shortest distance? It will not be that given by joining London & Chicago with a straight line on the page of the atlas. This is because the page on the atlas does not represent the curvature of the Earth. If you spend some time examining the locations of London & Chicago on a globe, you will see that the shortest distance is through the Arctic!

Some more thinking would reveal that the shortest distance between any 2 points on a globe will be through a “great circle” passing through both the points. A “great circle” being a line going around the Earth just like the Equator or the longitudes. The Equator is also a “great circle” which has been given the special name because it is situated equidistant from both the poles.

Plane Geometry on the Earth

The Earth is almost 8000 kms in diameter. Hence for distances typically covered by even major cities, the land surface can be taken as plane surfaces, where principles of plane geometry can be applied.

But the GPS system which we use today for navigating our streets, uses satellites orbiting the Earth at heights of several hundred kms. This system has to use spherical geometry to locate the different points on the Earth which is a sphere!

The Geometry of the Universe

Astronauts & cosmologists who study the far reaches of the Universe are sure that at the level of galaxies, the geometry of the Universe is not plane geometry. But they are not sure as to the nature of its geometry; whether it is Spherical, Hyperbolic or some other completely different geometry!