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: v2 · 2026-08-30 · History
Domain-specific #: 2123
Origin domain: metric geometry
Subdomain: radial–orbital path metrics
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.^[1]^[2] 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

\[ \delta(p_1,p_2)=\min\!\left(\lvert\varphi_1-\varphi_2\rvert,,2\pi-\lvert\varphi_1-\varphi_2\rvert\right), \qquad 0\le \delta\le\pi. \]

Two route forms compete. An inner-radius arc route travels radially until reaching the smaller radius, follows the origin-centered arc there, and completes any remaining radial difference. Its length is

\[ L_{\mathrm{arc}}=\lvert r_1-r_2\rvert+\min(r_1,r_2)\,\delta. \]

A through-origin route goes radially to the center and radially outward, with length

\[ L_{0}=r_1+r_2. \]

The Karlsruhe distance is the shorter:

\[ d_K(p_1,p_2)=\min\!\left(L_{\mathrm{arc}},L_0\right). \]

If both radii are positive, comparison gives \(L_{\mathrm{arc}}-L_0=\min(r_1,r_2)(\delta-2)\). The route therefore switches at exactly two radians: the arc route wins when \(\delta<2\), the origin route wins when \(\delta>2\), and both have equal length at \(\delta=2\). Hyman and Mayhew reproduce this formula and switching argument as the baseline for radial–orbital route-catchment models.^[3]

Structural Signature¶

The stable structure is:

planar points + distinguished origin + polar coordinates + radial and concentric-arc admissibility + path-length accumulation + minimization over admissible routes → an origin-dependent metric with an angle-controlled geodesic switch.

Sig role-phrases:

Planar carrier — the objects compared are points of the plane.
Distinguished origin — one fixed center defines all radii, rays, and concentric circles.
Polar separation — a pair is described by two radii and their smaller angular separation.
Radial admissibility — motion toward or away from the origin is allowed along any ray.
Orbital admissibility — angular motion is allowed on any circle centered at the origin.
Length functional — radial segments contribute ordinary radial length and circular arcs contribute radius times angle in radians.
Inner-radius candidate — any shortest route using an arc can place it at the lesser endpoint radius, avoiding unnecessary radial excursion.
Origin candidate — a second canonical route removes all angular cost by passing through the origin.
Minimum selector — distance is the lesser candidate length, producing the two-radian switching boundary.
Metric invariants — non-negativity, identity of indiscernibles, symmetry, and the triangle inequality hold because the rule is an intrinsic shortest-path distance.

The invariant is not merely “a polar formula.” It is shortest admissible path length under continuous radial and origin-centered orbital travel. Changing the center changes the metric. Allowing arbitrary straight chords, only a finite street graph, direction-dependent costs, or unequal radial and orbital speeds changes the object.

What It Is Not¶

Not Euclidean distance. Euclidean geometry allows the straight chord between any two points. Karlsruhe distance measures a constrained radial–orbital route and is generally at least the Euclidean chord length.
Not Manhattan distance in Cartesian coordinates. Manhattan distance restricts motion to two globally fixed orthogonal directions and is translation invariant. Karlsruhe directions rotate with position around a fixed center and are not translation invariant.
Not a discrete road-network distance. The idealization permits a radial ray and an orbital arc at every location. A real city with finitely many streets, one-way rules, congestion, or missing rings requires a network or travel-time model.
Not the city metric used for accelerated rectilinear transport networks. Computational-geometry “city metrics” can combine Manhattan off-network travel with a finite set of faster segments. That is a different construction despite the urban metaphor.
Not a geodesic as an individual path. The metric is the pairwise distance rule; a shortest radial–orbital route realizing that value is a geodesic under the rule.
Not the generalized KT or KT1 travel-time models. Those extensions introduce unequal speeds or a finite orbital and move the switching boundary. They use Karlsruhe distance as a baseline rather than synonyms for it.^[3]

