tractrix

tractrix

Tractrix

(from the Latin verb

trahere

‘pull, drag’) is the curve along which a small object (

tractens

) moves when pulled on a horizontal plane with a piece of thread by a puller (

tractendus

) which moves rectilinearly.

Let the object initially be in the

x

⁢

y

-plane on the

x

-axis in the point

(

a

,

0

)

and the puller in the origin;

a

is the of the pulling thread.  Then the puller begins to move along the

y

-axis in the positive direction.  The object follows drawing the path curve

y

=

y

⁢

(

x

)

so that the line determined by the thread is at every the

tangent

of the curve.  This condition gives in the point

(

x

,

y

)

the

differential equation

d

⁢

y

d

⁢

x

=

-

a

2

-

x

2

x

with the

initial condition

y

⁢

(

a

)

=

0

.  The solution is

y

=

∫

x

a

a

2

-

x

2

x

⁢

𝑑

x

or

y

=

±

(

a

⁢

ln

⁡

a

+

a

2

-

x

2

x

-

a

2

-

x

2

)

.

Here the minus alternative is for the case that the puller moves in the negative direction from the origin.  In fact, both branches, corresponding to both signs, belong to the tractrix.  The branches meet in the cusp point

(

a

,

0

)

.

The substitution

x

:=

a

⁢

cos

⁡

t

gives for the tractrix the parametric

x

=

a

⁢

cos

⁡

t

,

y

=

±

a

⁢

(

ln

⁡

1

+

sin

⁡

t

cos

⁡

t

-

sin

⁡

t

)

.

Another one is

x

=

a

cosh

⁡

u

,

y

=

±

a

⁢

(

u

-

tanh

⁡

u

)

,

where

cosh

and

tanh

are the

hyperbolic functions

cosinus hyperbolicus

and

tangens hyperbolica

.

Remarks

1.

It is obvious that the line, on which the puller goes, is the asymptote of the tractrix.  The curve thus has the property that its tangent, between the asymptote and the point of tangency, has the (

=

a

).

2.

The differential equation of the

orthogonal curves

of the tractrix is

d

⁢

y

d

⁢

x

=

x

a

2

-

x

2

,

whence they are the circles

x

2

+

(

y

-

C

)

2

=

a

2

.

3.

The

arc length

of one branch on the

interval

[

b

,

a

]

is simply

∫

b

a

1

+

(

d

⁢

y

d

⁢

x

)

2

⁢

𝑑

x

=

a

⁢

∫

b

a

d

⁢

x

x

=

a

⁢

ln

⁡

a

b

.

4.

The area

A

between the tractrix and its asymptote is

π

⁢

a

2

2

.   This may be calculated ordinarily as

A

=

2

⁢

∫

0

a

(

a

⁢

ln

⁡

a

+

a

2

-

x

2

x

-

a

2

-

x

2

)

⁢

𝑑

x

;

integrating by parts and using the area of a quarter-circle yield

A

=

2

⁢

[

a

⁢

/

x

=

0

a

⁡

x

⁢

ln

⁡

a

+

a

2

-

x

2

x

-

a

⁢

∫

0

a

x

⁢

d

d

⁢

x

⁢

(

ln

⁡

a

+

a

2

-

x

2

x

)

⁢

𝑑

x

-

π

⁢

a

2

4

]

and moreover

A

=

2

⁢

a

⁢

/

x

=

0

a

⁡

[

x

⁢

ln

⁡

(

a

+

a

2

-

x

2

)

-

x

⁢

ln

⁡

x

+

a

⁢

arcsin

⁡

x

a

]

-

π

⁢

a

2

2

=

2

⁢

a

⁢

(

0

-

0

+

a

⋅

π

2

)

-

π

⁢

a

2

2

=

π

⁢

a

2

2

(see this entry (

http://planetmath.org/GrowthOfExponentialFunction

) for

lim

x

→

0

+

⁡

x

⁢

ln

⁡

x

=

0

).  Another way to determine

A

is differential-geometric:  as the object draws the tractrix from above to down, the thread turns

180

o

and thus sweeps an area equal to a half-circle.

5.

The envelope of the

normal lines

of the tractrix, i.e. the

evolute

of the tractrix is the catenary (or “chain curve”)

x

=

a

⁢

cosh

⁡

y

a

.

Title

tractrix

Canonical name

Tractrix

Date of creation

2013-03-22 15:18:32

Last modified on

2013-03-22 15:18:32

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

26

Author

pahio (2872)

Entry type

Derivation

Classification

msc 51N05

Related topic

SubstitutionNotation

Related topic

Catenary

Related topic

EulersSubstitutionsForIntegration

Defines

tractrix

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