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Yun Wang

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A Simulation of Pursuit Curves

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JAVA logoNo 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.

Achilles 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 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.

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:

Journal coverAn 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 (1698-1758) 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 well-known Ladies' Diary (1743,1750 and 1752).

I.  Category One:  One dimensional pursuit in a plane with a linear track and uniform speeds.

Bernhard illustration

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.

Bernhart illustration #2"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.

Bernhard illustration #3

"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.

Bernhard illustration #4"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. 125-141.
Bernhart, Arthur, "Curves of Pursuit II," Scripta Mathematica,  23, 1957, pp. 49-65.
Bernhart, Arthur, "Polygons of Pursuit," Scripta Mathematica,  24, 1959, pp. 23-50. Bernhart, Arthur, "Curves of General Pursuit," Scripta Mathematica,  24, 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 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: < >

For a variety of Pursuit Problems:  >
For the evolute in JAVA:  < >

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. 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.
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Yun Wang
Java animation 2006

Jonathan Sahagun
Webpage update 2017