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Nappe with different parametrisations

Cartesian coordinate lines on a uniform square mesh when mapped to a nappe of a surface generate nonplanar quadrilaterals of very different areas, which illustrates the need for the Grammian term in the surface measure.

The same surface patch (actually a slightly smaller sub-patch here) can be parametrised using polar coordinates, leading to an approximation by different, and even more unequally sized patches.

The still images were produced by a small application called grapher that is distributed as part of Mac OSX, and the associated file is available for you to play with. You can find another example here. Of course it is also easy to produce similar plots in Maple or Mathematica or Matlab.