The circular tractrix (1965)

The American Mathematical Monthly, Vol. 72, No. 10 (Dec., 1965), pp. 1065-1071
Reprinted with permission of Mathematical Association of America, 1529 18th
Street, NW, Washington, DC 20036
THE CIRCULAR TRACTRIX
W. G. CADY, 127 Power St., Providence, Rhode Island
Abstract. After a historical sketch the equations are derived for tracld>: curves whose directrix
is a circle. Two types are distinguished. In one, the curve has the form of a spiral. In the other,
there is a succession of cusps forming a flower-like pattern; the conditions are considered under
which the curve can close upon itself. A mechanical device for drawing the curves i,� described.
At a meeting in Paris in 1693 Claude Perrault laid his watch on the table,
with the long chain drawn out in a straight line ([I} vol. 3). He showed that
when he moved the end of the chain along a straight line, keeping the chain
taut, the watch was dragged along a certain curve. This was one of the early
demonstrations of the tractrix. The line along which the chain is pulled is called
the directrix. Perrault did not know that the curve had already been recognized,
and its equation derived, by Sir Isaac Newton
[2]
in 1676. This is said to have
been the first example of the determination of a curve by the process of integra­
tion.
The theory of the tractrix was treated by J-I uygens and Leibniz, both of
whom considered the case in which the directrix was any arbitrary curve. It was
not until almost a century later that Euler [3] treated the problem so com­
pletely that little or nothing on the subject has appeared since. Among other
things Euler derived equations for the case in which the directrix was a circle
instead of a straight line.
Hitherto no one seems to have pointed out the fact that a tractrix can be
drawn not only by pulling (whence the name "tractrix"), but also by pushing.
That is, the string or chain can be replaced by a rigid bar; a mirror image of
the same curve is then obtained. Such a reverse tractrix might appropriately be
called a "trudrix," from the Latin trudere, to push. In the circular tractrix dis­
Cllssed here both pulling and pushing occur.
The ordinary tractrix, having a straight line as directrix, is the one treated
in textbooks. It may be called a linear tractrix. The circular tractrix, though
really more interesting, does not seem to have been considered at all in our day.
It is interesting mechanically and aesthetically as well as mathematically and
historically.
We now derive the equations for the circular tractrix in a form better
adapted for comparison with the actual curves than that given by Euler.
Fig. 1 shows a linear tractrix, and for comparison part of a circular tractrix.
As in the linear case, the curve is characterized by the fact that the length of the
tangent from any point on the curve to the directrix is constant.
The differential equation of the circular tractrix is easily derived with the
aid of Fig. 2, in which OA and AB are the driving and tracing arms. Initially
the arms are in line, with a=OA and b=AB. If 'Y is the angle between b and
a,
measured counter-clockwise from b, we have at the start 'Y=7r. Now let a be
rotated in the positive direction about the origin 0 to the position OA', through
1065
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1066
1066
[Decemnber
[December
THE CIRCULAR
CIRCULAR TRACTRIX
TRACTRIX
THE
O. The
The tracing
tracing point,
point, initially
initially at
at B, is
is dragged
dragged along
along the
the curve
curve to
to B',
B',
the angle
angle 0.
the
where A'B'==b.
A'B'=b. B'
B' thus
thus represents
represents any point
point on the
the curve,
curve, and b
b is
is tangent
tangent to
to
where
the curve
curve at B'.
B'. If aa were
were rotated
rotated in the
the negative
negative direction,
direction, the
the branch
branch of
of the
the
the
curve shown
shown below OB would be traced.
traced. The angular
angular amplitude
amplitude of
of the
the radius
radius
curve
vector at any point
point on the
the curve
curve is 4'.
t/;.
vector
B
|B
J)
/~~~~~
E
aQ
/~~
o
with
tractrixFBE
FBE with
a circular
as directrix,
directrix,and
line as
a straight
straightline
DBC with
with a
aind a
circculartractrix
tractrixDBC
FIG. 1.
