Consider a 2-dimensional surface embedded in 3 dimensional euclidean space.
ex: A plane, a sphere, a hyperboloid of 2 sheets (or 1 sheet), the graph of sin(x + y), 2 parallel planes etc...
If we consider the case of the plane and of the graph of sin(x + y) every point on these surfaces can be characterized by its distance from the "origin" of these graphs, on 2 perpendicular axis.
The hyperboloid, the sphere, 2 parallel planes can all be characterized by selecting a point in between them and utilizing 2 angles, to describe every point on both surfaces.
But consider a surface of 3 parallel planes...
Can every point on these three planes be described using 2 numbers? Disturbingly, yes...
Lets say we took the cartesian plane (a single 2d plane) and we drew three rays from the origin that clock at 0 degrees, 120 degress, and 240 degrees, thus seperating the cartesian plane into essentially 3 separate "areas". Let us say we select a point with 2 coordinates on this plane. It will land on one of the "areas". If the two rays that bound the area are assigned 0 degree and 360 degree then it is possible to create a functional mapping such that every possible degree value on one of our original planes (in polar coordinates) can be expressed for each of the three "areas" of the sliced cartesian plane, the distance from the origin is in this mapping is the same as the distance from the origin of the three original planes with polar coordinates.
Thus every individual point on this coordinate plane (2 coordinates) can map to a point on of the three planes earlier, such that there is a unique 1 to 1 mapping between every point on the three planes and the single cartesian plane.
This same argument can be extended for 4 planes, 5 planes etc...
But can it be proven that any collection of 2 dimensional surfaces be essentially mapped using exactly 2 coordinates? I think the answer is yes but I cannot generalize my argument to ALL curves.