study of billiards, geometrical optics. According to th

ciple, light propagates from point A to point B in the

time. In a homogeneous and isotropic medium, that i

geometry, this means that light "chooses" the straight

Consider now a single reflection in a mirror that

be a straight line / in the plane; see figure 1.8. Now

for a broken line AXB of minimal length where X G

position of point X, reflect point B in the mirror and

Clearly, for any other position of point X, the broken

longer than AXB. This construction implies that th

by the incoming and outgoing rays AX and XB with t

equal. We obtain the billiard reflection law as a cons

Fermat principle.

B A

X \ / \ / X

B'

Figur e 1.8. Reflection in a flat mirror

Exercise 1.12. Let A and B be points inside a plan

struct a ray of light from A to B reflecting in each sid

Let the mirror be an arbitrary smooth curve /; see

variational principle still applies: the reflection point

the length of the broken line AXB. Let us use calculu

reflection law. Let X be a point of the plane, and defi

f(X) = \AX\ + \BX\. The gradient of the function \A

vector in the direction from A to X, and likewise for