# Deriving an expression for minimum arc length along a 3D surface between any two points.

Consider a 3D surface, defined by the function $z = f(x, y)$. Assuming the surface is differentiable (no kinks), is there a function that expresses the minimum arc length traced along the surface between any two points $(x_0, y_0)$ and $(x_1, y_1)$?

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What you seem to want here is the length of the surface's geodesic going through your two points... – J. M. Jan 7 '12 at 1:46
Yes, there is such a function: namely, the minimum arc length function. But my guess is that you really mean to ask, "Is there a simple explicit formula for such a function?" in which case I would have to say no. – Jesse Madnick Jan 7 '12 at 3:33
The answer here is of course yes, as was pointed out above. Moreover, in general there is no "function" in the sense that the minimizing arcs do not have to be unique (globally). However, if $x$ is a point on the surface, then for all $y$ sufficiently close to $x$, there is a unique arc connecting $x$ and $y$ of minimum length. For more details, see on geodesics here: en.wikipedia.org/wiki/Geodesic – William Jan 7 '12 at 4:07