Mathematical Description

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Mathematical description[edit] quation[edit]

tenaries for different values of a

ree different catenaries through the same two points, depending horizontal force being and λ mass per unit length.

e equation of a catenary in Cartesian coordinates has the form[27]

where cosh is the hyperbolic cosine function. All catenary curves are similar to each other, having eccentricity = √2.

Changing the parameter a is equivalent to a uniform scaling of the curve.[30]

The Whewell equation for the catenary is[27]

Differentiating gives

and eliminating gives the Cesàro equation[31]

The radius of curvature is then

which is the length of the line normal to the curve between it and the x -axis.[32]

Relation to other curves[edit]

http://en.wikipedia.org/w/index.php?title=Catenary&action=edit&section=5

http://en.wikipedia.org/wiki/Cartesian_coordinate_system

http://en.wikipedia.org/wiki/Catenary#cite_note-Lockwood122-27

http://en.wikipedia.org/wiki/Hyperbolic_function

http://en.wikipedia.org/wiki/Similarity_(geometry)

http://en.wikipedia.org/wiki/Eccentricity

http://en.wikipedia.org/wiki/Parameter

http://en.wikipedia.org/wiki/Scaling_(geometry)

http://en.wikipedia.org/wiki/Catenary#cite_note-30

http://en.wikipedia.org/wiki/Whewell_equation

http://en.wikipedia.org/wiki/Ces%C3%A0ro_equation

http://en.wikipedia.org/wiki/Osculating_circle

http://en.wikipedia.org/wiki/Tangent#Normal_line_to_a_curve

http://en.wikipedia.org/wiki/File:Catenary-tension.png

http://en.wikipedia.org/wiki/File:Catenary-pm.svg

http://en.wikipedia.org/wiki/Tangent#Normal_line_to_a_curve

http://en.wikipedia.org/wiki/Osculating_circle

http://en.wikipedia.org/wiki/Ces%C3%A0ro_equation

http://en.wikipedia.org/wiki/Whewell_equation

http://en.wikipedia.org/wiki/Scaling_(geometry)

http://en.wikipedia.org/wiki/Parameter

http://en.wikipedia.org/wiki/Eccentricity

http://en.wikipedia.org/wiki/Similarity_(geometry)

http://en.wikipedia.org/wiki/Hyperbolic_function

http://en.wikipedia.org/wiki/Cartesian_coordinate_system

When a parabola is rolled along a straight line, the roulette curve traced by its focus is a

catenary.[33]

The envelope of the directrix of the parabola is also a catenary.[34]

The involute from the

vertex, that is the roulette formed traced by a point starting at the vertex when a line is rolled on a

catenary, is the tractrix.[33]

Another roulette, formed by rolling a line on a catenary, is another line. This implies that square

wheels can roll perfectly smoothly if the road has evenly spaced bumps in the shape of a series of

inverted catenary curves. The wheels can be any regular polygon except a triangle, but the catenary

must have parameters corresponding to the shape and dimensions of the wheels.[35]

Geometrical properties[edit]

Over any horizontal interval, the ratio of the area under the catenary to its length equals a, independent

of the interval selected. The catenary is the only plane curve other than a horizontal line with this

property. Also, the geometric centroid of the area under a stretch of catenary is the midpoint of the

perpendicular segment connecting the centroid of the curve itself and the x-axis.[36]

Science[edit]

A charge in a uniform electric field moves along a catenary (which tends to a parabola if the charge

velocity is much less than the speed of light c ).[37]

The surface of revolution with fixed radii at either end that has minimum surface area is a catenary

revolved about the x-axis.[33]

Analysis[edit] Model of chains and arches[edit]

In the mathematical model the chain (or cord, cable, rope, string, etc.) is idealized by assuming that it is

so thin that it can be regarded as a curve and that it is so flexible any force of tension exerted by the

chain is parallel to the chain.[38] The analysis of the curve for an optimal arch is similar except that the

forces of tension become forces of compressionand everything is inverted.[39]

An underlying principle is

that the chain may be considered a rigid body once it has attained equilibrium.[40]

Equations which define

the shape of the curve and the tension of the chain at each point may be derived by a careful inspection

of the various forces acting on a segment using the fact that these forces must be in balance if the chain

is in static equilibrium.

