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.
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.
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:
The familiar piecewise form follows immediately:
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) — more general patterns this builds on
-
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 → Metric → Function (Mapping)
Neighborhood in Abstraction Space¶
Karlsruhe Metric sits in a sparse region of the domain-specific corpus (81st percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (1565 abstractions)
Nearest neighbors
- Horocycle — 0.82
- Hadamard space — 0.82
- Reach (Mathematics) — 0.81
- Isotropic coordinates — 0.81
- Wiechel Projection — 0.81
Computed from structural-signature embeddings · 2026-09-08