I'm currently facing with the problem that I have to define a parabola with only two tangents in 2D-space. The clue for this task is, that I just know the x-positions ($x_1$ and $x_2$) as well as the two angles of the tangents ($\alpha_1$ and $\alpha_2$). The y-positions of the points are unknown. After the parabola is defined it will be moved up or down so that the curve encapsulates a specific area with the x-axis. The angles are given by tilt sensors at the given x-positions. The parabola will describe the condition of a floating body, so the y-positions will vary with trim and draught.
The parabola should be of the form:
$\overline F = \left( \begin{matrix} cos & -sin \\ sin & cos \end{matrix}\right) \cdot \left( \begin{matrix} t \\ at^2 \end{matrix} \right) \ and \ maybe\ \ + \left( \begin{matrix} x \\ y \end{matrix}\right) $
Unfortunately, I have no idea how to solve this. Can anybody give me a hint or a solution? I would highly appreciate this. Tank's.
Best regards
Nicolas
Visual hint:
If you know two tangents and the tangency points, you also know where a third point of the parabola lies and you can easily construct two other points on the parabola by exploiting a skew reference system.