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Let 
be the equilateral triangle with sidelength 1/3.
We construct tables T obtained from by
the so called string construction (see i.e. [11, 12]).
One takes an unstretchable string having length 
,
wraps it around , pulls it tight at a point, M,
and drags it around . The point M then traces the table.
Varying the parameter 
, we get a one-parameter family
of billiard tables 
. Each of these tables is composed of
piecewise elliptic arcs. (See Fig. 1.).
|
Fig. 1. Four examples of tables given by the string construction at
an equilateral triangle.
|
The corresponding billiard maps, 
,
form a natural one-parameter family of
twist maps. We investigate this family in the
present work.
We denote by 
the space of unit vectors, with
foot points in , directed inwards.
The space is a closed cylinder with natural coordinates
, where 
is the normalized
arc-length parameter on T, and where
measures the height in . The cylinder
is the phase space for the billiard map, 
,
of the billiard table T (Fig. 2.).
|
Fig. 2. The billiard map  and the phase space .
|
By construction, each billiard map 
has a canonical invariant circle,
. It is formed by the rays
supporting . Their orientation is induced by the positive orientation
of . The opposite choice of orientation yields an other
invariant circle 
.
The curve is the caustic
corresponding to the invariant circle 
,
for any 
[2].
|
Fig. 3. A family of billiard tables obtained by the string
construction. (  ). The invariant circles are
moving up in the phase space as increases.
|
Thus the family
, consists of
billiard tables with the same caustic .
|
Fig. 4. Some orbits for the parameter  .
|
|
Fig. 5. Some orbits for the parameter  .
|
|
Fig. 6. Some orbits for the parameter  .
|
By our choice of ,
the rotation number 
of satisfies
.
Next: A Family of Circle
Up: Billiards that share a
Previous: Billiards that share a
Oliver Knill, Jul 8, 1998