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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 traced by a pursuer that continually directs its motion toward a moving target's current position. The geometric test is local: before coincidence, the target lies on the forward tangent ray of the pursuer's path at the corresponding time. A speed rule, target trajectory and initial separation determine a particular curve; the tangent condition alone identifies the family. The curve is the resulting locus, not the target's path and not the controller that realizes the heading.[1][2]

Write pursuer position \(P(t)\) and target position \(T(t)\), with \(P(t)\ne T(t)\) and \(\dot P(t)\ne0\). The directed condition is \(T(t)-P(t)=\lambda(t)\dot P(t)\) for some \(\lambda(t)>0\). If the pursuer has specified positive speed \(v_P(t)\), its idealized motion can be written \(\dot P(t)=v_P(t)\,[T(t)-P(t)]/\|T(t)-P(t)\|\). This normalized form is a direct restatement of the original tangent and speed conditions, not a promise that every choice of \(T\), \(v_P\) and initial data yields capture. At coincidence its denominator is zero, so a separate endpoint convention is needed.[1]

The historical straight-line ship problem and a four-mouse cyclic model realize the same local geometry with very different global motion: in the first, target motion is prescribed; in the second, each target is also another pursuer. That persistence across settings is the reason to keep Pursuit Curve as an identity narrower than Curve rather than as a one-off puzzle.[1][2]

Structural Signature

Sig role-phrases: moving target \(T\) → noncoincident pursuer \(P\) → present-position forward-tangent alignment → speed/initial-data specification → pursued-path locus; capture is a separate outcome.

  • Target trajectory. At time \(t\), \(T(t)\) names the location actually aimed at in the idealized instantaneous model. It may be prescribed independently or evolve through a coupled pursuit system.[1]
  • Pursuer trajectory. \(P(t)\) must have a nonzero forward tangent on the interval being classified. Its image is the pursuit curve; target and pursuer share a time parameter so “current” has a definite meaning.[1]
  • Directed alignment. Positive collinearity of \(\dot P(t)\) and \(T(t)-P(t)\) is stronger than merely being on the same unoriented line. A negative multiple would point away from the target. In planar coordinates Malacka's cross-product condition gives collinearity, while orientation fixes the forward sense.[1]
  • Motion specification. A speed law, initial position and target law turn the geometric condition into a particular ODE or coupled system. A constant speed ratio is convenient in some classical analyses, not an invariant of the whole family: Bernhart's original general treatment explicitly permits arbitrary target tracks and varying speed ratios.[1][3]
  • Resulting locus. The curve is traced by the pursuer before capture, termination or loss of the regularity assumptions. Its shape can change with relative speeds and target motion while retaining the alignment test.[1][2]
  • Optional capture condition. Exact \(P(t_c)=T(t_c)\) or a declared nonzero capture radius is a model-specific endpoint. Existence of a pursuit path does not establish either event.[1][2]

What It Is Not

It is not any path in a chase. A pursuer may aim ahead of its quarry, maintain a bearing, or follow a delayed observation; those trajectories need not have the current target on the forward tangent. Azevedo and Pelluso study a finite-information-speed modification precisely because targeting a retarded position changes the classical assumption.[2]

It is not a generic pure-pursuit controller. The control rule produces a trajectory under vehicle dynamics; turn-rate limits or sensing delays can make the realized path depart from the ideal geometric condition. Robot path tracking toward a selected look-ahead point on a fixed reference path is also not automatically pursuit of a contemporaneously moving target.

It is not a capture theorem. Malacka defines capture separately and analyzes a straight-line case under declared initial data and \(k>1\). The fact that one configured case has a capture point does not make speed ratio by itself sufficient for arbitrary target motion. Nor is a cyclic four-mouse configuration required for a single pursuit curve.[1]

Scope of Application

The identity belongs primarily to geometry and dynamical systems. It can be used in idealized ship, aircraft or other moving-target models when the same-time positions and velocity direction are available. The analysis may be two-dimensional or extended to a different ambient space, but it must retain a meaningful tangent and directed line of sight. With an exogenous target path, it is an ODE for the pursuer; with cyclic pursuit, it is a coupled collection of motion equations.[1][2]

Its practical boundary is equally important. Sensors can report delayed or noisy positions; real vehicles cannot change heading instantaneously; a target may maneuver unpredictably. These departures may be modeled as approximations or alternative pursuit laws, but they cannot silently be called exact pursuit curves. The idealization is useful for comparing geometry, not an operational instruction or a performance guarantee.

