In the rectangular plane, let be points, such that
- is parallel to the horizontal axis;
- Point is positioned along the locus;
- , where point is the intersection point of two axes;
As shown on the right, for , overlaps . The following are all possible arrangements, involving and respectively, where is one of the acute angles of .
If the arc length of the locus bounded by is , input as your answer.
Bonus. Find the exact value of .