Three sites · all probability weights · exact transport

A triangle of distributions

Move the white vertices to change the geometry. Move the dark target β to change its weights. Each color is the cost of transporting a distribution α to β.

Logarithmic slider; enter other positive values within floating-point range. Colors show the raw cost Tₚ, without a p-th root. Changing p resets the color scale.

Try

Drag β or click inside to move the target. All three vertices remain draggable.

A colored triangle of probability distributions. Use the labelled handles to change the three support points and the target.

Target β b = (⅓, ⅓, ⅓)

x₁ x₂ x₃

Inspected α Tₚ = —

Hover to inspect; on touch, choose Inspect α and tap.

Ground costs |xᵢ − xⱼ|ᵖ

Reading the figure & keyboard controls

The filled triangle represents probability weights, not a spatial density. The contour lines are unlabeled and spaced by . They are computed from the exact three-by-three coupling problem, also when p > 1 and the ground costs fail the triangle inequality. For p ≥ 1, the plotted Tₚ equals Wₚᵖ; for 0 < p < 1, Tₚ is itself a metric. Lock the scale to compare costs while moving sites or weights; values above a locked maximum use the darkest color.

The physical axes have equal scale, and moving a vertex keeps b fixed. A small area and separation guard prevents collapse. Tab to any handle and use arrows (Shift for larger steps); Home restores that handle when feasible. In Inspect α mode, focus the plot and use arrows to move the inspected distribution; Home selects β. Tapping a handle without dragging also inspects it.