OBSERVATION CHAMBER / ACTIVE
Dynamical Billiard Observatory
Change the boundary and follow a point through free flight and mirror-like collisions. Compare its path with the same collisions on a boundary map.
LAB 91 / DYNAMICS / BILLIARD · Signal observation route
01 / FOLLOW THE FLIGHT
The boundary redirects the same moving point
Canvas 2D is unavailable. Controls and numeric collision readouts remain usable.
Click near the wall to choose a start. Focus the drawing: ← → start, ↑ ↓ angle, Space play/pause, S one collision, R reset.
Green: recorded path and start · amber: current free flight · dashed: next hit
03 / READ THE COLLISIONS
One point for each wall collision
The horizontal coordinate follows the boundary counterclockwise. The vertical coordinate p = v · t is the outgoing tangential velocity, between −1 and +1. Each dot comes from a collision in the flight view.
s = 0: rightmost point for circle/ellipse; lower-right arc junction for stadium. Ellipse arc length uses 2048 midpoint intervals; collision positions are solved analytically.
Same orbit, numeric readout
- Collisions
- 0
- Model time
- 0
- Speed |v|
- 1
- max ||v| − 1|
- 0
- Max boundary residual
- 0
- First near return (collisions)
- —
- Last collision (s/L, p)
- —
Near return means both position and velocity are within 10⁻⁶ of their initial values. A finite near return is not a proof of periodicity. No return does not prove chaos.
OBSERVATION GUIDE
Compare the boundary, then the trace
- Choose circle and angle 0°. Step twice: the point retraces a diameter. At 45°, four flights form a square.
- Keep the start and angle, switch to ellipse, then change its ratio. Watch the straight segments and collision-map dots change together.
- Choose stadium. Its straight walls join two semicircles. Compare this boundary with the smooth ellipse; use a report or PNG to keep one observation.
MODEL / LIMITS
Free flight, specular reflection
q(t + τ) = q(t) + τv
v′ = v − 2(v · n)n, |v| = 1
n is the outward unit normal. The next positive ray intersection is found from line or quadratic equations. Distance equals flight time at unit speed. There is no friction, spin, gravity, finite ball radius or scoring.
Unlike LAB 67’s prescribed map or LAB 86’s pendulum section, these map points are produced by the visible boundary collisions. The model draws at most 2048 recorded hits.
Near-tangent or failed intersections stop the orbit. These presets have no sharp corners; the stadium joins have the same normal on both sides. Playback stops if 24 collisions are reached in one frame, without advancing the remaining time. A delayed frame advances at most 0.05 model time units, slowing playback.
04 / RECORD AND VERIFY
Keep this observation
Numerical check details
Generated report
Self-tests use separate deterministic fixtures: circle returns, 10,000 collisions per shape, speed, boundary, reverse flights, stadium segments and arcs, invalid input and tangent stops. They do not replace the current observation. Model changes and playback clear generated output.
N. Chernov — Entropy, Lyapunov exponents and mean free path for billiards (§1)
American Mathematical Society — Bunimovich Stadium
OBSERVATION POLICY
This ideal point-particle model runs in your browser. Inputs, calculations, reports and images stay local. No camera, microphone, real table or visitor is measured.
Route
This chamber belongs to Signals & Networks.
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