Getting the center point of a quadratic Bezier Line with SKPath (Skia Sharp)

Question

I draw Bezier Lines with the QuadTo-Method on a SKPath. My requirement is to get a point on the rendered Bezier Line which is more or less in the middle of the line. I use this point to show a label on the line and to provide a hit point to interact with the line.
Currently, I shoehorned a simple algorithm which tries to resolve the points of the line vie GetFillPath and then try to detect the point which is nearest to the center of a thought straight line from the Bezier’s start and end.
This works, however, it feels extremely brutish. Is there a more sophisticated way to fulfill my requirement?

Answer 1

The quadratic Bezier has a pretty simple parametric formula to calculate points lying on it, you can find it in many places, e.g. https://en.wikipedia.org/wiki/B%C3%A9zier_curve

As per description of your problem it is important to notice, that you don't specifally want to calculate the point B(0.5), as the length of curves (P₀, B(0.5)) and (B(0.5), P₂) can be different for non-symmetric Beziers. What I would do is:

1. Flatten the curve to a polyline with an arbitrary precision p (selecting all the t params so that for every 3 consecutive t such that: t_k < t_k+1 < t_k+2 distance between line (P_k, P_k+2) and point B(t_k+1) < p)
2. Find k such that: L(P₀, ..., B(t_k)) <= L(P₀,..., P₂)/2 < L(B(t_k+1), ..., P₂), where L is the length of polyline.
3. Now with found k we know our middle point t_mid fulfills t_k <= t_mid < t_k+1. This can again be computed with an arbitrary precision.

For any higher degree Beziers you can use de Casteljau algorithm to calculate points for any given 0 <= t <= 1.