Shortest Path between Two Points in the Unit Disk Reflecting off the Circumference

Given two points within the unit disk labeled

S

(in green) and

E

(in red), what is the shortest path between the two points that reflects off the unit circle? You can drag the two points.

The central angle between the two points is

β

. The angle from the

x

axis to

S

is

ϕ

. The shortest path that touches the unit circle is at angle

*

θ

.