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The Pursuit Family of Curves
Possibly originating with Leonardo da Vinci
but first extensively investigated by Bouguer in 1732.
two beads move with related
velocities. When the ratio k of the two velocities
is greater than one ( k > 1 ), the pursuer travels
faster than the pursued. The question then becomes, "At
what point do the two meet?"
One particle travels along a specified
curve, while a second pursues it, with a motion always directed toward
the first. The velocities of the two particles are always in the
What is the "capture" point?
of the two beads representing two particles. The yellow bead
pursues the black bead in such a way that the yellow bead on the right
always moving toward the black bead. The yellow bead follows the
vector and is always directed toward the black bead.
animation is for the special case where the pursued always moves in a linear fashion. We find
examples in the literature of linear motion along the x-axis. Others have
considered vertical motion, either on the y-axis, or parallel to it.
Various forms of the equations for the pursued having linear motion are
found in the literature.
equations for a linear pursuit are sometimes applied where
is the ratio of the two
velocities. The pursued starts at rest and then moves along a
line at x = a, not x = 0 as in the above
illustrations. The equation of motion for the pursuer is then
solvable by first setting the first derivative equal to a
point p ( y' = p ).
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
Even readers of a mathematical bent in North America
enjoyed "Curves of Pursuit" in the Mathematical
Monthly (1859), one of the earliest publications of general
mathematical interest on this side of the
Atlantic. Professor O. Root of Hamilton College, Clinton, NY
published on one dog chasing three foxes. Other early
Americans who published on the topic
include Artemas Martin. Access to his mathematics collection at
the American University in Washington, DC has enriched a number of
NSF-MAA sponsored workshops.
Famous 20th century mathematicians who published on pursuit curves
include Georg Cantor, the father of set theory, and the top
British classical analyst, J. E. Littlewood of the famous team of Hardy
Littlewood. He composed and published an essay on "Lion and Man."
The NCB can only briefly write on the vast and rich literature of
pursuit curves. We conclude with the closing from Bernhart's series of
articles . . .
"And God said,
'Let there be light'; and there was light.' The Hebrew text uses
the same word or the command and its fulfillment. But we can
imagine the angelic architect asking for more details: 'What
path shall light follow in going from P to Q
?' And the answer might have been, 'Don't bother me with
such details. See that it makes the trip in a minimum time.' "
"Curves of General
Pursuit", Scripta Mathematica, 24,
p. 206, 1959.
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
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.
|Wagon, Stan, MATHEMATICA®IN
ACTION, W. H. Freeman and Co. ISBN
|Wagon, Stan, MATHEMATICA®
IN ACTION, 2nd ed., Springer-Verlag, 2000. ISBN
|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.
the thrills of
visiting the Sistine Chapel in Vatican City is seeing Raphael's The
of Athens. His famous frescoes are just outside the doorway
to the Chapel. Among the important mathematical figures
represented are Euclid (see the postage stamp on the right.), Ptolemy,
and Pythagoras. Click on the stamps to see a larger
view of Euclid and his students. In particular, The School of Athens is considered
one of the earliest and finest examples of perspective, a highly
geometrical illusion of giving distance its proper proportion on a
MATHEMATICA® animation contributed by