Pursuit Curve

Trace the path of a pursuer whose instantaneous forward tangent points toward a moving target's current position.

Version

v1 · 2026-10-03 · History

Domain-specific #

13535

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Differential Geometry, Ordinary Differential Equations → Mathematics

Core Idea

A pursuit curve is the path of a pursuer whose instantaneous velocity points toward a moving target's current position. Before coincidence, the target lies on the forward tangent ray at the corresponding point of the pursuer's path. With pursuer position \(P(t)\), target position \(T(t)\) and a specified positive speed \(v_P(t)\), the idealized relation is \(\dot P(t)=v_P(t)[T(t)-P(t)]/\|T(t)-P(t)\|\) wherever \(P(t)\ne T(t)\). The resulting locus is the curve; the heading rule and the target's own trajectory are different objects.[^ref-3b92a9ad325a]

Scope of Application

The same local relation appears in classical straight-line ship pursuit and four-mouse cyclic pursuit. In the former, the target track is prescribed; in the latter, each target is another moving pursuer, so the paths are coupled. Target law, speed and initial separation shape the particular path. Bernhart's general treatment permits variable speed ratios and arbitrary target tracks. Exact capture or a finite-radius endpoint must be separately specified and established; a speed ratio alone is not a universal capture test.[^{ref-3b92a9ad325a][}ref-2c7ee632c50d][^ref-db44897396bc]

Clarity

“Chasing” is not enough: lead aiming, a delayed observation or a path-tracking look-ahead point need not lie on the current-target tangent ray. Positive alignment matters, since a tangent line without orientation could point away. The formula only applies before \(P=T\), where its denominator is nonzero. A controller that seeks the relation is not identical to the geometric path it produces.[^{ref-3b92a9ad325a][}ref-2c7ee632c50d]

Manages Complexity

The directed-tangent test unifies otherwise different target motions while exposing which further data a concrete calculation requires. It lets one derive a single pursuer equation for a given target path or a coupled system for cyclic pursuit. The short name becomes misleading if target trajectory, speed law, initial data and capture convention are suppressed.[^ref-3b92a9ad325a]

Abstract Reasoning

Specify the same-time paths and initial configuration; check that each nonzero pursuer velocity is a positive multiple of target-minus-pursuer displacement. Solve or analyze the resulting motion law only under those assumptions. Then assess capture as an additional outcome, not part of path recognition. Malacka's straight-line and four-mouse examples illustrate how the local invariant survives while the global equations differ.[^ref-3b92a9ad325a]

Knowledge Transfer

What transfers between vessel and cyclic-mouse models is the instantaneous geometric relation, not a specific capture point or solution method. A delayed-information pursuit model changes the aim point to a past observed target position and therefore need not instantiate the same exact curve.

[^ref-3b92a9ad325a]: Zuzana Malacka, “Pursuit Curves and Ordinary Differential Equations,” Communications 14(1), 66–68 (2012), original article PDF, §§2–4. [^ref-2c7ee632c50d]: Thales Azevedo and Anderson Pelluso, “Space pirates: a pursuit curve problem involving retarded time,” American Journal of Physics 90 (2022), original arXiv version, abstract and §§I–II. [^ref-db44897396bc]: Arthur Bernhart, “Curves of General Pursuit,” Scripta Mathematica 24 (1959), original article scan, §D1.

Relationships to Other Abstractions

Current abstraction Pursuit Curve Domain-specific

Parents (1) — more general patterns this builds on

• Pursuit Curve is a kind of Curve Domain-specific

  A pursuit curve is a curve with an additional contemporaneous target-alignment constraint.

Hierarchy paths (2) — routes to 2 parentless roots

• Pursuit Curve → Curve → Continuity → Neighborhood → Topology

Show alternative path (1)

Neighborhood in Abstraction Space

Pursuit Curve sits in a sparse region of the domain-specific corpus (61^st percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Organizational & Operational Failure Modes (38 abstractions)

Nearest neighbors

• Target Fixation — 0.86
• Steering Law — 0.86
• Subjacency — 0.85
• Stellar encounter — 0.84
• Motion Lines — 0.84

Computed from structural-signature embeddings · 2026-10-08