pursuit curve

With the paths of the pursuer and pursuee parameterized in time, the pursuee is always on the pursuer's tangent. That is, given {{math|F(t)}}, the pursuer (follower), and {{math|L(t)}}, the pursued (leader), for every {{mvar|t}} with {{math|{{prime|F}}(t) ≠ 0}} there is an {{mvar|x}} such that

:$L(t)\; =\; F(t)\; +\; xF\text{'}\backslash !(t).$

File:BourguerCourbepoursuite1732.jpg's 1732 article studying pursuit curves]]

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.{{cite journal |last=Bouguer |first=Pierre |author-link=Pierre Bouguer |year=1732 |title=Sur de nouvelles courbes auxquelles on peut donner le nom de lignes de poursuite |language=fr |journal=Mémoires de mathématique et de physique tirés des registres de l'Académie royale des sciences |pages=1–15}}

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.{{cite book |last=Nahin |first=Paul J. |author-link=Paul J. Nahin |year=2007 |title=Chases and Escapes: The Mathematics of Pursuits and Evasion |publisher=Princeton University Press |pages=27–28 |isbn=978-0-691-12514-5}}

File:Radiodrome-params-colour.png

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

{{math|1 +  y′ ² {{=}} k² (a − x)² y′′ ²}}.

{{-}}

File:Four point pursuit curve.gif (the mice problem for n=4).]]

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.

• Logarithmic spiral
• Tractrix
• Circles of Apollonius#Apollonius pursuit problem
• Pursuit–evasion

{{reflist}}

{{Commons|Curve of pursuit}}

• [http://mathworld.wolfram.com/PursuitCurve.html Mathworld], with a slightly narrower definition that |L′(t)| and |F′(t)| are constant
• [http://www-history.mcs.st-and.ac.uk/Curves/Pursuit.html MacTutor Pursuit curve]

{{Differential transforms of plane curves}}

Category:Curves

Category:Pursuit–evasion