OBSERVATION CHAMBER / ACTIVE

OBSERVATION ROUTE

CURRENT: 70 OPTIMAL TRANSPORT DENSITY GEODESIC OBSERVATORY

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Optimal Transport Density Geodesic Observatory

Observe a balanced one-dimensional density transport plan through monotone coupling, displacement geodesics, W2 cost, entropy, and barycenters.

structure / density geodesic Observation Model

ROUTE 70 / STRUCTURE / DENSITY GEODESIC

WASSERSTEIN / DISPLACEMENT GEODESIC

How mass moves between two densities

Edit two probability densities on [0, 1]. SOLVE builds their one-dimensional monotone coupling, then places each cumulative-mass strip at x_t = (1 − t)x + ty.

PAUSED

Bars are piecewise-constant density bins. Curves are monotone displacement strips, not point particles or pixels.

All calculations stay local in this browser.

t: 0.00
source mass: —
target mass: —
current mass: —
W₂: —
quadratic cost C₂: —
plan entropy H(P): —
barycenter mean: —

DENSITY CONTROLS

Shape the two measures

READY

CHAMBER STATUS

SYSTEM: ONLINE
MODEL: W2
COUPLING: MONOTONE

TRANSPORT

t = 0.00

STEP moves t by 0.05. PLAY follows the displacement geodesic. Reduced motion disables PLAY.

PRESET DENSITIES

SEPARATED

EDIT DENSITY BINS

12 / [0, 1]

SOURCE ρ₀

1 2 3 4 5 6 7 8 9 10 11 12

TARGET ρ₁

1 2 3 4 5 6 7 8 9 10 11 12

Each array stores bin mass. SOLVE normalizes positive endpoints before forming the cumulative intervals.

GEODESIC TIME tt = 0.00

KEYBOARD

Focus the canvas: Space/P PLAY · Enter/S STEP · R RESET · ←/→ change t · Home/End endpoints.

OBSERVATION READOUT

What the coupling conserves

FINITE

monotone strips: —
current mean: —
marginal max error: —
noncrossing violations: —
geodesic identity error: —
midpoint barycenter error: —
model: ρ_t = ((1 − t)x + ty)_#P · W₂² = C₂

The plan is exact for the finite 1D bin model. Entropy is reported for P; it is not a regularization term in the W2 solve.

DETERMINISTIC SELF-TEST

Numerical acceptance

PASS

FIXED FIXTURE / PASS

CHECK	VALUE	LIMIT	RESULT

Observation report

OBSERVATION CONTRACT

Density space, not image morphing

The input is a pair of non-negative, unit-mass piecewise-constant densities. Their cumulative intervals define a monotone coupling. The canvas draws those strips and their displacement interpolation. It does not blend pixels, move point particles, solve advection, or diffuse a field.

WHAT TO WATCH

Cost, mass, and the midpoint

Marginal error stays at floating-point round-off while every strip keeps its mass.
W2 is the square root of the quadratic transport cost C2.
At t = 0.50, the displacement midpoint is the equal-weight Wasserstein barycenter.

MODEL NOTE

Monotone 1D coupling

P_ij = |[F₀(i), F₀(i)+aᵢ) ∩ [F₁(j), F₁(j)+bⱼ)|

The finite plan has at most 2N − 1 non-zero strips. N = 12 here, so solve and render stay bounded.

Formula note

C₂(P) = Σᵢⱼ Pᵢⱼ(xᵢ − yⱼ)², W₂ = √C₂, ρₜ = ((1−t)x + ty)_#P, and H(P) = −Σ Pᵢⱼ log Pᵢⱼ. The barycenter at t = 1/2 satisfies 4 C₂(ρ₀, ρ₁/₂) = C₂(ρ₀, ρ₁) for this displacement plan.

OBSERVATION GUIDE

Touchpoints for observation

Start with SEPARATED. SOLVE exposes the monotone strips and W2. Move the t slider or press STEP to watch the current density and barycenter.

Choose SPLIT or SKEW, then compare cost and plan entropy.
Edit a bin with a keyboard or touch. NORMALIZE keeps both endpoint masses at one.
Use REPORT for a local text record and PNG for the current canvas. Reduced motion keeps STEP available.

This is a finite balanced density model. It is not unbalanced OT, a particle animation, an image morph, advection, diffusion, or a physical measurement.

Runs inside the browser with no upload and no registration.

OBSERVATION POLICY

This chamber runs a deterministic balanced one-dimensional density model in your browser. Densities, coupling, calculations, reports, and images stay local. It does not measure visitors or physical transport.

Route

This chamber belongs to Structures & Symmetries.

STRUCTURE Back to Structures & Symmetries Route.