How to scale bezier handles to preserve curvature

Question

Given a cubic bezier path $P_0P_1 P_2 P_3$, when one handle is scaled how, to scale other handle to preserve same level of curvature at $P_0$? 


I tried to find $\Delta$ of $P_1$ using a second derivative, but it changes angle of the handle.

Then I found a curvature function. It is definitely possible to use the function to find $\Delta$, but it’s hard to solve analytically.

Is there any other helpful curvature equations?

Illustration of the question

Answer 1

To preserve the curvature vector (not just curvature magnitude) at $P_0$ when moving $P_2$, you will need to preserve the vertical distance between $P_2$ and line $\overline{P_0P_1}$. This means that $P_2$ can only be moved in the direction of $\vec{P_0P_1}$.

Therefore, if you want to move $P_2$ in a direction that is perpendicular to $\overline{P_0P_1}$ (as shown in your picture), the only way to preserve curvature vector at $P_0$ is to move $P_0$, $P_1$ and $P_2$ all together.