Serret-Frenet equations

Serret-Frenet equations

Let

I

⊂

ℝ

be an

interval

, and let

γ

:

I

→

ℝ

3

be
an

arclength parameterization

of an oriented space curve, assumed to
be

regular

(

http://planetmath.org/Curve

), and free of

points of inflection

. Let

T

⁢

(

s

)

,

N

⁢

(

s

)

,

B

⁢

(

s

)

denote the corresponding

moving trihedron

, and

κ

⁢

(

s

)

,

τ

⁢

(

s

)

the corresponding

curvature

(

http://planetmath.org/CurvatureOfACurve

)
and

torsion

functions (

http://planetmath.org/Torsion

). The following

differential

relations

, called the

Serret-Frenet equations

, hold
between these three vectors.

T

′

⁢

(

s

)

=

κ

⁢

(

s

)

⁢

N

⁢

(

s

)

;

(1)

N

′

⁢

(

s

)

=

-

κ

⁢

(

s

)

⁢

T

⁢

(

s

)

+

τ

⁢

(

s

)

⁢

B

⁢

(

s

)

;

(2)

B

′

⁢

(

s

)

=

-

τ

⁢

(

s

)

⁢

N

⁢

(

s

)

.

(3)

Equation

(

1

) follows directly from the

definition

of the
normal (

http://planetmath.org/MovingFrame

)

N

⁢

(

s

)

and from the definition of the
curvature (

http://planetmath.org/CurvatureAndTorsion

),

κ

⁢

(

s

)

. Taking the

derivative

of
the relation

N

⁢

(

s

)

⋅

T

⁢

(

s

)

=

0

,

gives

N

′

⁢

(

s

)

⋅

T

⁢

(

s

)

=

-

T

′

⁢

(

s

)

⋅

N

⁢

(

s

)

=

-

κ

⁢

(

s

)

.

Taking the derivative of the relation

N

⁢

(

s

)

⋅

N

⁢

(

s

)

=

1

,

gives

N

′

⁢

(

s

)

⋅

N

⁢

(

s

)

=

0

.

By the definition of torsion (

http://planetmath.org/CurvatureAndTorsion

), we have

N

′

⁢

(

s

)

⋅

B

⁢

(

s

)

=

τ

⁢

(

s

)

.

This proves equation (

2

).
Finally,
taking derivatives of the relations

T

⁢

(

s

)

⋅

B

⁢

(

s

)

=

0

,

N

⁢

(

s

)

⋅

B

⁢

(

s

)

=

0

,

B

⁢

(

s

)

⋅

B

⁢

(

s

)

=

1

,

and making use of (

1

) and (

2

)
gives

B

′

⁢

(

s

)

⋅

T

⁢

(

s

)

=

-

T

′

⁢

(

s

)

⋅

B

⁢

(

s

)

=

0

,

B

′

⁢

(

s

)

⋅

N

⁢

(

s

)

=

-

N

′

⁢

(

s

)

⋅

B

⁢

(

s

)

=

-

τ

⁢

(

s

)

,

B

′

⁢

(

s

)

⋅

B

⁢

(

s

)

=

0

.

This proves equation (

3

).

It is also convenient to describe the Serret-Frenet equations by using
matrix notation. Let

F

:

I

→

SO

⁡

(

3

)

(see -

special orthogonal
group

), the mapping defined by

F

⁢

(

s

)

=

(

T

⁢

(

s

)

,

N

⁢

(

s

)

,

B

⁢

(

s

)

)

,

s

∈

I

represent the Frenet frame as a

3

×

3

orthonormal
matrix. Equations (

1

) (

2

) (

3

) can be
succinctly given as

F

⁢

(

s

)

-

1

⁢

F

′

⁢

(

s

)

=

(

0

κ

⁢

(

s

)

0

-

κ

⁢

(

s

)

0

τ

⁢

(

s

)

0

-

τ

⁢

(

s

)

0

)

In this formulation, the above relation is also known as the structure
equations of an oriented space curve.

Title

Serret-Frenet equations

Canonical

name

SerretFrenetEquations

Date of creation

2013-03-22 12:15:13

Last modified on

2013-03-22 12:15:13

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

20

Author

rmilson (146)

Entry type

Theorem

Classification

msc 53A04

Synonym

Frenet equations

Synonym

Frenet-Serret equations

Synonym

Frenet-Serret formulas

Synonym

Serret-Frenet formulas

Synonym

Frenet formulas

Related topic

SpaceCurve

Related topic

Torsion

Related topic

CurvatureOfACurve

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