Be patient.
This file may be slow.
Instructions
required
 Scroll down the screen to find the buttons at the
bottom of the blank white box on the right.
Position both the red pursuer and green pursuee by first using the “Placement” button. Click on the
empty canvas to place the red pursuer. Click, hold, and drag to see the
green pursuee.
 Click on “Placement”
again to
place another pair of particles on the
canvas. Enjoy!


 Investigate the particles you have placed on the
screen.
 Your first particle will be colored red and will
pursue the second particle.
 Your second particle will be colored green and will
flee in the direction you set.
 If you double click on the particles it will bring up
control sliders.
 The pursuer will be able to modify the speed of the
pursuit.
 The pursuee will be able to modify the velocity of
the particle (in x and y directions).
 After placing the particles on the screen, click on
the "Start" button to begin the
applet.
 More particles may be placed on the screen in pairs
by again clicking on "Placement."
 If you wish to stop, simply click "Stop."
 If you wish to clear the particles, click "Clear."
 You may wish to place several particles on the
screen, simulate Zeno's ancient paradox, or try the other categories of
pursuit curves listed below.

Achilles
and the Tortoise
If the particles are placed at the bottom of the screen, and the
velocities are adjusted to be 50%, as in Zeno's Paradox, then
the
viewer can enjoy seeing Achilles overtake the Tortoise! (Ha!)
Calculus instructors often introduce the concept of convergence of
infinite series  in this case, a convergent geometric series
with r = 1/2  by
having students discuss Achilles and the
Tortoise.
The paradox is more or less as follows: Given a head start by the
tortoise, Achilles, the fastest runner in Greece, can never catch the
tortoise. During the time it takes Achilles to cover the original
distance, the tortoise has moved forward to a newer distance.
Zeno of Elea was a Pythagorean. His four paradoxes on the
divisibility of motion, time and space were preserved by Aristotle in
his Physics.

Historical Sketch:
An excellent overview of the history of pursuit curves is
found in a series of articles written by Arthur Bernhart (University of
Oklahoma) and published in Scripta
Mathematica in the 1950s. He organizes his review into
four categories: pursuit curves where the pursued moves along a
straight line; the chase takes place in a circular fashion; the race
among several competitors is in a polygonal fashion; and finally,
special cases involving dynamical pursuit with variable speeds, centers
of gravity, and other aberrant properties. This series of
articles
cuts across centuries of time, countries and languages.
A bit of historical background is fascinating. The
publications by Bernhart and several others often begin in antiquity
with Zeno's solution to the classic Achilles
and the Tortoise, mention the work of Leonardo da Vinci, and
then
move to a
Frenchman, Pierre Bouguer
(16981758) who expanded pursuit to two dimensions. Interest
crossed the border into Italy, where the problem became curva di caccia, and then
into
Germany where readers will find dachshunds in Hundekurven problems. Across
the English Channel a spider was pursuing a fly in the wellknown Ladies' Diary (1743,1750 and
1752).
I.
Category One: One
dimensional pursuit in a plane with a linear
track and uniform speeds.
Let the point Q move
along a given tract Q(t) while another point P moves always in the
direction PQ
on P(s). If the velocity
vector dP/ds has
the same sense as PQ,
the locus P(t) is called a curve of pursuit, otherwise
a curve of flight.

II.
Category Two:
Pursuit curves for a circular track.
"A
dog at the center of a
circular pond C makes
straight for a duck which is swimming along the edge of the pond.
If the rate of swimming of the dog is to the rate of swimming of the
duck as m : 1, determine the
equation of the curve of pursuit and the distance the dog swims to
capture the duck."
C
is the center of the pond, Q
is the "quacker," and the point of attack is K, which conveniently forms an
inscribed right triangle.
American Mathematical Monthly, 27 (1920), p.
31
A. S. Hathaway, Houston, Texas

III.
Category Three:
Problems of triangular pursuit.
"Three dogs are placed at the three vertices of an equilateral triangle;
they run one after the other. What is the curve described by each
of them?

IV.
Category Four:
Differential equations valid for
arbitrary track and variable speeds;
Miscellaneous problems sometimes confused with pure pursuit curves.
"Navigation:
Does one swimmer P
pursue another Q
when his course is
toward Q though
his heading is somewhat
upstream? If P
swims through the water medium at speed e, and the current flows with
speed f at an
angle φ with the desired course PQ, then P must head off course by a
correction angle ε in order to make good his course.
Important
References for this Specific Table 
Bernhart, Arthur,
"Curves of Pursuit," Scripta
Mathematica, 20,
1954, pp. 125141.

Bernhart, Arthur,
"Curves of Pursuit II," Scripta
Mathematica, 23,
1957, pp. 4965. 
Bernhart, Arthur,
"Polygons of Pursuit," Scripta
Mathematica, 24,
1959, pp. 2350. 
Bernhart, Arthur,
"Curves of General Pursuit," Scripta
Mathematica, 24,
1959, pp. 180206. 
The opportunities for
animation of pursuit curves are enormous. The NCB invites faculty
and students to try their hand at some of these problems as class
projects. Then hopefully you will add a "choice" effort to our
NCB MATH Archive collection as a sampler of a fun activity from your
campus.

Useful Links and Books

Click on the stamp to see
Zeno in Raphael's "School of Athens" near the Sistine Chapel in the
Vatican.

For more information on Pursuit Curves: < http://en.wikipedia.org/wiki/Curve_of_pursuit
>

For a variety of
Pursuit Problems: < http://mathworld.wolfram.com/topics/ApolloniusPursuitProblem.html
> 
For the evolute in
JAVA: < http://wwwhistory.mcs.stand.ac.uk/history/Curves/Pursuit.html
>
Note: The French scientist Pierre Bouguer attempted to measure
the density of the Earth by using a plumb line deflected
by the attraction of gravity. He collected data on the top of a
Peruvian
mountain. While he was more or less unsuccessful,
the thought that he would attempt this in South America in 1740 is
slightly amazing.

Gray, Alfred, Modern Differential Geometry of
Curves and Surfaces with MATHEMATICA^{®},
2nd ed., CRC Press, 1998, pp. 6669.

Weisstein, Eric W., CRC
Concise Encyclopedia of MATHEMATICS, CRC Press, 1999, p.1461.

Yates, Robert C., Curves
and Their Properties, NCTM, 1952, pp. 170171. 


JAVA
animation contributed by
Yun Wang
ywang80@gmail.com
2006.


