First I made the interpolation points as Bezier curves of higher order. This is not correct, see wrong.png. Then added via points - all became good, see good.png. If there is a sequence P0, P1, P2, P3, and P0,P3 - on curve, P1,P2 not on curve, It splits into two quadratic Bezier curve.
P0, P1, (P1+P2)/2
and
(P1+P2)/2, P2, P3