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.^[2]^[1] Later work on higher-order abstract Voronoi diagrams continues to list point sites in the Karlsruhe metric as a concrete application of general algorithms.^[4]

The second habitat is transportation geometry and stylized urban modeling. In an ideal radial–orbital city, the metric separates trips whose shortest route uses an orbital from those whose shortest route passes through the center. Hyman and Mayhew use it as a deliberately simple baseline, then relax equal-speed and everywhere-available-orbital assumptions to study more realistic route catchments.^[3]

It also serves education and visualization because the formula produces visible changes in distance circles, bisectors, and Voronoi cells. The metric makes clear that a Voronoi diagram depends on the chosen notion of distance, not merely the sites. Such use remains literal when the exact radial–orbital rule is computed.

The scope does not include any system that happens to have a hub and ring. A hub-and-spoke airline network, circular biological tissue, or radial organizational chart is not an instance unless its pairwise distance is actually defined by the Karlsruhe admissible paths and length rule.

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:

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

When one point is the origin, its polar angle is arbitrary, but the formula remains well defined because \(\min(r_1,r_2)=0\); the distance is simply the other radius. When both points lie on one ray, \(\delta=0\) and distance is their radial difference. When they lie on opposite rays, \(\delta=\pi>2\) and the shortest route passes through the origin.

The threshold is in radians, not degrees and not a fitted parameter. An arc of radius (m) and angle 2 has length (2m), exactly equal to going from one endpoint radially inward from radius (m) to the origin and back outward to radius (m). At the switch, shortest paths are nonunique: every choice of arc radius between zero and the smaller endpoint radius has the same total length.

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

\[ r_1+r_2+\rho(\delta-2). \]

This is linear in \(\rho\). If \(\delta<2\), it is minimized at the largest allowed value, \(\rho=\min(r_1,r_2)\); if \(\delta>2\), it is minimized at \(\rho=0\); if \(\delta=2\), every intermediate radius ties. Thus an entire route family reduces to one comparison.

That compression makes downstream constructions tractable. Distances to many sites can be compared to form Voronoi cells; equality of two distances defines bisectors; higher-order cells identify the (k) nearest sites. The metric is sufficiently non-Euclidean to test abstract algorithms beyond norm-induced geometry while remaining explicit enough for exact calculation. In transportation modeling, the same switch yields route catchments without simulating every detailed street link.

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. The triangle inequality follows from shortest-path composition: an admissible shortest route from (p) to (q) followed by one from (q) to (s) is an admissible route from (p) to (s), so the shortest (p)-to-(s) route cannot be longer.
Route-regime prediction. Angular separation alone determines whether an orbital or origin route wins under equal radial and orbital unit costs; endpoint radii determine the length within the chosen regime but not the switch.

These properties license diagnostics. If a proposed “Karlsruhe distance” is translation invariant, it has probably lost the distinguished center. If its switch depends on radius under equal unit costs and continuous orbital availability, the formula has been altered. If direction or congestion makes travel asymmetric, the result may cease to be a metric and become a directed cost.

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.

Transfer to real urban systems is conditional. A sufficiently dense radial–orbital network can motivate the model, but finite rings, speed differences, congestion, forbidden turns, and one-way movement violate its assumptions. Hyman and Mayhew's KT and KT1 extensions demonstrate the honest response: change the distance model and derive a new switching rule, rather than calling every radial city exactly Karlsruhe.^[3]

Outside geometry, “go around near the center or go through the hub” is only analogy. The portable skeleton belongs to Metric and Path: define admissible routes, aggregate their lengths, and minimize. The Karlsruhe name should remain with the exact polar path rule.

Examples¶

Canonical — two competing routes¶

Let \(p_1=(r_1,\varphi_1)=(2,0)\) and (p_2=(5,1)), with angle in radians. The angular separation is \(\delta=1\). The inner-radius arc route has length

\[ \lvert5-2\rvert+\min(2,5)(1)=3+2=5. \]

The through-origin route has length (2+5=7). Therefore (d_K(p_1,p_2)=5), realized by moving along the radius from 5 to 2 and then around a radius-2 arc through one radian. If only the angular separation changes to (2.5), the arc candidate becomes (3+2(2.5)=8), while the origin route remains 7; the shortest path switches through the center. The examples differ only in angle, isolating the two-radian regime change.

