A pursuit curve is a planar curve that describes the trajectory of a point (the pursuer) that moves at each instant with a velocity directed toward another moving point (the target). The speed of the pursuer may be constant or may vary with time, and the target’s motion may be prescribed arbitrarily. The resulting path is determined by a system of differential equations that relate the relative positions of the two points.
Definition
Let $\mathbf{p}(t)$ denote the position vector of the pursuer and $\mathbf{t}(t)$ that of the target at time $t$. A pursuit curve arises when the velocity of the pursuer satisfies
$$ \frac{d\mathbf{p}}{dt}=v_{p}(t),\frac{\mathbf{t}(t)-\mathbf{p}(t)}{|\mathbf{t}(t)-\mathbf{p}(t)|}, $$
where $v_{p}(t)$ is the speed of the pursuer (often taken as a constant). The target’s motion is given independently, typically by
$$ \frac{d\mathbf{t}}{dt}= \mathbf{v}_{t}(t), $$
with $\mathbf{v}_{t}(t)$ known.
Mathematical formulation
When the target moves along a straight line with constant speed $v_{t}$ and the pursuer moves with constant speed $v_{p}$ (with $v_{p}>v_{t}$), the pursuit curve can be expressed in polar coordinates $(r,\theta)$ relative to the initial position of the pursuer. For the classic “dog‑chasing‑rabbit” problem, where the rabbit runs along the x‑axis and the dog starts at the origin, the curve satisfies
$$ \frac{dr}{d\theta}= -\frac{r}{\tan\theta},\frac{v_{p}}{v_{p}-v_{t}\cos\theta}. $$
Closed‑form solutions exist only for special cases; otherwise, numerical integration is employed.
Classical problems
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Dog‑chasing‑rabbit: A dog at the origin pursues a rabbit that runs along a straight line at constant speed. The dog always points directly at the rabbit. The curve is a logarithmic spiral when the speeds are equal; otherwise it is a more general pursuit curve that terminates when the dog catches the rabbit (if $v_{p}>v_{t}$) or asymptotically approaches it (if $v_{p}=v_{t}$).
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Four‑bug problem: Four bugs start at the vertices of a square, each moving directly toward the next bug clockwise with equal speed. Their trajectories are logarithmic spirals that converge to the center of the square.
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Missile guidance: In proportional navigation, a missile continuously adjusts its heading to keep the line‑of‑sight rate to the target at zero, producing a pursuit curve that approximates optimal interception paths.
Solutions and properties
- For constant speeds and a target moving in a straight line, the pursuit curve can be reduced to a first‑order ordinary differential equation solvable by separation of variables.
- When the target follows a circular or more complex trajectory, the equations generally lack elementary solutions; analysts rely on numerical methods or series approximations.
- Pursuit curves are usually smooth, but may develop singularities (e.g., cusps) if the pursuer’s speed is insufficient to overtake a faster target.
Applications
- Biology: Modeling predator‑prey interactions, such as the flight paths of hawks pursuing insects.
- Robotics and control theory: Designing pursuit–evasion algorithms for autonomous agents and unmanned aerial vehicles.
- Computer graphics and animation: Generating realistic chase scenes by simulating pursuit dynamics.
- Physics and engineering: Analyzing relative motion in fluid dynamics and missile guidance systems.
Historical notes
The pursuit problem dates back to the 18th century. Pierre Bouguer (1732–1813) posed an early version involving a dog chasing a hare. Later, mathematicians such as Charles H. H. May (1905) and J. L. Synge (1936) provided systematic treatments using differential equations. The four‑bug problem, popularized by mathematician Martin Gardner in the mid‑20th century, brought broader attention to pursuit curves in recreational mathematics.
See also
- Differential equations
- Curve of pursuit (variant terminology)
- Pursuit–evasion games
- Logarithmic spiral
- Proportional navigation
References
- J. L. Synge, The Theory of Curves and Surfaces, Cambridge University Press, 1936.
- E. J. Haugland, “The Pursuit Curve”, American Mathematical Monthly, vol. 71, no. 3, 1964, pp. 305–310.
- R. Hartl, Mathematical Modeling in Biology, Springer, 2013, Chapter 7 (Predator‑prey pursuit).