normal curvatures

normal curvatures

Let us determine the

normal curvatures

(

http://planetmath.org/NormalSection

)

ϰ

of the

surface

z

=

z

⁢

(

x

,

y

)

(1)

in the origin, when (1) has the

continuous

1st and 2nd order

partial derivatives

in a neighbourhood of

(

0

,

0

)

and satisfies

z

⁢

(

0

,

0

)

=

z

x

′

⁢

(

0

,

0

)

=

z

y

′

⁢

(

0

,

0

)

=

0

.

(2)

It’s a question of the

curvature

(

http://planetmath.org/CurvaturePlaneCurve

) of the intersection curves of the surface (1) and planes containing the

z

-axis, which is the normal of the surface in the origin.

If the angle between the

z

⁢

x

-plane and a plane

τ

containing the

z

-axis is denoted by

φ

, when the line of

intersection

of the plane

τ

and the

x

⁢

y

-plane is represented by the equations

x

=

ϱ

cos

φ

,

y

=

ϱ

sin

φ

(

-

∞

<

ϱ

<

∞

)

,

then equation of the the

normal section

curve

C

φ

is

z

=

z

⁢

(

ϱ

⁢

cos

⁡

φ

,

ϱ

⁢

sin

⁡

φ

)

,

where

ϱ

is the abscissa and

z

the ordinate.  It follows that

d

⁢

z

d

⁢

ϱ

=

∂

⁡

z

∂

⁡

x

⁢

cos

⁡

φ

+

∂

⁡

z

∂

⁡

y

⁢

sin

⁡

φ

,

d

2

⁢

z

d

⁢

ϱ

2

=

∂

2

⁡

z

∂

⁡

x

2

⁢

cos

2

⁡

φ

+

2

⁢

∂

2

⁡

z

∂

⁡

x

⁢

∂

⁡

y

⁢

sin

⁡

φ

⁢

cos

⁡

φ

+

∂

2

⁡

z

∂

⁡

y

2

⁢

sin

2

⁡

φ

;

thus by (2), in the origin we have

d

⁢

z

d

⁢

ϱ

=

0

,

d

2

⁢

z

d

⁢

ϱ

2

=

a

⁢

cos

2

⁡

φ

+

2

⁢

b

⁢

sin

⁡

φ

⁢

cos

⁡

φ

+

c

⁢

sin

2

⁡

φ

,

where

a

,

b

,

c

the values of the

derivatives

∂

2

⁡

z

∂

⁡

x

2

,

∂

2

⁡

z

∂

⁡

x

⁢

∂

⁡

y

,

∂

2

⁡

z

∂

⁡

y

2

in the origin.

Using those values, we obtain for the normal curvature of

C

φ

in the origin the value

ϰ

⁢

(

φ

)

=

[

d

2

⁢

z

d

⁢

ϱ

2

(

1

+

(

d

⁢

z

d

⁢

ϱ

)

2

)

3

/

2

]

ϱ

=

0

=

a

⁢

cos

2

⁡

φ

+

2

⁢

b

⁢

sin

⁡

φ

⁢

cos

⁡

φ

+

c

⁢

sin

2

⁡

φ

.

(3)

This result gets a more illustrative form when we try to express it by using the least and the greatest value of

ϰ

⁢

(

φ

)

.  Instead to utilize the zeros of the derivative of the sum in (3), it’s simpler first to transfer to the double angle (

http://planetmath.org/DoubleAngleIdentity

),

ϰ

⁢

(

φ

)

=

a

+

c

2

+

a

-

c

2

⁢

cos

⁡

2

⁢

φ

+

b

⁢

sin

⁡

2

⁢

φ

,

(4)

and here to introduce an auxiliary angle

α

(

0

≤

α

<

π

) such that

a

-

c

2

:=

k

⁢

cos

⁡

2

⁢

α

,

b

:=

k

⁢

sin

⁡

2

⁢

α

.

This allows us to write (4) as

ϰ

⁢

(

φ

)

=

a

+

c

2

+

k

⁢

cos

⁡

2

⁢

(

φ

-

α

)

.

(5)

From this we see immediately that the curvature attains its greatest and least value

a

+

c

2

±

k

when

φ

=

α

and

φ

=

α

+

π

2

.

Accordingly, the corresponding

τ

, the

principal normal planes

, are

perpendicular

to each other; their normal section curves on the surface (1) in the origin are briefly called the

principal sections

.

The expression (5) of the normal curvature may still be edited.  Let us take a new

parameter

angle

φ

-

α

:=

θ

.  One can write

ϰ

⁢

(

φ

)

=

a

+

c

2

⁢

(

cos

2

⁡

θ

+

sin

2

⁡

θ

)

+

k

⁢

(

cos

2

⁡

θ

-

sin

2

⁡

θ

)

=

(

a

+

c

2

+

k

)

⁢

cos

2

⁡

θ

+

(

a

+

c

2

-

k

)

⁢

sin

2

⁡

θ

:=

ϰ

θ

.

So the final result, the so-called

Euler’s theorem (

http://planetmath.org/SecondFundamentalForm

)

, can be expressed in the form

ϰ

θ

=

ϰ

1

⁢

cos

2

⁡

θ

+

ϰ

2

⁢

sin

2

⁡

θ

.

(6)

Here, the

principal curvatures

ϰ

1

and

ϰ

2

are the greatest and the least value of the normal curvature, respectively, and

θ

is the angle between (

http://planetmath.org/AngleBetweenTwoPlanes

) the normal section plane corresponding

ϰ

1

and the normal section plane corresponding

ϰ

θ

. As it becomes clear in the parent entry (

http://planetmath.org/NormalSection

), the result (6) is true not only in the origin but at any point on a surface when the given

function

has the continuous 1st and 2nd derivatives in some neighbourhood of the point.

References

1

Ernst Lindelöf

:

Differentiali- ja integralilasku
ja sen sovellutukset II

.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1932).

Title

normal curvatures

Canonical name

NormalCurvatures

Date of creation

2013-03-22 17:26:27

Last modified on

2013-03-22 17:26:27

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Topic

Classification

msc 53A05

Classification

msc 26B05

Classification

msc 26A24

Related topic

SecondFundamentalForm

Related topic

MeusniersTheorem

Related topic

ErnstLindelof

Defines

principal normal plane

Defines

principal section

Defines

principal curvature

Defines

Euler’s theorem

Generated on Fri Feb 9 22:23:47 2018 by

LaTeXML