1. A linear
linear tractrix
FIG.
of equal
length.
GH are
are of
A'B', and
and GH
equial length.
AB, A'B',
lines AB,
The tangent
tanigentlines
a circle
of radius
as directrix.
directrix.The
radius a
a as
circle of
a
B'
0
A
B
arm aa about the
the origin
origin
the driving
drivingarm
of the
rotationof
by rotation
2. Part
generatedby
FIG.
tractrixgenerated
FIG. 2.
Part of
of a circular
circulartractrix
the point
the angle
point on
on
angle 0,
0, the
a has
has rotated
rotated through
throughthe
Wrhen
o.
a
0. The circle
circle with
is the
the directrix.
directrix.When
radius aa is
with radius
is the
of any
is 'Y.
the angular
any
and aa is
amplitudeof
angular amplitude
is at B', with
the
B' = b.
the curve
A 'B'
b. The angle
betweenbb and
angle between
curve is
with A'
T. '"t'is
=
= b.
= a, AB
AB =b.
point
on the
the curve.
curve. OA
OA =a,
pointon
an infinitesimal
infinitesimal
angle
Now suppose
angle
furtherpositively
rotatedfurther
throughan
a to
to be rotated
positivelythrough
suppose a
B'
while
the
moves
adO
a-circle,
small
distance
along
dO.
ves
through
the
small
distance
adO
along
the
a-circle,
while
B'
moves
dO.A'
A' mo
the
moves through
the angle
rotatedthrough
The arm
becomes rotated
angle
arm bb becomes
throughthe
a short
shortdistance
the curve.
curve. The
distance along
along the
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1965]
1965]
1067
THE
CIRCULAR TRACTRIX
TRACTRIX
THE CIRCULAR
-a
adO
adO
cos ("I
- 7r) =
d =
d(3
=- -cos
(y -db
(1)
a
- dO cos "I.
--dO
y.
b
,Bcause
in 80 and
and (3
-yto
to change
the amount
amount dy=ddO+df,
changes in
These
cause 'Y
change by
by the
d'Y=d8+d(3, or
or
These changes
(1),
from (1),
from
a
a
= dO
cos."I.
d-y=
dO---dO
d'Y
- - dO cos
(2)
b
b
the ratio
relationbetween
and 80 depends
depends only
only on
ratio of
of the
two
Evidently
between 'Yyand
the relation
the two
Evidently the
onithe
= p,
p, and
and find
findfrom
(2):
from (2):
arms.
write alb
We now
now write
arms. We
al/b=
dO
=
dO=
(3)
(3)
d'Y
dy
11-p- P cos "I
-y
differentforms
to whether
p> 1,
1,
assumes different
of (3) assumes
accordinigto
The
forms according
whether p>
integral of
The integral
or p
p
< 1, or
=1.
p<1,
p=1.
b, and
and
than b,
is greater
When
greater than
When p>
1, a is
p> 1,
-
,
"I))
sin lea
(a= cotalIn
0=
cotaln [�_22____
+ c,
sin -2(a +
sinHa
+ "I)
y)
(4)
___
(1/p), whence
=cos-I (lip),
a=cos-1
where a
=cos-1 (bla)
(bla) =cos-1
whence
(5)
cotaa =
cota
1
- 1)
-�-
V(p2
V (p2
-
-
I
.
1)
= r in
(4) and
and find
findthat
that C
Thereforefor
in (4)
all
C=O.
and 'Y
0=0 and
set 8=0
C, set
find C,
To find
To
=O. Therefore
for all
-y=7r
b, we
a>> b,
of 8,
when a
we have
have
0, when
values of
I
sin Ha
'(a- - "I))
sin
sin
= cot
o0 =
a In
cotcn
(6)
sin
+
i
I
) .
