Given
- a circle with radius $r$ centered on the origin at (0,0)
and, in polar coordinates:
a start point, at radial distance $p$ (with $ 0 < p < r$) from the origin, and angle $\alpha$
an end point on the circle boundary at radial distance $r$ from the origin and angle $\beta$
how can I construct a curve segment that satisfies the following constraints:
- it starts at the start point, aligned with angle $\alpha$ (radially outwards)
- 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
?
Without loss of generality, the angle $\alpha$ can be assumed to be 0, because only the angle difference between $\alpha$ and $\beta$ matters.
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 only works when $ p = 0 $.
An example of curves that fulfill the first three constraints, and most the fourth, but only the straight lines the last (and happen to have their start point at the center).
