|No longer functional.
Due to modern web browsers dropping support for the Netscape Plugin Application Programming Interface (NPAPI), this animation for pursuit is no longer available. Sorry for the inconvenience.
But, this invites a great opportunity for you to write code that does illustrate Achilles and the Tortoise.
and the Tortoise
Zeno's 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.
If the images of Achilles and the Tortoise are placed at the bottom of the screen, and the
velocities are adjusted to be 50%, as in Zeno's Paradox, then
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
Zeno of Elea was a Pythagorean. His four paradoxes on the
divisibility of motion, time and space were preserved by Aristotle in
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
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
move to a
Frenchman, Pierre Bouguer
(1698-1758) who expanded pursuit to two dimensions. Interest
crossed the border into Italy, where the problem became curva di caccia, and then
Germany where readers will find dachshunds in Hundekurven problems. Across
the English Channel a spider was pursuing a fly in the well-known Ladies' Diary (1743,1750 and
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
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.
Pursuit curves for a circular track.
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."
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.
A. S. Hathaway, Houston, Texas
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
Differential equations valid for
arbitrary track and variable speeds;
Miscellaneous problems sometimes confused with pure pursuit curves.
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.
References for this Specific Table
"Curves of Pursuit," Scripta
1954, pp. 125-141.
"Curves of Pursuit II," Scripta
1957, pp. 49-65.
"Polygons of Pursuit," Scripta
1959, pp. 23-50.
"Curves of General Pursuit," Scripta
1959, pp. 180-206.
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
Useful Links and Books
Click on the stamp to see
Zeno in Raphael's "School of Athens" near the Sistine Chapel in the
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://www-history.mcs.st-and.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
mountain. While he was more or less unsuccessful,
the thought that he would attempt this in South America in 1740 is
|Gray, Alfred, Modern Differential Geometry of
Curves and Surfaces with MATHEMATICA®,
2nd ed., CRC Press, 1998, pp. 66-69.
|Weisstein, Eric W., CRC
Concise Encyclopedia of MATHEMATICS, CRC Press, 1999, p.1461.
|Yates, Robert C., Curves
and Their Properties, NCTM, 1952, pp. 170-171.
Java animation 2006
Webpage update 2017