piecewise C2 cubic Bezier spline

Question

I am implementing c2 interpolating splines.

Assuming the first segment's 4 points are p0, p1, p2, p3. And second segment's 4 points are q0, q1, q2, q3.

To achieve C2 continuity, I'm using the following conditions:

q0 = p3
q1 = q0 + (p3 - p2)
q2 = (p3 - 2 * p2 + p1) + 2 * q1 - q0

However, when I apply this setup, I notice that the distance between the control points in each successive segment seems to increase compared to the previous segment’s control points. Is this lengthening effect expected with C2 continuity?

My result

So I’d like to know if this lengthening effect is expected behavior. If it’s not, how can I correct it?

Thank you.