Pursuit curve

Simple pursuit curve

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.

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

$L(t)=F(t)+xF^{\prime }(t).\,$

Single pursuer

Curves of pursuit with different parameters

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 1+y' ² = k² (a−x²) y" ².

Multiple pursuers

Curve of pursuit of vertices of a square (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.

See also

• Logarithmic spiral
• Tractrix
• Circles of Apollonius#Apollonius pursuit problem

External links

• Mathworld, with a slightly narrower definition that |L′(t)| and |F′(t)| are constant
• MacTutor Pursuit curve