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Within say a surface whose points {x 1 , x 2 , … } are obtained by evaluating an expression e as a function of parameters p (so that for example e = {x, y, f[x, y]} , p = {x, y} for a Plot3D surface) the metric turns out to be given by
(Transpose[#] . # &) [Outer[D, e, p]]
In ordinary Euclidean space a defining feature of geometry is that the shortest path between two points is a straight line.