Two mirrors are inclined to each other by an angle u.

A ray of light is incident on one of the two mirrors with a direction parallel to another and after four reflections exactly retraces its optical path.

Calculate the angle of inclination of the two mirrors.



If the beam is incident in a direction parallel to another mirror, this will then make an angle u with the first mirror.

In addition, the triangle is isosceles BHD for the law of reflection.
Whence: B H ̂ D = 180 ° − 2 u

Because the line in H is normal to the mirror we can get the angle v:

v = B H ̂ D − 90 ° = 90 ° − 2 u

If we consider now the triangle AHD, for the exterior angle theorem, it must be:

90 ° + w = 2 v + u → w = 2 ( 90 ° − 2 u ) + u − 90 ° = 90 ° − 3 u

Finally one can observe that w must be equal to u because complementary angles to a same angle.

Whence:

u = 90 ° − 3 u → 4 u = 90 ° → u = 90 ° 4 = 22.5 °

N° 4 Reflection