Karlsruhe Metric

A planar path metric with a distinguished origin in which travel is restricted to radial segments and origin-centered circular arcs, so shortest routes switch at two radians between an inner-radius arc route and a route through the origin.

Version

v3 · 2026-09-19 · History

Domain-specific #

2123

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Metric Geometry, Radial Orbital Path Metrics → Mathematics

Aliases

Moscow metric

Core Idea

The Karlsruhe Metric is a distance on the Euclidean plane after one point has been designated as the origin and allowable travel has been restricted to two local directions: along rays through that origin and along circular arcs centered on it. It is also called the Moscow metric in computational-geometry literature. The city names evoke radial streets and orbital avenues, but the mathematical object is an idealized continuous path geometry, not the measured road network of either city.

Let planar points (p_i) have polar coordinates \((r_i,\varphi_i)\), and define their smaller angular separation by.

How would you explain it like I'm…

Circle-or-Middle Walking

Imagine a pretend town with a center point. You may only walk on straight roads that go out from the center, or on round roads that circle around it. To get somewhere, you either go partway around a circle, or walk all the way to the middle and back out. The Karlsruhe Metric is the length of whichever way is shorter.

Ring-and-Spoke Distance

The Karlsruhe Metric measures distance on a flat map with one special point, the center. You may only move along straight lines through the center or along circles around it. One way to reach a point is to walk in or out to the smaller circle, go around it, then walk the rest of the way in or out. Another way is to walk to the center and straight back out. The distance is whichever of these two trips is shorter, so if two places are far apart around the circle, going through the center wins.

Radial–Arc Shortest Distance

The Karlsruhe Metric (also called the Moscow metric) is a distance on the plane where travel is limited to rays through an origin and circular arcs centered on it. Using polar coordinates (radius r, angle φ), let δ be the smaller angle between two points. The arc route costs |r1 − r2| + min(r1, r2)·δ: travel radially to the smaller radius, go around the arc, and finish radially. The through-origin route costs r1 + r2. The distance is the smaller of the two. Comparing them shows a clean switch: when δ is less than 2 radians the arc wins, when it is more than 2 radians going through the center wins. Despite the city names, it is an idealized geometry, not a real road map.

 

The Karlsruhe Metric is a distance on the Euclidean plane relative to a designated origin, where admissible paths consist only of radial segments and origin-centered circular arcs. Using polar coordinates (r_i, φ_i), define the angular separation δ = min(|φ1 − φ2|, 2π − |φ1 − φ2|), so 0 ≤ δ ≤ π. The inner-radius arc route travels radially to the smaller radius, follows the arc there, and completes the remaining radial difference, giving L_arc = |r1 − r2| + min(r1, r2)δ. The through-origin route has length L_0 = r1 + r2. The metric is d_K = min(L_arc, L_0). For positive radii, L_arc − L_0 = min(r1, r2)(δ − 2), so the optimal route switches at exactly δ = 2 radians, with a tie there. Despite the city names (it is also called the Moscow metric), it is an idealized continuous geometry, not a model of any actual road network. It serves as a baseline for radial–orbital route-catchment models.

Scope of Application

The primary mathematical habitat is metric and computational geometry. The metric is a concrete example of a non-Euclidean planar distance whose bisectors and Voronoi regions differ from Euclidean ones. Klein's work on concrete and abstract Voronoi diagrams and Dehne and Klein's wavefront framework place the Moscow/Karlsruhe metric among “nice metrics” for which generalized Voronoi machinery can be studied. Later work on higher-order abstract Voronoi diagrams continues to list point sites in the Karlsruhe metric as a concrete application of general algorithms.

Clarity

The outer-minimum formula is the safest definition because it handles the origin and the switch without hidden cases:

\[ d_K=\min\left(r_1+r_2,\ \lvert r_1-r_2\rvert+\min(r_1,r_2)\delta\right). \]

The familiar piecewise form follows immediately:

\[ d_K(p_1,p_2)= \begin{cases} \lvert r_1-r_2\rvert+\min(r_1,r_2)\delta, & 0\le\delta\le2,\\ r_1+r_2, & 2<\delta\le\pi. \end{cases} \]

Manages Complexity

An unrestricted search over paths could look infinite-dimensional. Any radial–orbital path may change direction repeatedly or use several arcs. The Karlsruhe structure collapses that search to two canonical candidates. If a route uses an arc at radius \(\rho\), its cost contains radial access plus \(\rho\delta\). Moving an arc outside the smaller endpoint radius only adds radial travel. For \(0\le\rho\le\min(r_1,r_2)\), the cost is

Abstract Reasoning

Several properties follow directly from the structure:

• Rotation invariance about the origin. Rotating both points by the same angle preserves radii and angular separation.
• Failure of translation invariance. Translating both points usually changes their radii and angle relative to the fixed origin, so their Karlsruhe distance can change.
• Positive homogeneity. Scaling both points from the origin by \(\lambda\ge0\) multiplies distance by \(\lambda\).
• Metric axioms. Non-negativity, symmetry, and identity follow from the formula.

Knowledge Transfer

The metric transfers literally wherever the same mathematical preconditions are imposed: a plane, a fixed center, continuous radial and concentric-arc travel, additive path length, and shortest-route minimization. It can be used as a mathematical model of an idealized city, as the distance underlying a Voronoi diagram, or as a benchmark for abstract geometric algorithms without changing meaning.

Relationships to Other Abstractions

Current abstraction Karlsruhe Metric Domain-specific

Parents (1) — outgoing typed relations

• Karlsruhe Metric is a kind of Metric Prime

  The Karlsruhe Metric strictly instantiates Metric (prime:metric).

Hierarchy path (1) — routes to 1 parentless root

• Karlsruhe Metric → is a kind of Metric → is a kind of Function (Mapping)

Neighborhood in Abstraction Space

Karlsruhe Metric sits in a sparse region of the domain-specific corpus (71^st percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2769 abstractions)

Nearest neighbors

• Ellipse — 0.85
• Vincenty's formulae — 0.84
• Gnomonic Projection — 0.84
• Euclidean Rotation — 0.83
• Skew coordinates — 0.82

Computed from structural-signature embeddings · 2026-10-25