Beautiful geometry: Laser bouncing of walls of a semicircle

Question

Consider a semicircle with diameter $AB$. A beam of light exits from $A$ at a $58^{\circ}$ to the horizontal $AB$, reflects off the arc $AB$ and continues reflecting off the "walls" of the semicircle until it returns to point $A$.

How many times does the beam of light reflect of the walls of the semicircle (not including when it hits $A$ at the end)?

Note: "walls" of the semicircle refer to the diameter $AB$ unioned with the arc $AB$.

Can the laser bounce of segment $AB$ should it hit the diameter? — imranfat, Mar 14 '16 at 03:58
That makes it an interesting question...What are your thoughts on this? — imranfat, Mar 14 '16 at 03:58
A Geogebra construction might give some insight into a problem such as this. — John Wayland Bales, Mar 14 '16 at 04:19
Well I find that, if we replace the semicircle with a circle, we get that the beam hits the circle at points which forms 64 degree arcs. — math_lover, Mar 14 '16 at 04:22
We know that when the beam hits a wall, it reflects so that the angle of incidence equals the angle of reflection. Also, if we know the how many times the beam contacts the diameter $AB$ we can solve the problem by replacing the semicircle with the full circle and then reflecting over the diameter that number of times. — math_lover, Mar 14 '16 at 04:27
If we have a full circle instead of a semicircle, after hitting the wall $n$ times the beam is at point $P$ on the circumference with $\angle AOP=64n$. The beam first returns to $A$ when $360|64n$ or $n=45$. The beam crosses the diameter $8$ times. Going back to the original problem with the semicircle, the answer is $45 +8=53$, or something likes this. — math_lover, Mar 14 '16 at 04:38
Actually, the beam crosses the diameter 16 times, so the answer is 61. — math_lover, Mar 14 '16 at 04:45
Possible duplicate of Reflected rays /lines bouncing in a circle?. The only apparent difference is that this question considers a semicircle instead; however, reflecting the semicircle case across the $x$-axis gives back the circle case. — Semiclassical, Mar 14 '16 at 06:20

score 1 · Answer 1 · answered Mar 14 '16 at 05:47

1

I cheated and used GeoGebra. It appears to reflect 29 times (unless my old eyes miscounted) until it comes down vertically to point C. At that point it will reverse path for a total of 59 reflections until it returns to A.

answered Mar 14 '16 at 05:47

John Wayland Bales

21,814

score 1 · Accepted Answer · answered Mar 14 '16 at 17:03

By reflection across the diameter we can map what happens to the beam inside inside the semicircle to a full circle; them beam contacting the diameter and bounces off in the semicircle is equivalent to the beam passing through the diameter of the full circle.

We know that when the beam hits a wall, it reflects so that the angle of incidence equals the angle of reflection. By a simple angle-chase, we get that the beam hits the circle at points which forms 64 degree arcs.

After hitting the wall $n$ times the beam is at point $P$ on the circumference with $∠AOP=64n$. The beam first returns to $A$ when $360|64n$ or $n=45$. Thus the beam contacts the arc $AB$ 44 times, because we don't count when it hits $A$ at the end. The beam crosses from the upper half-circle to the lower half-circle or vice verca $64*45/180=16$ times, however this counts includes when the beam hits $A$ at the end, so it actually crosses $15$ times. The total numbner of contacts with the walls is then $44+15=59$.

Beautiful geometry: Laser bouncing of walls of a semicircle

2 Answers2