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a.) an origin,

b.) three everywhere non-coplanar curves, call them c1, c2, c3.

Where "everywhere non-coplanar" means that at any given choice of parameters t1, t2, t3, then c1'(t1), c2'(t2), c3'(t3) form a linearly independent set.

The requirement (a.) is necessary in order for us to locate distinct points. The requirement (b.) gives us a way in which to measure the position of the points. I suppose this could work by saying that when you specify a point p = (p1,p2,p3), a three-tuple, then that means the point lies at the intersection of the planes perpendicular to the velocity vectors c1'(p1), c2'(p2), c3'(p3).

Similarly, (b.) also gives us a way to provide a basis at every point, since the basis at point p=(p1,p2,p3) can be written as c1'(p1), c2'(p2), c3'(p3).

Does this seem correct, and if not, what are the general requirements for a coordinate system?