Calculations to prove the non-Euclidean nature of 3-D space

[KedarMhaswade]

TL;DR Summary

This post asks about the precise conversion between the 3-d Cartesian and Spherical Polar coordinates. The problem is from Anthony French's book: Newtonian Mechanics.

In Anthony French's book, Newtonian Mechanics, while explaining the non-Euclidean nature of the 3-d space, he poses a problem (I have rephrased it slightly):

1. Suppose you are on Earth's equator (r = 6,400 km) at the prime meridian (point I).
2. You first walk along the equator 1000 miles east and reach another meridian at point J. You then walk 1000 miles north along that meridian and reach a point K (red arrows).
3. You go back to point I and walk along the prime meridian 1000 north and reach a point Q on some small circle. You then walk 1000 miles east along that small circle and reach a point R (blue arrows).

Are K and R the same point? If not, what is the distance between those two points?

Here is the figure (zoomed in):

To simplify somewhat, we assume a sphere and therefore no geodesy is involved. I assume the physics convention of the Spherical Polar coordinates, i.e. the coordinates are: $(r,\theta,\varphi)$ where $r$ is the radius, $\theta$ is the polar angle, and $\varphi$ is the azimuthal angle.

My plan was to calculate Cartesian coordinates for K and R and determine the distance between the two. I guess I can make some progress on calculations, since the determination of $r$ ($6400 km$)and $\theta_1$ and $\theta_2$ appears straightforward. However, I am unsure how to calculate the $\varphi$ for the points K and R. For instance, in case of point K, $\varphi$ is different from the angle $A_1$ whose radian measure is $\frac{1000\cdot 2\pi}{c}$ where $c$ is Earth's circumference.

 

[jbriggs444]

As a minor correction, the space defined by the surface of the [hypothetical ideally spherical] Earth is, two dimensional, not three dimensional. [Roughly speaking, the "dimension" of a space is the number of coordinates you need to specify a position within that space. Although you could use three coordinates (e.g. x, y and z), you only need two (e.g. latitude and longitude)].

Be that as it may, I have a concern with the drawing. As I understand it, both point Q and point R are 1000 miles north of the equator. Would that not put them on the same line of latitude?

 

[KedarMhaswade]

Be that as it may, I have a concern with the drawing. As I understand it, both point Q and point R are 1000 miles north of the equator. Would that not put them on the same line of latitude?

Good point! I think you mean points K and R. I agree, they are at the same latitude. Will correct that.

 

[KedarMhaswade]

For the record however, the book also shows a figure (pp. 60) where points K and R appear on different latitudes.

 

[jbriggs444]

Good point! I think you mean points K and R. I agree, they are at the same latitude. Will correct that.

Yes. It should be clear that Q, K and R are all on the same line of latitude. It is a lesson not to trust drawings too much.

 

[KedarMhaswade]

I couldn't edit my post on this PF web interface . I redrew the figure. Hope it clarifies.