creating an aligned component to another compound angled component

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This might be a curly one.... I am working on a chair brace located at the top of the chair. The 2 side frames of the chair are both inclined in toward the center slightly, as well as reclined toward the back. This creates a compound angle that the brace needs to align to. The brace consists of a curved back that is attached to 2 circular steel rods that fit down inside the tops of the frames. I need to make the brace such that at the points where it fits down inside the frames it must be aligned parallel to the axes of the frames on each side, as well as be perpendicular to the frames when viewed from the orthogonal side views (left and right)
I have constructed an arc and then extruded it, but it is incorrect. What it actually requires is a loft through 3 profiles. The 2 ends need to be aligned to the axes of the frames, while the middle profile needs to ensure it perpendicular orientation to the frames when viewed from the sides.
I hope I have explained that ok.... Would love some help.