Pursuit curve
In geometry, a curve of pursuit is a curve constructed by analogy to having a point or points representing pursuers and pursuees; the curve of pursuit is the curve traced by the pursuers.
DefinitionEdit
With the paths of the pursuer and pursuee parameterized in time, the pursuee is always on the pursuer's tangent. That is, given Template:Math, the pursuer (follower), and Template:Math, the pursued (leader), for every Template:Mvar with Template:Math there is an Template:Mvar such that
- <math>L(t) = F(t) + xF'\!(t).</math>
HistoryEdit
The pursuit curve was first studied by Pierre Bouguer in 1732. In an article on navigation, Bouguer defined a curve of pursuit to explore the way in which one ship might maneuver while pursuing another.<ref>Template:Cite journal</ref>
Leonardo da Vinci has occasionally been credited with first exploring curves of pursuit. However Paul J. Nahin, having traced such accounts as far back as the late 19th century, indicates that these anecdotes are unfounded.<ref>Template:Cite book</ref>
Single pursuerEdit
The path followed by a single pursuer, following a pursuee that moves at constant speed on a line, is a radiodrome.
It is a solution of the differential equation Template:Math.
Multiple pursuersEdit
Typical drawings of curves of pursuit have each point acting as both pursuer and pursuee, inside a polygon, and having each pursuer pursue the adjacent point on the polygon. An example of this is the mice problem.
See alsoEdit
ReferencesEdit
External linksEdit
- Mathworld, with a slightly narrower definition that |L′(t)| and |F′(t)| are constant
- MacTutor Pursuit curve