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
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