Let the path followed by the chain be given parametrically by r = ( x , y ) = ( x (s), y (s))

where s represents arc length and r is the position vector . This is the natural parameterizationand has

the property that

where u is a unit tangent vector .

http://en.wikipedia.org/wiki/Parabola

http://en.wikipedia.org/wiki/Roulette_(curve)

http://en.wikipedia.org/wiki/Conic_section#Eccentricity.2C_focus_and_directrix

http://en.wikipedia.org/wiki/Catenary#cite_note-Yates_13-33

http://en.wikipedia.org/wiki/Envelope_(mathematics)

http://en.wikipedia.org/wiki/Involute

http://en.wikipedia.org/wiki/Tractrix

http://en.wikipedia.org/wiki/Square_wheel

http://en.wikipedia.org/wiki/Regular_polygon

http://en.wikipedia.org/wiki/Electric_charge

http://en.wikipedia.org/wiki/Electric_field

http://en.wikipedia.org/wiki/Speed_of_light

http://en.wikipedia.org/wiki/Surface_of_revolution

http://en.wikipedia.org/wiki/Mathematical_model

http://en.wikipedia.org/wiki/Curve

http://en.wikipedia.org/wiki/Tension_(physics)

http://en.wikipedia.org/wiki/Compression_(physical)

http://en.wikipedia.org/wiki/Static_equilibrium

http://en.wikipedia.org/wiki/Parametric_equations

http://en.wikipedia.org/wiki/Arc_length

http://en.wikipedia.org/wiki/Position_vector

http://en.wikipedia.org/wiki/Unit_speed_parametrization

http://en.wikipedia.org/wiki/Unit_tangent_vector

http://en.wikipedia.org/wiki/File:CatenaryForceDiagram.svg

http://en.wikipedia.org/wiki/Unit_tangent_vector

http://en.wikipedia.org/wiki/Unit_speed_parametrization

http://en.wikipedia.org/wiki/Position_vector

http://en.wikipedia.org/wiki/Arc_length

http://en.wikipedia.org/wiki/Parametric_equations

http://en.wikipedia.org/wiki/Static_equilibrium

http://en.wikipedia.org/wiki/Compression_(physical)

http://en.wikipedia.org/wiki/Tension_(physics)

http://en.wikipedia.org/wiki/Curve

http://en.wikipedia.org/wiki/Mathematical_model

http://en.wikipedia.org/wiki/Surface_of_revolution

http://en.wikipedia.org/wiki/Speed_of_light

http://en.wikipedia.org/wiki/Electric_field

http://en.wikipedia.org/wiki/Electric_charge

http://en.wikipedia.org/wiki/Regular_polygon

http://en.wikipedia.org/wiki/Square_wheel

http://en.wikipedia.org/wiki/Tractrix

http://en.wikipedia.org/wiki/Involute

http://en.wikipedia.org/wiki/Envelope_(mathematics)

http://en.wikipedia.org/wiki/Roulette_(curve)

Diagram of forces acting on a segment of a catenary from c to r . The forces are the tension T0 at c, the

tension T at r , and the weight of the chain (0, −λgs). Since the chain is at rest the sum of these forces must be

zero.

A differential equation for the curve may be derived as follows.[41]

Let c be the lowest point on the

chain, called the vertex of the catenary,[42]

and measure the parameter s from c. Assume r is to the

right of c since the other case is implied by symmetry. The forces acting on the section of the chain

from c to r are the tension of the chain at c, the tension of the chain at r , and the weight of the chain.

The tension at c is tangent to the curve at c and is therefore horizontal, and it pulls the section to theleft so it may be written (−T 0, 0) where T 0 is the magnitude of the force. The tension at r is parallel to

the curve at r and pulls the section to the right, so it may be written T u=(T cos φ, T sin φ), where T is

the magnitude of the force and φ is the angle between the curve at r and the x -axis (see tangential

angle). Finally, the weight of the chain is represented by (0, −λgs) where λ is the mass per unit

length, g is the acceleration of gravity and s is the length of chain between c and r .

