Curvilinear coordinates

Roy Ernest Ballard

1 Position and coordinates

Coordinates are called curvilinear when coordinates are constant not over a plane but over some curved line or surface. For example any line drawn on the surface of a sphere is curved.
A one-dimensional creature living on a circle cannot measure distances other than  along the perimeter. If this creature were capable of rational thought it might want to calculate distances using axes at right angles but this requires an extra dimension, an abstract idea if you are thoroughly 1-D.  In higher dimensions the same puzzle is the subject of Riemannian geometry and the general theory of relativity.....
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https://drive.google.com/file/d/0BxPw8lRke1HBWmtubEhuQ1RuNkU/view?usp=sharing