these points. Let X G Ft and let u be the Finsler unit
to the trajectory of light from A to X. An infinitesimal
Huygens principle states that the tangent space to the
parallel to the tangent space to the indicatrix TUS(X)
figure 1.11.
Figure 1.11. Huygens principle
We are in a position to deduce the billiard reflection
geometry. To fix ideas, let us consider the two-dimensi
Let I be a smooth curved mirror (or the boundary of a
and AXB the trajectory of light from A to B. As us
that point X extremizes the Finsler length of the brok
Theorem 1.14. Let u and v be the Finsler unit vectors
incoming and outgoing rays. Then the tangent lines to
S(X) at points u and v intersect at a point on the ta
at X; see figure 1.12 featuring the tangent space at po
Proof. We repeat, with appropriate modifications, th
the Euclidean case. Consider the functions f{X) = \AX
\BX\ where the distances are understood in the Finsl
and r\ be tangent vectors to the indicatrix S(X) at p
One has, for the directional derivative, Du(f) = 1 sin
to the trajectory of light from A to X. On the othe
Huygens principle, £ is tangent to the front of point
through point X. This front is a level curve of the fun
Dz(f) = 0. Likewise, Dv(g) = 0 and Dv(g) = - 1 .
Previous Page Next Page