Commentary 2.3 Coordinate transformations plane and sphere

The last part of Exercise 2.04 asks us to perform a coordinate transformation. We really want a transformation from a curved space to a curved space, so I decided to try to find a coordinate transformation for the surface of a sphere. This was quite hard. So I started with rotating the plane, R². That came in useful later to calculate the transformation for a sphere.

Consider a point C which has coordinates (x , y) in S and (x' , y') in S', S' is rotated by an angle θ to S. It's quite easy to work out x' and y' from x and y and the answer was given at (1.3.1) in the book.

I then wanted to get a coordinate transformation for the surface of a sphere. The coordinates are longitude, latitude as usual and the rotated coordinates have 0 latitude rotated up by θ along 0 longitude as pictured below.

First of all I tried to do this directly like the rotating plane. But I got in a mess. I then decided to combine three transformations:

A: Transform polar coordinates into Cartesian: S -> S"

B: Rotate the Cartesian coordinates: S" -> S"' (already done for plane)

C: Convert back to polar coordinates: S"' -> S'

The answer was

x² is longitude, x³ is latitude, x¹ has disappeared - it was the distance from the centre which is always 1 on the surface of our sphere.

So I now hope that I have the equipment to tackle the last part of exercise 2.04.

Read all the details at 

Commentary 2.3 Coordinate transformations plane and sphere.pdf