Clarity

The phrase “always heads toward the target” hides three questions. Which target position? The current one in the defining rule, not an extrapolated future or received past position. Which direction? The forward tangent ray, not an unoriented tangent line. Which object? The pursuer's locus, rather than the guidance algorithm or target trajectory. Answering all three prevents the classical line-of-sight relation from being conflated with other pursuit methods.[1][2]

The seed's expression \(L(t)=F(t)+xF'(t)\) is not a safe general definition: an unspecified scalar and possible zero tangent leave time, direction and applicability unclear. Positive collinearity \(T-P=\lambda\dot P\) at noncoincident regular instants states the geometric invariant without asserting that a single constant offset works at every time.

Manages Complexity

The local alignment rule compresses many distinct pursuit stories into one test. A linear target track, a curved target track and a cyclic chain can each be checked by looking at same-time displacement and tangent direction. That simplification makes it possible to derive coordinates or numerical trajectories from a compact ODE instead of arguing from the visual impression of “chasing.”[1]

The compression must not erase global conditions. Speed, initial separation, target law and capture convention affect the resulting shape and outcome. A model that records only “pursuit curve” but omits those data cannot support a claim about interception time, curvature or even whether exact coincidence occurs.[1][2]

Abstract Reasoning

First choose a common time parameter and specify \(T(t)\), \(P(t_0)\) and a positive pursuer speed law. Next form the instantaneous displacement \(T(t)-P(t)\) wherever it is nonzero, normalize it if a speed is prescribed, and solve or analyze the resulting equation. At each regular point, test both collinearity and forward orientation; mere nearness of the pursuer to the target track is insufficient.[1]

Then separate path classification from endpoint analysis. The tangent relation establishes the former. The latter needs the actual dynamics, initial data and a definition of capture. In a coupled cyclic system, the “target law” for one pursuer cannot be inserted as an independent input without simultaneously solving the next pursuer's motion. This distinction explains why a single simple geometric signature can lead to different mathematical problems.[1]

Knowledge Transfer

The tangent test transfers from the classical two-vessel construction to a many-agent cyclic construction because in both, each relevant velocity is directed toward a contemporaneous moving point. The target's trajectory need not be the same kind of curve, and the mathematical method of solution need not carry over. What transfers is the local directed relation, not the particular solution or capture claim.[1][2]

Transfer to engineering is conditional. An ideal vehicle path may approximate the pursuit-curve model if its realized velocity continuously aligns with current target position. Sensing latency, turn constraints or aiming at a look-ahead waypoint break that literal mapping and require a different model. The analogy “respond to the latest state” can inspire a more portable abstraction, but without geometrical velocity and line of sight it is no longer a pursuit curve.

Examples

Straight-line vessel pursuit

In the classical pirate-and-merchant configuration, the merchant travels along a straight line while the pirate starts to one side and keeps pointing its instantaneous velocity at the merchant's present location. The pirate's route bends as that line of sight changes. Malacka's worked case assumes a vertical target track, offset initial positions and a particular speed-ratio regime; Azevedo and Pelluso explicitly review the same contemporaneous-target baseline before studying delayed information.[1][2]

Mapped back: moving target \(T\) → merchant on a vertical line; pursuer \(P\) → pirate from an offset start; present-position alignment → each pirate tangent ray passes through the current merchant position; motion specification → straight-line target law, speed rule and initial offset; locus → pirate's bent path, not merchant's line; capture → separately analyzed only under the stated assumptions.

Four-mouse cyclic pursuit

Malacka's second worked case starts four mice at the corners of a square. Each runs toward the next mouse, which is moving because it in turn follows its neighbor. Each mouse therefore traces a pursuit curve, but the four curves must be treated as a coupled system. The symmetry and solution method differ from the one-target straight-line case; the instantaneous directed-tangent requirement does not.[1]

Mapped back: moving target \(T\) → the next mouse for each pursuer; pursuer \(P\) → that mouse's own position; present-position alignment → each velocity aims at its neighbor's concurrent location; motion specification → square initial configuration, cyclic assignment and speed law; locus → each mouse's interdependent path; capture → any meeting outcome requires the configured model, not the word “pursuit.”

