M087 · Geometry / collision maps

Chaotic Billiards Arena

Launch two nearby rays into a circle, ellipse, stadium or square with a central scatterer. Compare analytic collisions, integrable caustics, boundary sections and tangent growth; separate regular bouncing from sensitive trajectories without changing the elastic reflection law.

Reviewed collision $\text{Pending}$

Retained collisions $\text{Pending}$

Primary elapsed time $\text{Pending}$

Boundary arclength fraction $\text{Pending}$

Tangential unit-speed component $\text{Pending}$

Collision-index pair distance $\text{Pending}$

Finite-time growth per collision $\text{Pending}$

Valid tangent prefix $\text{Pending}$

Initial local area determinant $\text{Pending}$

Derivative refinement disagreement $\text{Pending}$

Maximum tangential reflection defect $\text{Pending}$

Maximum local speed-squared defect $\text{Pending}$

Circle or ellipse invariant $\text{Pending}$

Maximum invariant drift $\text{Pending}$

One-flight reversal defect $\text{Pending}$

Primary record status $\text{Pending}$

OBSERVE / COMPARE / VERIFY

Can geometry alone create chaos?

Table: circle / ellipse / stadium / Sinai

\mathcal B

2.00\,

Shape: ellipse stretch / straight half-length / disk radius

\eta

0.80\,

Launch boundary arclength fraction

s_0/\mathcal P

0.14\,

Initial tangential velocity component

p_0

0.37\,

Initial position perturbation exponent

\log_{10}\epsilon

-6.00\,

Requested collisions

N

600.00\,

Seeded atlas orbits

N_{\rm atlas}

10.00\,

Reviewed collision or kick

n/N

0.00\,\%

Drag the orange ring in the boundary section to edit launch position and tangential velocity; both sliders also support keyboard input. Playback advances collision number, not common physical time: the nearby trajectory has its own flight times. Grey atlas paths retain at most 500 collisions each; the apparatus shows the last 40 flights. Growth uses locally unwrapped boundary coordinates and a tangent vector renormalized every collision. A failed nearby or tangent record stops independently; the primary path may remain valid. Global drift readouts summarize the complete retained record.

Ideal point particle, unit speed, frictionless specular reflection and fixed analytic boundaries. The Sinai table is a square with one circular scatterer and permits long free corridors. Exact corners and near-grazing encounters have no reliable smooth continuation and stop the affected record. Finite-time tangent growth is a diagnostic, not a certificate of ergodicity.

Start with the essentials

Focus question: Can geometry alone create chaos?
One-sentence intuition: Conservation, local stability and a finite statistical picture answer different questions.

Core mathematical model

Same reflection law, different dynamics

\mathbf v_{+}=\mathbf v_{-}-2(\mathbf v_{-}\cdot\mathbf n)\mathbf n,\qquad |\mathbf v_{+}|=|\mathbf v_{-}|

Each free flight ends at the nearest admissible analytic boundary intersection. The normal may point either way; the reflection is unchanged. Corners and grazing encounters are singular.

Use the invariant collision measure

q=\frac{s}{\mathcal P},\qquad p=\mathbf v\cdot\mathbf t,\qquad T^{*}(dq\wedge dp)=dq\wedge dp

The boundary is oriented with the accessible interior on its left. The outer wall is counterclockwise, the obstacle clockwise. Concatenated arclength uses the total perimeter. The ellipse uses high-order arclength quadrature, not its polar angle as a canonical coordinate.

A circular caustic is an invariant

L=xv_y-yv_x,\qquad r_{\rm caustic}=|L|

The unit-radius circle has constant angular momentum. Every chord is tangent to the same concentric caustic; a triangular orbit is periodic, while an irrational boundary rotation fills a one-dimensional section. Neither dense chords nor a complicated image imply chaos.

Confocal ellipse and hyperbola

\lambda=b^2v_x^2+a^2v_y^2-(xv_y-yv_x)^2,\qquad \frac{x^2}{a^2-\lambda}+\frac{y^2}{b^2-\lambda}=1

The minor semiaxis is one. The invariant distinguishes elliptic from hyperbolic caustics; the focal separatrix is the degenerate limit. The caustic is an analytic prediction, while the fourth plot audits its invariant along computed collisions.

Finite pairs and tangent vectors differ

\lambda_N=\frac1N\sum_{j=0}^{N-1}\log\|DT(z_j)\widehat{\delta z}_j\|,\qquad \widehat{\delta z}_{j+1}=\frac{DT(z_j)\widehat{\delta z}_j}{\|DT(z_j)\widehat{\delta z}_j\|}

The initial tangent points along boundary position. Every step renormalizes it. A separate finite nearby ray is never reset and can saturate. Growth is per collision, not per unit physical time. The metric uses shortest boundary-coordinate distance and tangential velocity difference.

Audit a smooth local map

\det DT\approx1,\qquad D_hT(z)=\frac{T(z+h\mathbf e_i)-T(z-h\mathbf e_i)}{2h}

The displayed determinant uses central differences with locally unwrapped coordinates. Halved-difference disagreement is numerical evidence, not a rigorous bound. Derivatives stop across a collision branch change or a curvature seam. A finite positive growth rate does not prove global ergodicity.

Common difficulties

Complex is not chaotic

Typical misconceptionA dense circular trajectory must be chaotic.

Better mental modelIts constant caustic and one-dimensional section show regular motion.

Choose collision coordinates

Typical misconceptionA uniform angle parameter always preserves section area.

Better mental modelUse actual arclength and tangential unit-speed component; ellipse polar angle is not arclength.

Collision count is not time

Typical misconceptionThe two paths are compared at identical physical times.

Better mental modelThey are paired by collision index and have separate flight durations.

Singular directions need honesty

Typical misconceptionThe solver should invent a bounce at a corner.

Better mental modelThe reflection normal is ambiguous; stop and change the initial state. A tangent audit can stop before the primary trajectory does.

Run the experiment

01
Start with a triangle
Select the periodic triangle and review the record.
What to observe: The same three impact points repeat; the circular caustic stays fixed.
02
Change the confocal family
Compare elliptic and hyperbolic caustic presets.
What to observe: Different caustic families remain integrable; their invariant drift stays small.
03
Change only geometry
Compare the stadium sensitivity and parallel-wall bouncing.
What to observe: One table contains exceptional regular trajectories alongside sensitive typical motion.
04
Test the diagnostic limits
Try Sinai, vary the perturbation, extend the record, then select the corner.
What to observe: Finite pairs saturate, tangent growth can continue, and an undefined collision remains stopped.