Back to . .
.
Curve
Bank Home
Classics
Index Page
Deposit #59

The Pursuit Family of Curves
Possibly originating with Leonardo da Vinci
but first extensively investigated by Bouguer in 1732.


This
section
features the
Pursuit Curves.
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
same ratio.
Thus,
the
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?"
What is the "capture" point?


Observe
the
pathways
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
is
always moving toward the black bead. The yellow bead follows the
linear green
vector and is always directed toward the black bead.
Our MATHEMATICA^{®}
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 xaxis. Others have
considered vertical motion, either on the yaxis, or parallel to it.

Various forms of the equations for the pursued having linear motion are
found in the literature.
Cartesian Form I

Cartesian From II

Differential
equations for a linear pursuit are sometimes applied where
k
is the ratio of the two
velocities. The pursued starts at rest and then moves along a
vertical
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
particular
point p ( y' = p ). 

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).
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
NSFMAA 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
and
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.' "
Arthur Bernhart,
"Curves of General
Pursuit", Scripta Mathematica, 24,
p. 206, 1959.

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


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.

Wagon, Stan, MATHEMATICA^{®}IN
ACTION, W. H. Freeman and Co. ISBN
0716722291.

Wagon, Stan, MATHEMATICA^{®}
IN ACTION, 2nd ed., SpringerVerlag, 2000. ISBN
0387986847.

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

Yates, Robert C., Curves
and Their Properties, NCTM, 1952, pp. 170171. 
One of
the thrills of
visiting the Sistine Chapel in Vatican City is seeing Raphael's The
School
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
plane surface.



MATHEMATICA^{®} animation contributed by
Luis Garcia
2005.


