Find the control point of quadratic Bezier curve having only the end-points

Question

How to explicitly find the control point $C_0(x_0,y_0)$ of quadratic Bezier curve if I have only its end-points $C_1(x_1,y_1)$ and $C_2(x_2,y_2)$?

Guess

This should be done using the fact that the tangent passing through $C_1$ and $C_2$ meets at $C_0$. So, from $$y=m_1x+b_1~~ {\rm and}~~ y=m_2x+b_2, $$ with $$ m_1=\frac{y_0-y_1}{x_0-x_1},~~ m_2=\frac{y_0-y_2}{x_0-x_2},~~ b_1=y_1-m_1x_1,~~ b_2=y_2-m_2x_2. $$ Therefore $$ m_1x_0+b_1=m_2x_0+b_2~~ {\rm or}~~ x_0=\frac{b_2-b_1}{m_1-m_2}, $$ which is nothing else but identity. Am I doing something wrong?

Edit

The end-points are located in an ellipse.

Answer 1

You can't get the control point from the end points. A quadratic Bézier curve is defined by all three points. If you have only the end points, you can arbitrarily choose a control point to define a quadratic Bézier curve.