Mapped back: two polar points → distinguished origin → radial/orbital candidates → additive lengths → minimum selector → angle-controlled switch.

Applied / in practice — Karlsruhe-metric Voronoi diagrams¶

Given planar sites (S), assign every query point (x) to the site minimizing (d_K(x,s)). The resulting Voronoi cells reflect the origin-dependent path geometry rather than Euclidean chords. Dehne and Klein use the Moscow metric as an example beyond convex distance functions in their generalized wavefront treatment, and Bohler, Klein, and Liu explicitly include point sites under the Karlsruhe metric in the scope of higher-order abstract Voronoi algorithms.^[1]^[4] The application is not decorative: changing the metric changes bisectors, nearest-site regions, degeneracies, and the algorithmic assumptions required to construct them.

Mapped back: point sites → Karlsruhe pairwise distance → equal-distance bisectors → nearest-site minimization → metric-specific Voronoi subdivision.

Structural Tensions¶

T1: Geometric simplicity versus urban realism. Continuous radial and orbital availability yields a clean formula and an exact switch, but real networks contain only selected streets, speeds, capacities, and controls. Diagnostic: Is the task to expose radial–orbital morphology or to predict operational travel on a particular network?

T2: Center privilege versus translation invariance. The distinguished origin makes radial structure expressible and is the source of the metric's identity. It also means an ordinary translation can change distance. Diagnostic: Does the modeled system contain a real privileged center, or has one been inserted merely for computational convenience?

T3: Distance versus travel time. The standard metric assumes equal unit cost per length in radial and orbital directions. Faster ring roads or slower central routes change which path is best. Diagnostic: Is the measured quantity geometric length or time under direction-specific speeds?

T4: Unique value versus nonunique shortest paths. The distance value is always determined, but at \(\delta=2\) an entire family of arc radii ties. Algorithms that require a single path need a tie rule beyond the metric. Diagnostic: Does the downstream method require only distance, or a uniquely realized geodesic?

T5: Named autonomy versus reduction to Metric. Every instance satisfies the generic metric axioms, yet the origin, admissible-path family, formula, switch, and Voronoi consequences form a recurrent specialized package. Diagnostic: Is the question about distance axioms generally, where prime:metric suffices, or about this exact radial–orbital geometry?

Structural–Framed Character¶

The Karlsruhe Metric is structural-leaning. It is a formal distance function whose value and invariants do not depend on approval, institutional authority, or interpretive convention once the origin and rule are fixed. It is not human-practice-bound: abstract points and paths suffice. Its formula is recognized and calculated rather than negotiated.

The entry remains domain-specific because its vocabulary and recognition tests are unusually pinned: polar radii, one distinguished origin, radial segments, concentric arcs, and a two-radian switch. The city metaphor supplies the name and motivation, but the mathematical identity survives without either city. Conversely, carrying only “hub versus ring” to another field imports an analogy, not the metric. Its portable skeleton is the live Metric prime—an axiom-respecting pairwise distance—and the related shortest-path operation. Its character: a formal, structural metric species whose distinctive geometry remains tied to polar path constraints.

Structural Core vs. Domain Accent¶

This section decides why Karlsruhe Metric is a domain-specific abstraction rather than a prime.

What is skeletal. A set of objects receives a nonnegative symmetric pairwise distance satisfying identity and triangle inequality. The distance can be understood as the minimum accumulated length among admissible paths. Those roles are substrate-portable and belong to Metric, with Path supplying a related route interpretation.