Ha + "I)
is
than b,
b, and
and the
the integral
is smaller
smaller than
When
integral is
< 1,
1, a is
When p <
=
o0 =
(7)
(7)
2
2
V(lI- - p2)
PI)
tan-'
tan-1
( 11, /(��)
(I
1 -p1
- tan�22/- ) + C.
+
P
tan
+ c.
=0
= 7rr as
Then
as before.
and 'Y
C, set
findC,
To find
0 and
=
before. Then
To
set 80=
-y
----
?rb
7r
7r
7rb
C
-·
== ---=
C =
v(l
v(b2
- p2)
- a2)
\V(b2 -/(1-p2)
therefore
is therefore
(7) is
of (7)
The final
finalform
The
form of
(8)
(8)
Va
=
o0 =
1
1
- p2)
v(l
(1 p2)
{22 tan-' ((A11/(�)
)
1 + )tan �
2 +7 }
2
1
tan-1
n
•
I - P
tan
+ 7r
= b,
the integral
when p=l,
and the
is simply
integral is
Finally
b, and
Finally when
p = 1, a =
simply
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.
1068
1068
(9)
(9)
[Deceinber
[December
THE
THE CIRCULAR
CIRCULAR TRACTRIX
TRACTRIX
0e =
'Y
-
cot
cot 2
.
Here,
Here, as
as is
is easily
easily verified,
verified, the
the constant
constant of
of integration
integration vanishes.
vanishes.
1. We
1. If
Case 1.
We consider
consider first
first the
the case
case in
in which
which a>
a> b,
b, and
and therefore
therefore p>
p> 1.
If the
the
2
driving
driving arm
arm aa continues
continues to
to rotate
rotate counter-clockwise,
counter-clockwise, the
the curve
curve shown
shown in
in Fig.
Fig. 2
and
(6) passes
and expressed
expressed by
by (6)
passes inside
inside the
the a-circle
a-circle and
and forms
forms aa spiral,
spiral, approaching
approaching
illustratedin
asymptotically
asymptotically aa circle
circle of
of radius
radius R=a2-b2.
R =a2- b2• This
This is
is illustrated
in Fig.
Fig. 3,
3, in
in which
which
the
b ==AB
AB are
OA and
and b
are initially
initially in
in line.
line. When
When aa rotates
rotates clockwise,
clockwise, one
one
the arms
arms a ==OA
\
~~~~b
B
thanthe
thetracing
arm.
armlonger
thedriving
FIG.
with
withthe
tracingarm.
longerthan
FIG. 3.
3. Circular
tractrix
drivingarm
Circulartractrix
second
A second
branch
around the
the R-circle.
R-circle.A
is traced
as shown,
traced as
of the
the curve
curve is
branch of
shown,spiraling
spiralingaround
at the
the
if at
if a
a rotated
counter-clockwise.Moreover,
branch
rotated counter-clockwise.
traced if
would be traced
Mloreover,if
branch would
= b,two
B'A'
morespiral
of B, so
so that
that B'
start
A' =b,
two more
B' instead
instead of
is located at B'
spiral
the point
start the
point B is
These
branches
at B'.
B'. These
froman
an inwardly-facing
become traced,
cusp at
branches become
inwardly-facingcusp
startingfrom
traced, starting
In all
all these
these cases
cases
fromthe
theinside.
inside. In
branches
theR-circle
R-circleasymptotically
branchesapproach
asymptoticallyfrom
approach the
finalvalue
The
the final
a approaches
value 'Y=27r-cos-1(bja).
the
and a
the angle
between bb and
y=2w - cos-'(b/a). The
approaches the
angle 'Yybetween
course be
be anywhere
on the
the a-circle,
a can
can of
of course
arm a
forthe
a-circle,
starting
the driving
anywhereon
drivingarm
startingpoint
point for
would
B' would
which
located in
in coincidence
coincidencewith
with A,
If A'
had been
been located
A' had
is the
the directrix.
directrix.If
which is
A, B'
0 and
A.
and A.
have fallen
fallenbetween
between 0
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1965]
1965]
1069
1069
THE
TRACTRIX
CIRCULAR TRACTRIX
THE CIRCULAR
Case
extreme cases of
when a
1.
a bb and
firstis wThen
of (6). The first
two extremie
There are two
Case III.I. There
anidp 1.
