curvature determines the curve

curvature determines the curve

The

curvature

(

http://planetmath.org/CurvaturePlaneCurve

) of

plane curve

determines uniquely the form and of the curve, i.e. one has the

Theorem.

If

s

↦

k

⁢

(

s

)

is a continuous real function, then there exists always plane curves satisfying the equation

κ

=

k

⁢

(

s

)

(1)

between their curvature

κ

and the arc length

s

.  All these curves are

congruent

(

http://planetmath.org/

Congruence

).

Proof.

Suppose that a curve

C

satisfies the condition (1).  Let the value

s

=

0

correspond to the point

P

0

of this curve.  We choose

O

as the origin of the plane.  The

tangent

and the normal of

C

in

O

are chosen as the

x

-axis and the

y

-axis, with

positive

directions the directions of the tangent and normal vectors of

C

, respectively.  According to (1) and the definition of curvature, the equation

d

⁢

θ

d

⁢

s

=

k

⁢

(

s

)

for the direction angle

θ

of the tangent of

C

is valid in this

coordinate system

; the

initial condition

is

θ

=

0

when

s

=

0

.

Thus we get

θ

=

∫

0

s

k

⁢

(

t

)

⁢

𝑑

t

:=

ϑ

⁢

(

s

)

,

(2)

which implies

d

⁢

x

d

⁢

s

=

cos

⁡

ϑ

⁢

(

s

)

,

d

⁢

y

d

⁢

s

=

sin

⁡

ϑ

⁢

(

s

)

.

(3)

Since

x

=

y

=

0

when

s

=

0

, we obtain

x

=

∫

0

s

cos

⁡

ϑ

⁢

(

t

)

⁢

𝑑

t

,

y

=

∫

0

s

sin

⁡

ϑ

⁢

(

t

)

⁢

𝑑

t

.

(4)

Thus the function

s

↦

k

⁢

(

s

)

determines uniquely these functions

x

and

y

of the parameter

s

, and (4) represents a curve with definite form and .

The above reasoning shows that every curve which satisfies (1) is congruent (

http://planetmath.org/Congruence

) with the curve (4).

We have still to show that the curve (4) satisfies the condition (1).  By differentiating (

http://planetmath.org/HigherOrderDerivatives

) the equations (4) we get the equations (3), which imply

(

d

⁢

x

d

⁢

s

)

2

+

(

d

⁢

y

d

⁢

s

)

2

=

1

,  or

d

⁢

s

2

=

d

⁢

x

2

+

d

⁢

y

2

which means that the parameter

s

represents the arc length of the curve (4), counted from the origin.  Differentiating (3) we get, because

ϑ

′

⁢

(

s

)

=

k

⁢

(

s

)

by (2),

d

2

⁢

x

d

⁢

s

2

=

-

k

⁢

(

s

)

⁢

sin

⁡

ϑ

⁢

(

s

)

,

d

2

⁢

y

d

⁢

s

2

=

k

⁢

(

s

)

⁢

cos

⁡

ϑ

⁢

(

s

)

.

(5)

The equations (3) and (5) then yield

d

⁢

x

d

⁢

s

⁢

d

2

⁢

y

d

⁢

s

2

-

d

⁢

y

d

⁢

s

⁢

d

2

⁢

x

d

⁢

s

2

=

k

⁢

(

s

)

,

i.e. the curvature of (4), according the parent entry (

http://planetmath.org/CurvaturePlaneCurve

), satisfies

|

x

′

y

′

x

′′

y

′′

|

=

k

⁢

(

s

)

.

Thus the proof is settled.

References

1

Ernst Lindelöf

:

Differentiali- ja integralilasku
ja sen sovellutukset I

.  WSOY. Helsinki (1950).

Title

curvature determines the curve

Canonical name

CurvatureDeterminesTheCurve

Date of creation

2016-02-22 16:14:25

Last modified on

2016-02-22 16:14:25

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Theorem

Classification

msc 53A04

Synonym

fundamental theorem of plane curves

Related topic

FundamentalTheoremOfSpaceCurves

Related topic

ErnstLindelof

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