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In Euclidean space, There is a classic Theorem claims that: The length of every rectifiable curve can be approximated by sufficiently small straight line segment with ends on the curve.

Now, The question is that: Is there a similar thing happen on plane of constant curvature? I.e., Can the length of every rectifiable curve on the plane of constant curvature be approximated by sufficiently small geodesic line segment?

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Yes, this is normally how the length of the curve would be defined, if it is not assumed to be smooth. –  yasmar Nov 11 '11 at 18:26
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