The same surface admits many different parametrizations. For example, the coordinate z-plane can be parametrized as

for any constants a, b, c, d such that ''ad'' - ''bc'' != 0, i.e. the matrix is invertible.

Local differential geometry

The local shape of a parametric surface can be analyzed by considering the Taylor expansion of the function that parametrizes it. The arc length of a curve on the surface and the surface area can be found

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