integral representation of length of smooth curve

integral representation of length of smooth curve

Suppose

γ

:

[

0

,

1

]

→

ℝ

m

is a

continuously differentiable

curve. Then the definition of its length
as a

rectifiable curve

L

=

sup

⁡

{

∑

i

=

1

n

∥

γ

⁢

(

t

i

)

-

γ

⁢

(

t

i

-

1

)

∥

:

0

=

t

0

<

t

1

<

⋯

<

t

n

=

1

,

n

∈

ℕ

}

is equal to its length as computed in

differential geometry

:

∫

0

1

∥

γ

′

⁢

(

t

)

∥

⁢

𝑑

t

.

Proof.

Let the

partition

{

t

i

}

of

[

0

,

1

]

be arbitrary. Then

∑

i

=

1

n

∥

γ

⁢

(

t

i

)

-

γ

⁢

(

t

i

-

1

)

∥

=

∑

i

=

1

n

∥

∫

t

i

-

1

t

i

γ

′

⁢

(

t

)

⁢

𝑑

t

∥

(

fundamental theorem of calculus

)

≤

∑

i

=

1

n

∫

t

i

-

1

t

i

∥

γ

′

⁢

(

t

)

∥

⁢

𝑑

t

(

triangle inequality

for

integrals

)

=

∫

0

1

∥

γ

′

⁢

(

t

)

∥

⁢

𝑑

t

.

Hence

L

≤

∫

0

1

∥

γ

′

⁢

(

t

)

∥

⁢

𝑑

t

.
(By the way, this also shows that

γ

is rectifiable in the first place.)

The

inequality

in the other direction is more tricky.
Given

ϵ

>

0

, we know that

∫

0

1

∥

γ

′

⁢

(

t

)

∥

⁢

𝑑

t

can be approximated up to

ϵ

by a

Riemann sum

of the form

∑

i

=

1

n

∥

γ

′

⁢

(

t

i

-

1

)

∥

⁢

(

t

i

-

t

i

-

1

)

provided the partition

{

t

i

}

is fine enough,
i.e. has mesh width

≤

Δ

for some small

Δ

>

0

.
We want to approximate

γ

′

⁢

(

t

i

-

1

)

with

[

γ

⁢

(

t

i

)

-

γ

⁢

(

t

i

-

1

)

]

/

(

t

i

-

t

i

-

1

)

, but this only works
if

t

i

-

t

i

-

1

is small.

To get the precise estimates, use

uniform continuity

of

γ

′

on

[

0

,

1

]

to obtain a

δ

>

0

such that

∥

γ

′

⁢

(

τ

)

-

γ

′

⁢

(

t

)

∥

≤

ϵ

whenever

|

τ

-

t

|

≤

δ

.
Then for all

0

<

h

≤

δ

and

t

∈

[

0

,

1

]

,

∥

γ

⁢

(

t

+

h

)

-

γ

⁢

(

t

)

h

-

γ

′

⁢

(

t

)

∥

≤

1

h

⁢

∫

t

t

+

h

∥

γ

′

⁢

(

τ

)

-

γ

′

⁢

(

t

)

∥

⁢

𝑑

τ

≤

h

h

⁢

ϵ

=

ϵ

.

Let the partition

{

t

i

}

have a mesh width less than both

δ

and

Δ

. Then
setting

h

=

t

i

-

t

i

-

1

successively in each summand,
we have

∫

0

1

∥

γ

′

⁢

(

t

)

∥

⁢

𝑑

t

≤

∑

i

=

1

n

∥

γ

′

⁢

(

t

i

-

1

)

∥

⁢

(

t

i

-

t

i

-

1

)

+

ϵ

≤

∑

i

=

1

n

∥

γ

⁢

(

t

i

)

-

γ

⁢

(

t

i

-

1

)

∥

t

i

-

t

i

-

1

⁢

(

t

i

-

t

i

-

1

)

+

∑

i

=

1

n

ϵ

⁢

(

t

i

-

t

i

-

1

)

+

ϵ

=

∑

i

=

1

n

∥

γ

⁢

(

t

i

)

-

γ

⁢

(

t

i

-

1

)

∥

+

2

⁢

ϵ

≤

L

+

2

⁢

ϵ

.

Taking

ϵ

→

0

yields

∫

0

1

∥

γ

′

⁢

(

t

)

∥

⁢

𝑑

t

≤

L

.
∎

We remark that

L

=

∫

0

1

∥

γ

′

⁢

(

t

)

∥

⁢

𝑑

t

is true for

piecewise smooth

curves

γ

also, simply by adding together the results for each smooth

segment

of

γ

.

Title

integral representation of length of smooth curve

Canonical name

IntegralRepresentationOfLengthOfSmoothCurve

Date of creation

2013-03-22 15:39:39

Last modified on

2013-03-22 15:39:39

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

11

Author

stevecheng (10074)

Entry type

Derivation

Classification

msc 51N05

Related topic

ArcLength

Related topic

Rectifiable

Related topic

TotalVariation

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