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Pursuit Curve


If A moves along a known curve, then P describes a pursuit curve if P is always directed toward A and A and P move with uniform velocities. Pursuit curves were considered in general by the French scientist Pierre Bouguer in 1732, and subsequently by the English mathematician Boole.

Under the name "path minimization," pursuit curves are alluded to by math genius Charlie Eppes in the Season 2 episode "Dark Matter" of the television crime drama NUMB3RS when considering the actions of the mysterious third shooter.

The equations of pursuit are given by

 (A-P)/(|A-P|)·(P^.)/(|P^.|)=1,
(1)

which specifies that the tangent vector at point P is always parallel to the line connecting A and P, combined with

 P^.·P^.=1,
(2)

which specifies that the point P moves with constant speed (without loss of generality, taken as unity above). Plugging (2) into (1) therefore gives

 ((A-P)·P^.)/(|A-P|)=1.
(3)
Pursuit curve for a straight vertical path

The case restricting A to a straight line was studied by Arthur Bernhart (MacTutor Archive). Taking the parametric equation of A(t)=(0,t) and the equation of the point P to be P=(x,y), the equations of motion for this problem are given by

 1/(sqrt(x^2+(t-y)^2))[0-x; t-y]·[x^.; y^.]=1
(4)

and

 x^.^2+y^.^2=1.
(5)

Squaring and rearranging (4) gives

 [(t-y)y^.-xx^.]^2=x^2+(t-y)^2.
(6)

Expanding gives

 x^2(x^.^2-1)+2x(y-t)x^.y^.+(t-y)^2(y^.^2-1)=0,
(7)

which can be simplified using (5) to

 x^2y^.^2+2x(t-y)x^.y^.+(t-y)^2x^.^2=0.
(8)

But this is a perfect square,

 [xy^.+(t-y)x^.]^2=0,
(9)

so taking the square root of both sides gives

 xy^.+(t-y)x^.=0.
(10)

This equation can be converted into one for y as a function of x by dividing both sides by x^., giving

 xy^'+t-y=0,
(11)

where y^'=y^./x^.=(dy/dt)/(dx/dt)=dy/dx. To eliminate t, note that the arc length traveled by P is given by

 s=intsqrt(1+y^('2))dx=tv=t,
(12)

since v=1 is a constant for this problem, so (11) becomes

 xy^'-y-intsqrt(1+y^('2))dx=0.
(13)

Differentiating then gives a second-order ordinary differential equation

 xy^('')-sqrt(1+y^('2))=0
(14)

that can be solved analytically to yield

 y=c_1+c_2x^2-(lnx)/(8c_2).
(15)

As expected, the solution in involves two arbitrary constants c_1 and c_2 whose values are fixed by the initial conditions.

The initial conditions for a particle starting at (x_0,y_0) at time t=0 are given by

y(x_0)=y_0
(16)
(dy)/(dx)|_(x_0)=(y_0)/(x_0).
(17)

Plugging these into (15), solving for c_1 and c_2, and plugging the results back into (15) gives the full solution

 y=1/4[(y_0+r_0)eta+(y_0-r_0)lneta+3y_0-r_0],
(18)

where

eta=(x/(x_0))^2
(19)
r_0=sqrt(x_0^2+y_0^2).
(20)

The point at which the y-component of P's motion changes direction (corresponding to the minimum of y(x), where P turns and begin following the chase point from behind it) can be found by differentiating (18) with respect to x, setting equal to 0, and solving for x. The result is

 x^*=x_0sqrt((r_0-y_0)/(r_0+y_0)).
(21)

Plugging in and simplifying gives the corresponding y-coordinate,

 y^*=(x_0y_0+2r_0y_0-r_0x_0)/(2(r_0+x_0))+1/4(y_0-r_0)ln((r_0-x_0)/(r_0+x_0)).
(22)

It is also possible to express the solution in closed form for x(t) and y(t). Plugging (◇) into (◇) and solving for t gives

 t=xy^'-y=1/4[(eta-1)(r_0+y_0)+(r_0-y_0)lneta],
(23)

which can be inverted in terms of the Lambert W-function to obtain

 x(t)=x_0sqrt((W(chie^(chi-4t/(r_0-y_0))))/chi),
(24)

where

 chi=(r_0+y_0)/(r_0-y_0).
(25)

Plugging this in to the equation for y(x) then gives

 y(t)=1/4[3y_0-r_0+(y_0-r_0)ln((W(chie^(chi-4t/(r_0-y_0))))/chi)+(y_0+r_0)(W(chie^(chi-4t/(r_0-y_0))))/chi].
(26)
Outside the circleFurther outside the circle
Outside the circle so curve cuts insideInside the circle

The figures above show various pursuit curves for A moving about a circle at constant speed.

The problem of n mice (or dogs) starting at the corners of a regular polygon and running towards each other is called the mice problem.


See also

Apollonius Pursuit Problem, Brocard Points, Mice Problem, Tractrix, Trawler Problem, Whirl

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References

Barton, J. C. and Eliezer, C. J. "On Pursuit Curves." J. Austral. Math. Soc. Ser. B 41, 358-371, 2000.Bernhart, A. "Curves of Pursuit." Scripta Math. 20, 125-141, 1954.Bernhart, A. "Curves of Pursuit-II." Scripta Math. 23, 49-65, 1957.Bernhart, A. "Polygons of Pursuit." Scripta Math. 24, 23-50, 1959.Bernhart, A. "Curves of General Pursuit." Scripta Math. 24, 189-206, 1959.MacTutor History of Mathematics Archive. "Pursuit Curve." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Pursuit.html.Simmons, G. F. Differential Equations, With Applications and Historical Notes. New York: McGraw-Hill, 1972.Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, p. 327, 1958.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 201-202, 1991.Yates, R. C. "Pursuit Curve." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 170-171, 1952.

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Pursuit Curve

Cite this as:

Weisstein, Eric W. "Pursuit Curve." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PursuitCurve.html

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