Ground Metrics

SDOT transports with the cost $c(x,y)=|x-y{|}^2$. Three independent choices change it, each in its own place:

	what changes	where it is stated
The space	${\mathbb{R}}^d$, or its quotient by a group $G$ of isometries: a torus, a Klein bottle, a cone, a crystal... The cost becomes $\min_{g \in G}	x - g( y )
The cost of a distance	$\phi (|x-y|)$, for all the diracs or per dirac ( radial functions around the diracs )	roadmap
The mass balance	equal masses, or not: what is not transported goes to a global sink ( partial transport, cells capped by balls of radius $\sqrt{w_i}$ )	target_mass of the distributions: see Partial Transport

They compose: each one is stated on its own, and they apply together. A partial transport on a torus, where 30 diracs share half of the unit square:

f = sdot.SumOfDiracs( positions, target_mass = 0.5 )        # the mass balance: half of the square is transported
g = sdot.Box()

p = sdot.plan( f, g, periodicity = True )                   # the space: a torus
p.add_to_viz( viz )

Each dirac takes the part of its cell that lies in a ball around it, in full color; the rest of the cell, faded, goes to the sink. The space is a torus, so the cells are drawn whole around their diracs, and again, fainter, moved by the translations of the torus: a ball that crosses a side of the square ( dashed ) comes back on the other one.

Open in a separate page ↗

Page
Periodicity	quotients of ${\mathbb{R}}^d$ by a group of isometries: how to give the group, where the cells are, how to look at them
Partial Transport	a source lighter than the target: how to state it, what each dirac takes, how to use the result

Reproducing the pages

The figures of these pages are written by the scripts next to them, in docs/guide/ground-metrics/:

python docs/guide/ground-metrics/index_figures.py
python docs/guide/ground-metrics/periodicity_figures.py                 # all of them
python docs/guide/ground-metrics/partial_figures.py power masses        # or some of them