1
$\begingroup$

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 $(x_1, y_1)$?

  • 0
    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: http://en.wikipedia.org/wiki/Geodesic2012-01-07

0 Answers 0