From r,0,0 to r,0,90 is not an angle change of 90 degrees. It is a raw distance of 90 traveled.
0,0,90 -> 0,90,90 -> 0,0,0
Forget the first value to simplify.
0,90 -> 90,90 -> 0,0
Three points. Now imagine those were cartesian values. It would look like this -
(0,90) X--------X (90,90) | / | / | / X (0,0)
What is that? It's a triangle. What are the sums of the interior angles? 180.
From r,0,0 to r,0,90 is not an angle change of 90 degrees. It is a raw distance of 90 traveled.
Full stop, that's where you went wrong. The units of R are units of length, but the other two coordinates are not, they are instead in units of angle. That is precisely what distinguishes polar coordinates from Cartesian coordinates. In Cartesian all three coordinates are units of length, as you continued to use in that example
Also it may be true that r vs the angles are of different units. However the angles are of the same unit type. You can compare them like the way I mentioned.
Ya that's the whole point. You warp the understanding of the world in polar coordinates. Where once converted you think of them "as cartesian". Now you effectively got an earth of just two dimensions. Hench, it's flat. In polar coordinates, yes the surface of the earth is essentially a 2 dimensional surface.
The thing that would kill the flat polar coordinate earth theory would be if the planet wasn't hollow. If the planet actually was hollow, it truly would be flat in polar coordinates.
You are warping the understanding of the world in something close to polar but not polar. Polar uses a radius (length) coordinate and two angular ones.
Regardless, you aren't getting that rotating about the R axis is the same as rotating about the vertical Cartesian axis on flat land, in that they both need 360 degrees to complete a revolution
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