Given three pairwise distinct points $p_1, p_2, p_3 \in \mathbb{R}^2$, I'd like to find a function $f: \mathbb{R} \to \mathbb{R}^2$ with at least $f \in C^1$ such that $f(0) = p_1, f(1) = p_3, f'(1) = a$ for a given vector $a$ "pointing" inside the convex hull of $p_1, p_2, p_3$ s.t. $(p_2 - p_3) \cdot a \leq 0$ and $(p_1 - p_3) \cdot a \geq 0$. Furthermore $f(t)$ should stay inside the convex hull spanned by $p_1, p_2, p_3$ for all $t \in (0,1)$.
The above problem is very similar to the Bezier spline with the only difference of the specific prescription of a derivative at one of the end point $p_3$ which does not necessarily point in the direction $(p_2-p_3)$.
As I'm quite new to spline interpolation, I'd be very much indebted to any help regarding this problem.