Structural Tensions

  • Present-state simplicity versus anticipatory interception. Immediate line-of-sight alignment needs no future target forecast and gives a crisp geometric path test. A lead strategy could serve a particular interception objective better, but it changes the path's defining direction. The cost of the simple identity is that it does not itself optimize capture. Diagnostic: Is the velocity pointed at current \(T(t)\), or a predicted intercept point, and what performance objective has actually been proved?[1][2]
  • Single-curve tractability versus coupled-motion fidelity. Prescribing the merchant's track permits one pursuer equation. In the mice case every “target” is dynamically another pursuer, so retaining the real geometry requires a coupled system and joint initial conditions. Treating those targets as fixed inputs simplifies calculation but misstates the case. Diagnostic: Does \(T(t)\) remain independent of the pursuer system, or must all trajectories be solved together?[1]

Structural–Framed Character

  • Evaluative weight: The identity is descriptive geometry, not a judgment that this path is efficient, optimal or desirable.
  • Human-practice dependence: Ships and guidance applications motivate examples, but no institution or human purpose is needed for the tangent relation itself.
  • Institutional origin: The historical ship problem supplied a named research lineage; its vessel conventions do not define every pursuit curve.[2]
  • Vocabulary travel: “Pursuit” travels to games and control, yet only uses retaining current-target directed-tangent geometry instantiate this entry.
  • Import versus recognition: A path is recognized by checking its same-time tangent relation, not by importing a controller label or historical story.

Its character: Near the structural end of the domain-specific spectrum: the identity is a formal, repeatable geometric relation, but differentiable trajectories, temporal target location and forward tangency are constitutive mathematical framing. It is therefore not a Prime-level generic “following” pattern.

Structural Core vs. Domain Accent

The core is a relation among a moving reference point, a responding trajectory and a time-indexed directed tangent. Under a speed law it becomes an ODE or coupled system. The mathematical domain accent is not decoration: remove time correspondence, line-of-sight geometry or nonzero forward tangent and the recognition test fails.[1]

A portable “respond to the current state rather than predict a future state” skeleton may be a future-prime question, but this draft does not claim that Prime identity or infer one from vocabulary. Live Curve is the defensible full-signature upward genus. Live Integral Curve is related through differential-equation solution trajectories, yet its specified vector-field framing is not automatically a strict parent of a nonautonomous or jointly coupled pursuit system.

This entry is a proposed strict kind of Curve; the extra current-target tangent constraint gives it an independent test. Continuity is a broad prerequisite inherited through curve formation, not a replacement for that geometric identity. Integral curve is a useful ODE neighbor, not an asserted upward edge. No canonical DAG relationship is changed here.

Relationships to Other Abstractions

Local relationship map for Pursuit CurveParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Pursuit CurveDOMAINDomain-specific abstraction: Curve — is a kind ofCurveDOMAIN

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

Neighborhood in Abstraction Space

Pursuit Curve sits in a sparse region of the domain-specific corpus (61st 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

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

Not to Be Confused With

  • Target path: \(T(t)\) is the input or coupled other trajectory; \(P(t)\) traces the pursuit curve.
  • Pure-pursuit guidance implementation: A controller may seek this ideal relation while actual vehicle dynamics prevent exact realization.
  • Look-ahead path tracking: A robot following a reference path's chosen point is not necessarily aiming at a moving agent's current position.
  • Lead or delayed-information pursuit: Future or retarded target positions alter the tangent condition.[2]
  • Cyclic pursuit: A multi-agent instance, not a requirement for a single pursuit curve.[1]
  • Capture: A possible endpoint established only under additional assumptions, not the curve's defining condition.[1]

References

[1] Zuzana Malacka, “Pursuit Curves and Ordinary Differential Equations,” Communications 14(1), 66–68 (2012), original publisher DOI 10.26552/com.C.2012.1.66-68; full original article PDF. Definition and equations on p.66 §2; straight-line example pp.66–67 §3; four-mouse example pp.67–68 §4. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v ↩w ↩x

[2] Thales Azevedo and Anderson Pelluso, “Space pirates: a pursuit curve problem involving retarded time,” American Journal of Physics 90 (2022), author-original arXiv version, abstract and §§I–II. The paper distinguishes the contemporaneous-target Bouguer baseline from its retarded-time extension; the latter's capture result is not generalized here. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m

[3] Arthur Bernhart, “Curves of General Pursuit,” Scripta Mathematica 24 (1959), original article scan, §D1 pp.189–190, explicitly allowing arbitrary target track and variable speed ratio under same-direction separation and pursuer velocity. registry ↩