I want to describe a Turbine Airfoil using B-Splines. For the leading edge (blue) of the airfoil I'm using an ellipse arc which is then followed by b-spline curve which describes the airfoil body (red). The B-spline degree is $p=3$, has 5 control points $P_0,P_1,\ldots,P_4$ and it is clamped at $P_0$ and $P_4$. Since the b-spline is clamped then it is also tangent to the first and last leg of its control polyline. Therefore, in order to get $C_1$ continuity at $P_0$ I made the first leg of the polyline $(P_0,P_1)$ in the same direction of the tangent line to the ellipse at $P_0$. However, the geometry must have at least $C_2$ continuity. I don't know if there is a simple way to match the curvatures of the ellipse and the b-spline at $P_0$ to get the required continuity. Please, notice that I have flexibility in defining the control points of the b-spline, I can add more or relocate then if necessary.
Thank you for your assistance
This paper should solve your problem.
https://www.sciencedirect.com/science/article/pii/0167839687900021
It shows how to construct a cubic curve (i.e. a spline with a single segment) that has prescribed positions, tangents, and curvatures at its start and end points. So, the curve will have 4 control points, not 5.
The approach is conceptually very simple — you just write down equations expressing the constraints, and solve them. Solving requires numerical methods, but nothing very difficult.