The radius
This is the
4. The
R in
in Fig.
Fig. 33
illustratedby
by Fig.
Fig. 4.
ra.diusR
(9), illustrated
expressedby (9).
the case expressed
has become
the
curve converges
the curve
upon the
zero, so that
that the
convergesasymptotically
asyinptoticallyupon
become equal to zero,
This
curve
was
called
by
R.
Cotes
the
Tract
rix
Com
pole.
plicata
(see
[4]
the
Tractrix
curve
by
Complicata
pole.
p.
202).
p.
=
=
and
driving
withdriving
tractrix
FIG. 4.
4. Circular
Circular
FIG,
tractrix with
and tracing
tracing arms
arms equal.
equal.
a becomes
infinite,
becomes infin',
Case
when, with
with bb finite,
te,
finite,a
The other
other extreme
is when,
extremecase is
III. . The
Case III
tractrix,and
linear tractrix,
familiarlinear
forthe
the familiar
that for
and
becomes that
The equation
equation becomes
so that
co. The
so
that p = C/J.
the four
become the
above become
tractrix mentioned
the four
of the
the circular
branches of
the
four branches
circular tractrix
mentioned above
four
of the
linear tractrix.
branches of
the linear
tractrix.
branches
=
A
I
arm.
thetracing
tracing
thanthe
arm
shorter
driving
withthe
thedriving
tractrix
FIG. 5.
5. Circular
Circular
FIG.
tractrix with
arm shorter
than
arm.
< 1.
1.
b, so
so that
that p <
arm b,
tracingarm
the tracing
than the
Case IV.
arm a
a is
is shorter
shorterthan
IV. The
The driving
drivingarm
Case
5,
in Fig,
Fig. 5,
shown in
formshown
of the
the form
(8), yielding
by (8),
This is
is the
the case
This
case expressed
expressed by
yielding curves
curves of
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1070
1070
[December
[Decebnber
THE CIRCULAR
CIRCULAR TRACTRIX
THE
TRACTRIX
with
a succession
of positive
with a
with petals
successionof
positiveand
and negative
negative cusps,
cusps, like
like a
a flower
flowerwith
petals spread
spread
out into
into a
a circle.
The radial
radial difference
out
circle. The
between positive
difference
between
positiveand
and negative
negativecusps is
is 2a.
Figures 3,
3, 4,
4, and
and 55 are
are reproduced
reproduced from
actual curves
curves traced
with the
Figures
from actual
traced with
the
apparatus described
described below.
apparatus
below.
With the
the aid
aid of
of Eq.
With
the following
relationscan
can be derived.
Eq. (8)
(8) the
followingrelations
derived.Since,
Since, at each
positive cusp,
is an
an extension
positive
cusp, bb is
extension of
a, it
it follows
follows that
between two
successive posi­
of a,
that between
two successive
positive cusps
the angular
angular rotation
rotationof
of 'Yy is
is 27r;
tive
cusps the
the angular
angular rotation
rotation()0 of
of the
the arm
arm aa is
27r;the
27r/V/(l
-p2);
and the
the angular
angular change
in the
27r
/ V (1p2) ; and
change in
vector is
the radius
radius vector
is
(10)
(10)
dl==
Ii1/!
271'
27r
-
V(l
- P)
NAI -PI)
= 271'
2r =
2r
- 271'
11-V(1-p2)
- v(l - p2)
-
V(l
- p2)
-\(1PI)
271'
27r
.
