# How to get the arc length of a real-valued function in 3D.

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:

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 $\gamma:[a,b]\to\mathbb{R}^2$? –  wj32 Nov 1 '12 at 0:17
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. –  Zchpyvr Nov 1 '12 at 0:19
mathworld.wolfram.com/Geodesic.html –  wj32 Nov 1 '12 at 0:26
Also, what kind of "slice" are you referring to? A straight line may not give you the shortest distance. –  wj32 Nov 1 '12 at 0:29
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. –  Zchpyvr Nov 1 '12 at 0:29

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.

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