Connect two points, given their angles, with a maximum radius I want to connect two points on a plane. Both of the points have their angles given. The angles are representing the direction vector in which the curve should pass through this point.
One of this point lies on the center of the coordinate system. The other one can lie in any other location of the space.
For example: 
$$
\begin{aligned}
P_0 &= (0,0) & \vec P_0 &= \begin{bmatrix} 1 \\ 0 \end{bmatrix} & \gamma &= 90°
\\
P_1 &= (3,3) & \vec P_1 &= \begin{bmatrix} 0 \\ 1 \end{bmatrix} & \gamma &= 0°
\end{aligned} 
$$
The resulting minimum radius of the curve should be as big as possible between the two points. In addition the length of the path should also be as short as possible. There should be no addition of extra straight paths or very big loops to stretch the radius.
For now I tried to access this problem with Beziér curves, Hermite curves and ellipses but for none I got the sufficient solution.
 A: If the input data don't imply an inflexion, then you can use a rational quadratic Bezier curve to solve the problem. Let the start and end points be $\mathbf{P}_0$ and $\mathbf{P}_1$, and let $\mathbf{P}_a$ be the point of intersection of the end tangent lines (the "a" stands for "apex").
The rational quadratic Bezier curve defined by this data is:
$$
\mathbf{C}(t) = \frac{ (1-t)^2 \mathbf{P}_0 + 
                         2wt(1-t) \mathbf{P}_a +
                              t^2 \mathbf{P}_1 } 
                         { (1-t)^2  + 2wt(1-t) +t^2 }
$$
You can use the weight $w$ to adjust the curvature of the curve. Using $w=0$ will give you a straight line, and large values of $w$ will give you a curve that passes very close to $\mathbf{P}_a$ and has a sharp turn in this region.
With some work, I expect you can calculate the minimum radius of curvature as a function of $w$, and then you can choose the value of $w$ that maximizes this minimum.
A simpler approach is to use some heuristic rule to define $w$. For example, one choice that I have used in numerous applications is:
$$
w = \sqrt{\tfrac12(1 + \mathbf{U} \cdot \mathbf{V})}
$$
where $\mathbf{U}$ and $\mathbf{V}$ are unit tangent vectors (i.e. unit vectors in the directions of $\mathbf{P}_a - \mathbf{P}_0$ and $\mathbf{P}_1 - \mathbf{P}_a$ respectively).
If the input data are symmetrical, this will give you a curve that it exactly circular. In other cases, you will get some other conic section curve (rational quadratics are always conics).
The picture below shows some example curves. The red one has $w$ computed via the heuristic formula I gave.