What is domain-bound. Karlsruhe geometry adds the plane, distinguished origin, polar coordinates, continuous radial and concentric-arc motion, ordinary path length, two canonical route types, and the \(\delta=2\) switch. Remove the center or allow arbitrary chords and it is no longer the same metric. Replace continuous rings with a finite transport network or unequal speeds and a different distance results.

Why this does not clear the prime bar. The named metric is one mathematically important species, not a substrate-neutral operation recurring with the same vocabulary across unrelated domains. Literal transfer is confined to geometries that deliberately install this exact rule. Broader lessons—make the distance choice explicit, constrain paths, minimize cost—already travel through Metric and Path. Cross-domain “Karlsruhe-like” descriptions discard the formula and become metaphor.

The Karlsruhe Metric strictly instantiates Metric (prime:metric). Its objects are planar points, its function is (d_K), it is nonnegative and symmetric, zero only for identical points, and triangle-respecting as an intrinsic shortest-path distance. This is the minimal proposed DAG parent.

It is related to Path (prime:path) because admissible routes realize the distance, but the live Path prime is graph-relational and a Karlsruhe route is a continuous geometric curve rather than necessarily an edge sequence. A second strict parent would therefore overstate the catalog match. The metric is also used with Voronoi diagrams, but “used to construct” is not subsumption.

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 (81^st percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

Euclidean metric. It measures straight-line chord length and has no privileged origin. Tell: May the route cut directly across a ring rather than follow radial and orbital directions?
Manhattan or taxicab metric. It permits motion along globally fixed Cartesian axes and is translation invariant. Karlsruhe's local directions rotate around its center. Tell: Are admissible directions fixed everywhere or defined relative to the origin?
City metric. In computational geometry this often means quickest travel using a finite accelerated transport network plus an ambient off-network metric. Tell: Is there a finite list of faster segments, or a continuous radial–orbital field?
Geodesic. A geodesic is a shortest path under a metric; Karlsruhe Metric is the whole pairwise distance rule and may admit multiple geodesics. Tell: Is the object one realizing route or the function comparing every pair?
KT and KT1 metrics. These transport extensions alter speeds or restrict orbital availability, producing different switching angles and catchments. Tell: Are radial and orbital unit costs equal and are arcs available at every radius?
Karlsruhe Accurate Arithmetic. That is an unrelated computer-arithmetic project sharing only the place name. Tell: Does the context concern reliable numerical computation or polar shortest paths?

References¶

[1] Frank K. H. A. Dehne and Rolf Klein, “The Big Sweep: On the Power of the Wavefront Approach to Voronoi Diagrams,” Algorithmica 17, no. 1 (1997): 19–32. registry ↩a ↩b ↩c

[2] Rolf Klein, Concrete and Abstract Voronoi Diagrams, Lecture Notes in Computer Science 400 (Springer, 1989). https://doi.org/10.1007/3-540-52055-4. registry ↩a ↩b

[3] Geoffrey Hyman and Les Mayhew, “The Properties of Route Catchments in Orbital–Radial Cities,” Environment and Planning B: Planning and Design 27, no. 6 (2000): 843–863. https://doi.org/10.1068/b26102. registry ↩a ↩b ↩c ↩d

[4] Cecilia Bohler, Rolf Klein, and Chih-Hung Liu, “An Efficient Randomized Algorithm for Higher-Order Abstract Voronoi Diagrams,” Algorithmica 81, no. 6 (2019): 2317–2345. https://doi.org/10.1007/s00453-018-00536-7. registry ↩a ↩b

[5] Rolf Klein, “Abstract Voronoi Diagrams and Their Applications,” in Computational Geometry and Its Applications, Lecture Notes in Computer Science 333 (Springer, 1988), 148–157. https://doi.org/10.1007/3-540-50335-8_31. registry

[6] Franz Aurenhammer and Rolf Klein, “Voronoi Diagrams,” in Handbook of Computational Geometry, ed. J.-R. Sack and J. Urrutia (North-Holland, 2000), 201–290. https://doi.org/10.1016/B978-044482537-7/50006-1. registry