Semi-Annular Spiral Billiard with Periodic Orbits

start position

start angle

time steps

set number of loops in table

1

2

3

4

5

connect table ends

yes

no

set

th

i

table-width on x axis

th

i

square root

log

th

i

prime

colorize path by bounce angle

yes

no

This Demonstration shows spiral-shaped billiard tables patched together using annular pieces. Starting at zero radians on the positive

x

axis, and repeating every

π

radians counter-clockwise around the spiral, periodic orbits occur where tangents to a table's inner and outer curved boundaries are parallel for the same spiral angle. To construct a table, users can choose the number of loops around the spiral and the rate at which the width of the spiral increases.

The more interesting cases (when "connect table ends" is "yes") form the spiral into a doubly connected region with the table's central and peripheral straight ends "seamed" together. In the doubly connected cases, all non-periodic trajectories are chaotic and time-irreversible. Since limited precision rounding errors produce information loss, when trajectories cross from a longer to shorter straight table edge (at the "seam"), backtracking calculations of earlier bounce positions contain minute errors. A table's two straight table edges form the table seam. All non-periodic trajectories cross the table seam.