evolute

evolute

The locus of the

center of curvature

of a

plane curve

is called the

evolute

of this curve.

The

coordinates

of the center of curvature belonging to the point

P

=

(

x

,

y

)

of the curve

γ

are

ξ

=

x

-

ϱ

⁢

sin

⁡

α

,

η

=

y

+

ϱ

⁢

cos

⁡

α

,

(1)

where

ϱ

is the

radius of curvature

in

P

and

α

is the

slope angle

of the

tangent line

of the curve in

P

.  So (1) may be regarded as the equations of the evolute of

γ

.

If the plane curve is given in the parametric form

x

=

x

⁢

(

t

)

,

y

=

y

⁢

(

t

)

,  the corresponding

parametric equations

of the evolute are

ξ

=

x

-

(

x

′

⁣

2

+

y

′

⁣

2

)

⁢

y

′

x

′

⁢

y

′′

-

x

′′

⁢

y

′

,

η

=

y

+

(

x

′

⁣

2

+

y

′

⁣

2

)

⁢

x

′

x

′

⁢

y

′′

-

x

′′

⁢

y

′

.

In the spexial case that the curve is given in the form

y

=

y

⁢

(

x

)

these equations can be written

ξ

=

x

-

(

1

+

y

′

⁣

2

)

⁢

y

′

y

′′

,

η

=

y

+

1

+

y

′

⁣

2

y

′′

.

For examining the properties of the evolute we choose for parameter the

arc length

s

, measured from a certain point of the curve; then in (1) the quantities

x

,

y

,

ϱ

,

α

and thus

ξ

and

η

are functions of

s

. We assume that all needed

derivatives

exist and are

continuous

.

Differentiating (1) with respect to

s

, we obtain

d

⁢

ξ

d

⁢

s

=

d

⁢

x

d

⁢

s

-

ϱ

⁢

d

⁢

α

d

⁢

s

⁢

cos

⁡

α

-

d

⁢

ϱ

d

⁢

s

⁢

sin

⁡

α

,

d

⁢

η

d

⁢

s

=

d

⁢

y

d

⁢

s

-

ϱ

⁢

d

⁢

α

d

⁢

s

⁢

sin

⁡

α

+

d

⁢

ϱ

d

⁢

s

⁢

cos

⁡

α

,

and recalling that

d

⁢

x

d

⁢

s

=

cos

⁡

α

,

d

⁢

y

d

⁢

s

=

sin

⁡

α

and

ϱ

⁢

d

⁢

α

d

⁢

s

=

1

it yields

d

⁢

ξ

d

⁢

s

=

-

d

⁢

ϱ

d

⁢

s

⁢

sin

⁡

α

,

d

⁢

η

d

⁢

s

=

d

⁢

ϱ

d

⁢

s

⁢

cos

⁡

α

.

(2)

If

d

⁢

ϱ

d

⁢

s

≠

0

in the point

(

x

,

y

)

of

γ

, the derivatives

d

⁢

ξ

d

⁢

s

and

d

⁢

η

d

⁢

s

do not vanish simultaneously, and so the evolute has in the corresponding point

(

ξ

,

η

)

a tangent line with the slope

d

⁢

η

d

⁢

s

:

d

⁢

ξ

d

⁢

s

=

-

1

tan

⁡

α

.

Since the of this is the slope of the

normal line

of the given curve

γ

, we have the

Theorem 1.

The normal line of the curve in a point

(

x

,

y

)

,  where

d

⁢

ϱ

d

⁢

s

≠

0

, is the tangent line of the evolute, having as tangency point the corresponding center of curvature

(

ξ

,

η

)

.  Thus the evolute is the

envelope

of the normal lines of the curve.

We shall calculate the arc length

σ

of the evolute corresponding the arc of the curve

γ

which is passed through when the parameter

s

grows from

s

1

to

s

2

; we assume that

ϱ

and

d

⁢

ϱ

d

⁢

s

are then continuous and distinct from zero. According the arc length formula,

σ

=

∫

s

1

s

2

(

d

⁢

ξ

d

⁢

s

)

2

+

(

d

⁢

η

d

⁢

s

)

2

⁢

𝑑

s

.

Using the equations (2) and the fact that the sign of

d

⁢

ϱ

d

⁢

s

does not change, we can write

σ

=

∫

s

1

s

2

(

d

⁢

ϱ

d

⁢

s

)

2

⁢

𝑑

s

=

∫

s

1

s

2

|

d

⁢

ϱ

d

⁢

s

|

⁢

𝑑

s

=

|

∫

s

1

s

2

d

⁢

ϱ

d

⁢

s

⁢

𝑑

s

|

=

|

/

s

1

s

2

⁡

ϱ

|

=

|

ϱ

2

-

ϱ

1

|

,

where

ϱ

1

and

ϱ

2

are the corresponding of

γ

. We have proved the

Theorem 2.

The of an arc of the evolute is equal to the difference of the of the given curve touching the arc of the evolute in its

end points

, provided that

ϱ

and

d

⁢

ϱ

d

⁢

s

are continuous and do not change their sign on the arc of the curve.

References

1

Ernst Lindelöf

:

Differentiali- ja integralilasku
ja sen sovellutukset I

. Toinen painos. WSOY, Helsinki (1950).

Title

evolute

Canonical name

Evolute

Date of creation

2013-03-22 17:35:06

Last modified on

2013-03-22 17:35:06

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Topic

Classification

msc 53A04

Related topic

ConditionOfOrthogonality

Related topic

ArcLength

Related topic

BolzanosTheorem

Related topic

SubstitutionNotation

Defines

evolute

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