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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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    By "distance" do you mean shortest distance or something that is specified by a path $\gamma:[a,b]\to\mathbb{R}^2$?2012-11-01
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    No, I'm talking about the shortest path between two points in $\mathbb{R}^3$ **along** the surface of the graph. It's as if I'm taking a slice of the surface and calculating the arc length of that slice.2012-11-01
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    http://mathworld.wolfram.com/Geodesic.html2012-11-01
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    Also, what kind of "slice" are you referring to? A straight line may not give you the shortest distance.2012-11-01
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    arggg... Is it really as complicated as a geodesic? I guess so... darn. Why didn't I see it before.... Thank you for pointing that out.2012-11-01
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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.