= -radians,
radians,
=
m
m
-
where
where
(11)
(-)/(
m=
_ p2)
-
_ a2)
v(b2
\I(b2-
1 - V(l-p2)
_ p2)
b - v(b2 - a2)
b-V/(b2-a2)
In general
the quantity
general the
In
from zero
infinity,
quantity m,
m, which
which may
may have any
any value from
zero to infinity,
is an
an irrational
irrationalnumber.
number. This
This means
means that
in general
is
general the
that in
the circular
circular tractrix
tractrixdoes
not become
closed curve
forany
any finite
finiterotation
rotationof
of the
the arm
arm a.
It is
only when
not
become aa closed
curve for
a. It
is only
when
is a
m is
a rational
rational fraction
that the
the curve
itself.
m
fraction that
curve closes
closes upon
upon itself.
= 0,
= a,
m=
When m
the curve
and the
one positive
one negative)
When
a, and
curve has
has one
0, bb=
and
positive (or
(or one
negative) cusp and
a spiral
spiral converging
on a
a point,
as we
a
converging on
we have
have already
point,as
already seen.
seen. When
When m
m is
is an
an integer
integerwe
have
a closed
closed curve
curve with
m positive
positivecusps,
cusps, completed
completedin
in one
one revolution
revolutionof
of the
the
have a
with m
amplitude 1/;.
4'. The
The curve
curve then
then has
m-foldsymmetry.
symmetry.
amplitude
has m-fold
= mdm
m is
When m
is a
a rational
m=
rationalfraction
it may
in lowest
When
be written
written in
lowest terms
fractionit
may be
termsas
as m
m1/M2,
2'
m1and
where
and m2
are integers.
the driving
is rotated,
where ml
positive
m2are
as the
arm is
integers.Then,
Then, as
drivingarm
rotated, ml
positive
ml
cusps
is completed
cusps become
become described,
while the
completed in
described,while
the covering
coveringoperation
operationis
in m2
m2revolu­
revolutions of
of the
the curve.
tions
curve.
m is
If m
small, ml«m
is nearly
as large
as b,
and there
thereare
are few
If
is very
very small,
large as
few cusps
nearly as
b, and
Ml<<M2,
2, aa is
in many
5 illustrates
this,
for
in
this
figure
many revolutions
ofthe
the curve.
in
this
in
revolutions of
curve. Fig.
Fig. 5
illustratesthis,for
m - 5/2.
figurem"",S/2.
The
negative
cusps
come
inside
the
a-circle.
The negative cusps come inside the a-circle.
a <<b, and
If m
m is
is large,
large,ml»m2,
thereare
are many
small "petals"
in few
and there
If
few revolurevolu­
many small
"petals" in
ml>>m2,a«b,
= 1,
tions. If
If m2
m2=
1, the
the curve
on itself
itselfin
in one
one revolution.
tions.
curve closes
closes on
revolution.
Returningto
to Figure
Figure 5,
see that
5, we
that the
the curve
we see
curve apparently
closes upon
Returning
apparently closes
upon itself
itself
= 2.5. Actually
with
m1= 5, m2=2,
so that
with ml=S,
m=2.S.
curve was drawn
with aa=31.8
= 31.8
that m
the curve
drawn with
m2= 2, so
Actually the
mm, bb=46.2
The discrepancy
is to
to be at­
=46. 2 mm,
from which,
which, by (11), m
= 2.64. The
mm,
atmm, from
m=2.64.
discrepancyis
to faulty
tributed
faulty mechanical
with possible
errors in
in the
tributed to
mechanical tracking,
tracking,together
the
together with
possible errors
meatsurement
of aa and
b.
measurement
of
and b.
A mechanical
ratio of
A
circular tractrix
with any
mechanical device
fordrawing
tractrixcurves
curves with
of
device for
drawing circular
any ratio
to aa is
is pictured
in Fig.
flat wooden
about 28
28 cm
bb to
pictured in
wooden base
base about
cm square,
square, carefully
carefully
Fig. 6.
6. A
A flat
carriesa
a bridge
is a
a clearance
of about
leveled,
thereis
about 9 cm.
