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I want to divide a cubic bezier curve, with 4 points, start, end and 2 control points, into segments that are not bigger then a certain distance. So, am looking for a computationally quick way to approximate the arc length of curve.

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Use subdivision and stop recursion when the line segment joining the two endpoints is small enough. Not the same thing as arc length, but will probably be good enough.

If you want to approximate the arc length of a Bézier curve, see my answer in this question.

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Instead of looking at the sub-segment length, you could look at the curvature of the sub-Bezier and stop recursion when the curvature is below some threshold. A couple simple dot-product operations and square roots are needed to check the cosines of the angles involved. – Todd Lehman Nov 29 '13 at 8:12

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