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The definition of arc length of a parametric function is given by $\int|r'(t)|dt=\int\sqrt{[x'(t)]^2+[y'(t)]^2+[z'(t)]^2} dt$

So I guess what I'm asking is how do I use a function like $z=\cos(x)+\sin(y)$ with this definition? I am aware that $z(x,y)$ is a surface, but is it possible to find the distance between two points through the surface using this definition? If not, then how do I go about doing so?


Example:

<span class=z=\cos(x)+\sin(y)"> If I were an ant along this surface, how would I find the distance needed to travel between one of the peaks and wells in this graph?

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    I meant taking a line curve from the surface and finding the arc length of that line curve.2012-11-01

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Answering to get this out of the “unanswered” queue:

Finding the shortest distance between two points on an arbitrary surface is asking for a geodesic connecting these points. Computing geodesics can be quite complicated, depending on how your surface is given. There is literature available on the subject, which I won't copy into this answer.