The chain is in equilibrium so the sum of three forces is 0, therefore

and

and dividing these gives

It is convenient to write

which is the length of chain whose weight is equal in magnitude to the tension

at c.[43]

Then

is an equation defining the curve.

The horizontal component of the tension, T cos φ = T 0 is constant and the

vertical component of the tension, T sin φ = λgs is proportional to the length of

chain between the r and the vertex.[44]

Derivation of equations for the curve[edit]

The differential equation given above can be solved to produce equations for

the curve.[45]

From

the formula for arc length gives

Then

http://en.wikipedia.org/wiki/Differential_equation

http://en.wikipedia.org/wiki/Tangential_angle

http://en.wikipedia.org/wiki/Catenary#cite_note-Routh_Art_443_318-44

http://en.wikipedia.org/wiki/Arc_length#Finding_arc_lengths_by_integrating

http://en.wikipedia.org/wiki/Tangential_angle

http://en.wikipedia.org/wiki/Differential_equation

and

The second of these equations can be integrated to give

and by shifting the position of the x -axis, β can be taken to

be 0. Then

The x -axis thus chosen is called the directrix of the

catenary.

It follows that the magnitude of the tension at a

point T = λgy which is proportional to the distancebetween the point and the directrix.

[44]

The integral of expression for dx /ds can be found

using standard techniques giving[46]

and, again, by shifting the position of the y -axis, α

can be taken to be 0. Then

The y -axis thus chosen passes though the

vertex and is called the axis of the catenary.

These results can be used to eliminate s giving

Alternative derivation[edit]

The differential equation can be solved

using a different approach.[47]

From

it follows that

and

Integrating gives,

http://en.wikipedia.org/wiki/List_of_integrals_of_irrational_functions

and

As before,

the x and y -axes can

be shifted so α and β

can be taken to be 0.

Then

and taking the

reciprocal of both

sides

Adding and

subtracting

the last two

equations

then gives the

solution

and

Determiningpar ameter s[edit]

In

gener al the

para

meter

a an

d the

positi

on of

the

axisand

direct

rix

are

not

given

but

must

be

deter

mine

d

from

other

infor

matio

n.

Typic

ally,

the

infor

matio

n

given

is

that

the

caten

ary is

susp

ende

d at

given

point

s P 1

and

P 2 an

d

with

given

lengt

h s.

The

equat

ion

can

be

deter

mined in

this

case

as

follow

s:[48]

Rela

bel if

nece

ssary

so

that

P 1 is

to the

left

of P 2

and

let h

be

the

horiz

ontal

and v

be

the

vertic

al

dista

nce

from

P 1 to

P 2. Tr

ansla

te the

axes

so

that

the

verte

x of

the

caten

ary

lies

on

the y-

axis

and

itsheigh

t a is

adjus

ted

so

http://en.wikipedia.org/wiki/Translation_(geometry)

the

caten

ary

satisfi

es

the

stand

ard

equat

ion of

the

curve

a

n

d

le

t

h

e

c

o

r

d

i

n

a

t

e

s

o

f

P

1

a

n

d

P

2

b

e

(

x

1,

y

1)

a

n

d

(

x

2,

y

2)

r

e

s

p

e

c

ti

v

e

ly

.

T

h

e

c

u

r

v

e

p

a

s

e

s

t

h

r

o

u

g

ht

h

e

s

e

p

o

i

n

t

s

,

s

o

t

h

e

d

if

f

e

r

e

n

c

e

o

f

h

e

i

g

h

t

is

http://en.wikipedia.org/wiki/Numerical_analysis

http://en.wikipedia.org/wiki/Suspension_bridge

http://en.wikipedia.org/wiki/Golden_Gate_Bridge

http://en.wikipedia.org/wiki/Davies_Gilbert

http://en.wikipedia.org/wiki/Gaspard-Gustave_Coriolis

http://en.wikipedia.org/wiki/Spring_(device)

http://en.wikipedia.org/wiki/Elasticity_(physics)

http://en.wikipedia.org/wiki/Spring_(device)

http://en.wikipedia.org/wiki/Hooke%27s_Law

http://en.wikipedia.org/wiki/Stiffness

http://en.wikipedia.org/wiki/Limit_(mathematics)

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