Given

• a circle with radius $r$ centered on the origin at (0,0)

and, in polar coordinates:

• a start point, $p$ for the radial distance of the point from the origin, and angle $\alpha$

• an end point on the circle boundary at radial distance $r$ and angle $\beta$

how can I construct a curve segment that satisfies the following constraints:

• it starts at the start point, aligned with (as in: tangential to) angle alpha
• it ends at the end point, aligned with angle beta (radially outwards)
• the curve stays within the circle area (intersects the circle boundary only once, at the end point)
• between the start and end points, the radial distance of points on the curve from the (0,0) origin is non-decreasing (ideally, strictly increasing)
• the maximum curvature is minimized

?

Does a function or algorithm for this exist?

This is a variation of this question that only considered the start point being placed at the center of the circle. The solution there does not work here.

An example of curves that fulfill the first three constraints, and most the fourth, but not none the last (and happen to have their start point at the center).