	mean	median
effective p	—	—
circuity	—	—

Same A to B. Very different path.

Houston · Greenspointreal OSRM route

C = d_route / d_straight
= 4,487 m / 303 m ≈ 14.8

Barcelona · Eixamplereal OSRM route

C = d_route / d_straight
= 1,046 m / 806 m ≈ 1.30

Houston's C ≈ 14.8 isn't typical sprawl — a single uncrossable barrier forces the route the long way around. Eixample's C ≈ 1.30 reflects Cerdà's 1859 plan: a 45°-rotated grid with chamfered octagonal corners, cut by the Diagonal and Meridiana avenues.

02

One family of distances, one parameter p.

p = 1 · Taxicab (L¹)stair-step

d₁ = |Δx| + |Δy|
= 3 + 4 = 7

p = 2 · Euclidean (L²)straight line

d₂ = √( Δx² + Δy² )
= √( 9 + 16 ) = √25 = 5

p is a continuous knob, not just the three named cases. Above 2 the unit ball squares off — diagonal shortcuts beat any right-angle grid. Real street networks settle at some effective p in this family; the next slide is how we read it off the routing data.

03

Probe the network, then solve for effective p.

1

Sample. Place N = 48 destinations uniformly on a Euclidean ring of radius R around origin x, then route to each on the OpenStreetMap graph via OSRM. Reject rays whose snapped endpoint lies more than 0.25·R from the requested point — the origin is on a disconnected component or in a road-sparse patch.

2

Summarize per cell. Across the N rays that survived the step-1 snap filter, take two summaries of d_route / d_euclid: the mean — every direction weighted equally — and the median — robust to outlier rays. Call them C̄(x; R) and C̃(x; R). Use the snapped endpoint coordinates for d_euclid — the requested ring points would bias the ratio.

3

Invert to effective p. For a pure L^p norm at scale R, each summary becomes a closed-form function of p alone — C̄(p) for the mean and C̃(p) for the median (next slide). Both are strictly monotone, so each cell's measured value inverts uniquely to an effective p:

p eff, mean (x) = C̄ -1 ( C̄(x; R) )

p eff, med (x) = C̃ -1 ( C̃(x; R) )

The toolbar picks which one colors the map.

04

From mean or median circuity to effective p.

mean — C̄(p)

median — C̃(p)

For each cell, effective p is the p at which the closed-form curve hits the cell's measured circuity — one closed-form lookup, no fitting. Mean weights every ray equally, so a single outlier drags p_eff down — best for spotting cells where one barrier (freeway, river edge) skews the average. Median is robust to outliers and tracks the typical-direction routing experience — best for reading the network's dominant character, not its isolated quirks.

05 · Off you go

Ready to explore?

Pick a city to start, or jump straight in. The City, Mode, and Radius chips up top let you compare side by side.

City click one to jump straight in

Mode which network do you walk / drive on?

Walking foot graph — alleys, footpaths, park cut-throughs

Driving car graph — major roads only, no pedestrian shortcuts

Radius how far each probe reaches from each cell

200 – 400 m a short walk — picks up block-scale geometry, sensitive to local cul-de-sacs

800 – 1,600 m a 10–20 min walk — picks up neighborhood-scale grid structure

1 – 3 km (driving) across-town routing — captures arterial layout and freeway topology

Per-cell Lp fit

Same A to B. Very different path.

One family of distances, one parameter p.

Probe the network, then solve for effective p.

From mean or median circuity to effective p.

Ready to explore?

Per-cell L^p fit