By the Lagrange multipliers principle, X is a critica

only if Vf(X) is orthogonal to I. The sum of the uni

A to X and from B to X is perpendicular to I if and o

BX make equal angles with /. We have again obtain

reflection law. Of course, the same argument works if

smooth hypersurface in multi-dimensional space, and

geometries other than Euclidean.

Figur e 1.9. Reflection in a curved mirror

The above argument could be rephrased using a dif

ical model. Let I be wire, X a small ring that can m

wire without friction, and AXB an elastic string fixe

and B. The string assumes minimal length, and the eq

dition for the ring X is that the sum of the two equal

along the segments XA and XB is orthogonal to I. T

equal angles condition.

1.3. Digression. Huygens principle, Finsler me

billiards. The speed of light in a non-homogeneous ani

depends on the point and the direction. Then the traje

are not necessarily straight lines. A familiar example i

going from air to water; see figure 1.10. Let c\ and CQ

of light in water and in air. Then c\ Co, and the traj

is a broken line satisfying Snell's law

cos a _ Co

cos ft c\