Algorithm to find the center of a Bezier curve I need to find the center of a Bezier curve to rotate it. I have a list of all points (control points, beginning and end, all points on the curve itself). How would I go about finding it's center?
Bézier curves are mathematical entities and have no clearly defined center. One can in fact define many different things as the center of the Bézier curve. I have tried to depict some of the possible centers in image 1. More than this do exist.



Image 1: Some of the possible centers of a single span Bézier curve

In practice nearly all graphics applications geared for drawing use the center of the local bounding box (BB) as their center. Animation software usually have an additional concept of pivot so they use the ask user approach, if no input is made they often revert to BB center or simply local coordinate center. This is probably because the BB needs to be calculated anyway and getting its center is pretty easy to do (see A Primer on Bézier Curves). 

The center of gravity metrics are also somewhat natural especially in an animation context, though nastier to compute. Easiest being to discretize the data and do the computation on the discrete input. This said some closed form solutions are possible for the curve center of gravity, but it's not a very nice equation to formulate and simplify.

Then we have the on the curve points: the midpoint by arc length and the point where the $t$ parameter is 0.5. In my mind the $t$ param is often problematic though easy to compute, and it loses meaning when you chain multiple Béziers after each other for a polybézier. The center of length of course is only natural as long as the curve is not closed.

We also define other possible centers, the center could be at the curve center of gravity of the hull, the average of the control points or the control cage BB center. Although in practice these do not seem to work out very well.

Please note: Although the curve in picture 1 shows the BB center quite near to some natural centers this this is not always the case for more complex curves and especially polybéziers.