Arc Length of Bezier Curves

Question

I'm not sure if this is more of a math.stackexchange.com (or physics??) question, but here it is:

How can I find out the distance traveled by a Bezier Curve? For example, the distance traveled by a Linear Bezier Curve is:

distance = sqrt(pow(x[1] - x[0], 2) + pow(y[1] - y[0], 2));

But what of a Quadratic, Cubic, Nth-Degree Bezier Curve?

My goal is to figure out the "resolution" beforehand, so I don't have to waste time checking if the next point is touching the previous point.

Answer 1

For cubic ones a simple way is to split up the curve into N segments and then compute the distances between each segment and sum those up, as Williham said, this in non trivial, but diving is a rather simple approach.

Also, as soon as you need more than just the length (e.g. get a point at 30% of the curves length), arc-length parameterization will come into play, so you should take a look at that.

I posted a fairly long answer on one of my own questions about béziers which explains that in simple code:
Moving ships between two planets along a bezier, missing some equations for acceleration

Answer 2

While this question is a better fit for math.stackexchange.com, the short version is that except for trivial and/or degenerate cases; solving for the exact length of an arbitrary bezier curve is non-trivial.

If you can make do with an approximation, quadratic curves are relatively easy to deal with (source), cubic curves less so.