Figure 1.12. Finsler billiard reflection

Let w be the intersection point of the tangent lin

points u and v. Then w = u + a£)=v + bri where a, b

It follows that Dw(f) = l,Dw(g) = - 1 and Dw(f +

is tangent to the mirror Z, then X is a critical point o

/ -f g, Finsler length of the broken line AXB. This

Finsler reflection law.

Of course, if the indicatrix is a circle, one obtains t

of equal angles. For more information on propagatio

Finsler geometry, in particular, Finsler billiards, see [2

1.4. Digression. Brachistochrone. One of the mos

lems in mathematical analysis concerns the trajectory

going from one point to another in least time, subject

tional force. This curve is called brachistochrone (in G

time"). The problem was posed by Johann Bernoulli

the 17th century and solved by him, his brother Ja

L'Hospital and Newton. In this digression we describ

of Johann Bernoulli who approached the problem fro

view of geometrical optics; see, e.g., [44] for a historic

Let A and B be the starting and terminal points

curve, and let x be the horizontal and y the vertic

convenient to direct the y axis downward and assum

coordinate of A is zero. Suppose that a point-mass dro

distance y. Then its potential energy reduces by mgy