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

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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
.
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