Through
leveled, carries
bridge below
below which
which there
clearance of
cm. Through
the
center of
of the
the bridge
the vertical
turned by
a handle
passes the
drivingshaft,
the center
bridge passes
vertical driving
shaft, turned
by a
handle
that plays
The driving
a graduated
so as
as to
to be
at any
plays over
over a
dial, so
any desired
angle. The
that
graduated dial,
be set
set at
desired angle.
driving
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1965]
19651
THE CIRCULAR
CIRCULAR TRACTRIX
TRACTRIX
THE
1071
1071
in this
shaft,and
and by
means of
of a
a set
screw can
this shaft,
arm
hole in
by means
set screw
can be
be
a hole
througha
arm aa passes through
clamped
at any
any desired
desired length.
length.
clamped at
Fastened to
free end of
yoke bent
bent out of
of a
strip of
of
a strip
of the
the driving
drivingarm
arm is
is a
a yoke
to the
the free
at the
the lower
of which
a
shaft,at
a thin
thintracing
The yoke
yoke carries
metal. The
sheet
sheet metal.
carries a
tracing shaft,
lower end
end of
which is
is a
arm should
should be
as close
close to
to the
the
This tracing
tracingarm
arm b.
b. This
tracingarm
the tracing
block for
holding the
be as
forholding
block
and clamped
clamped by
a set
set screw.
screw.
at any
any length
set at
it can
paper
can be
be set
length and
by a
as practicable;
practicable; it
paper as
to hold
the lead
fordrawing
drawing
is a
a block
of this
this arm
arm is
At
block with
with vertical
vertical bore
bore to
hold the
lead for
end of
At the
the end
the paper
paper
it is
is pressed
pressed onto
be used
used;; it
should be
of lead
lead should
the
onto the
type of
the curve.
The softest
softesttype
curve. The
weight of
4 ounces.
ounces.
of 33 or
or 4
by a weight
any friction
and from
play, and
The device
free as
from loose
loose play,
from any
friction
as free
as possible
possible from
must be as
device must
and paper.
paper.
except that
between ttracing
point and
racingpoint
that between
except
iriving
shaft
curves.
tractrixcurves.
drawing circular
fordrawing
FIG. 6.
6. Machine
Machine for
FIG.
circular tractrix
References
References
1. M.
M. Cantor,
Cantor, Vorlesungen
Vorlesungentiber
uber Geschichte
Geschichte der
der Mathematik,
Mathematik, Leipzig,
Leipzig, 1908,
1908, Vol.
Vol. 3,
3, pp.
pp. 214,
1.
214,
Vol. 4,
215,
786; Vol.
4, p.
p. 508.
508.
2
15, 786;
The Correspondence
Correspondenceof
of Isaac
Isaac Newton,
Newton, ed.
ed. by
by H. W. Turnbull,
Turnbull, Cambridge
Cambridge Univ.
Univ. Press,
Press,
2. The
New York,
York, 196
1961,
Vol. 3,
329.
3, pp.
pp. 148.
160,329.
New
1, Vol.
148, 160,
L. Euler,
Euler, Opera Omnia,
Omnia, Ser.
Ser. II,
II, Vol.
Vol. 7,
7, pp.
pp. 120-147,
3. L.
120-147, Lausanne, 1958; also Nova
Nova Acta
Academiae
Scientiarum Imperialis
Imperialis Petropolitanae
Petropolitanae II,
II, pp. 3-27,
�ademiae Scientiarum
A
3-27, 1788.
1788. Euler considers
considers also the
the
of the
the tractrix,
in which
general case
case of
tractrix,in
which the
the directrix
directrixmay
findsit
general
may be of
of any form,
form,but
but finds
it insoluble.
insoluble.
Loria, Spezielle
Spezielle algebraische
algebraische und
und transzendente
transzendente ebene
4. G. Loria,
ebene Kurven,
Kurven, vol.
vol. II, Teubner,
Teubner,
Leipzig, 1911.
1911.
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