In polar coordinates as you travel the surface of the earth the radius coordinate would stay relatively constant while the two angular coordinates change according to your position. A constant coordinate is not enough to say it is flat though.
Start traveling on the surface in a straight line for a large distance, such as the distance required to go from the north pole to the equator. Then stop and turn left 90 degrees, then travel straight again the same distance. Then stop and turn left 90 degrees and travel straight for the 3rd time. Then do it again for the 4th time. The surface is flat if you end up at your starting position even if you tracked your R polar coordinate and noted it hardly changed. The earth is curved so you won't be in the same spot after the 4th straight line...in fact, if you use the extreme example of pole-to-equator distance, you will end up at the starting position after the 3rd straight line, and will be somewhere wildly different after the 4th.
Yes but you're in polar coordinates now. Those 90 degree angles in cartesian, are now 60 degree angles. You made a triangle.
Wtf are you smoking?
Polar coordinates are going to have an r from the core and two angles. To make the three turns you were suggesting in cartesian, would be three 60 degree turns in the difference in the polar cordinate's two angle values.
Well 60 if you walked equal distance for the first three walks. Otherwise either way the sum of the interior angles in polar cordinates will be 180 not 270.
To make it more clear... The 3 points you end up standing on. Make a plane out of this. Now draw lines connecting the 3 dots. The angles between these lines will add up to 180 not 270. It's a triangle. That plane will be perpendicular to the core. These are the angles you turned in polar cordinates.
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