Find the control point of quadratic Bezier curve having only the end-points 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.
 A: As Joriki says, you can't get the other control point with no other information.
However, if you know a point on the line you can work it out.  The formula for a quadratic Bezier function taken from https://en.wikipedia.org/wiki/Bezier_curve is:
$P(t) = (1-t)^2P0 + 2(1-t)tP1 + t^2P2$
If you know any two control points, and a separate point on the line at a known time you can calculate the third control point.  Say you know the start and end points P0 and P2, and know the curve passes through P(0.5) at t=0.5:
$$
P(0.5) = (1-0.5)^2P0 + 2(1-0.5)0.5P1+0.5^2P2\\
P(0.5) = 0.25P0 + 0.5P1 + 0.25P2\\
P1 = 2P(0.5)-0.5P0 - 0.5P2
$$
For a cubic Bezier if you know the start and end points you can find the middle two control points using the method described here: https://web.archive.org/web/20131225210855/http://people.sc.fsu.edu/~jburkardt/html/bezier_interpolation.html
That requires you to know the points that the curve goes through at t=1/3 and t=2/3.  I'm afraid I haven't tried generalising it to any points in time.
A: 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.
