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.
No, if you rotate an object about the R axis, it still takes 360 degrees to make a full revolution. Degrees still mean the same thing in both Cartesian and Polar coordinate systems.
Start at polar coordinate r,0,0 (north pole). Travel to equator by increasing one of the angular coordinates to 90deg. Now spin counter clockwise 90 degrees, which is you rotating 90 degrees about the R axis at that point. Now travel along the equator by moving forward again, which increases the other angular coordinate.
Try all of that in a 3d graphing calculator, then do it with 60 degree rotation about the R axis instead of 90 and notice how the forward travel afterward increases both angular coordinates, not just one. You'll find that 3 straight lines with 60 degree turns in between do not land you at the same spot on a sphere.
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
(post is archived)