# Semi-Annular Spiral Billiard with Periodic Orbits

Semi-Annular Spiral Billiard with Periodic Orbits

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

x

π

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.