aarjaneiro/planetmath huggingface dataset:
classical differential geometry

classical differential geometry

Curves in

ℝ

2

•

inflexion point

•

singular points of plane curve

•

isocline

•

curvature

(

plane curve

)

•

circle of curvature

•

curvature determines the curve

•

curvature of Nielsen’s spiral

•

osculating curve

•

orthogonal curves

•

isogonal trajectory

•

parallel curves

•

properties of parallel curves

•

evolute

•

evolute of cycloid

•

Serret-Frenet equations

in

ℝ

2

(

http://planetmath.org/SerretFrenetEquationsInMathbbR2

)

•

famous curves

in the plane

•

arc-parametrizations

•

envelope

•

determining envelope

•

catacaustic

Curves in

ℝ

3

•

Serret-Frenet equations in

ℝ

3

(

http://planetmath.org/SerretFrenetFormulas

)

•

space curve

•

level curve

•

curvature (

http://planetmath.org/CurvatureOfACurve

) and

torsion

(

http://planetmath.org/Torsion

) of a space curve

•

moving trihedron

Surfaces in

ℝ

3

•

level curve,

level surface

•

surface of revolution

•

surface normal

•

normal section

•

normal curvatures

•

Meusnier’s theorem

•

mean curvature at surface point

•

first fundamental form

(

http://planetmath.org/FirstFundamentalForm

)

•

second fundamental form

•

sphere map and

shape operator

•

Gaussian curvature

and

mean curvature

•

geodesic

•

Gauss-Bonnet theorem

•

standard

connection

in

ℝ

3

•

Gauss equation

The space

ℝ

3

•

ortho-normal frame fields in

ℝ

3

(or non constant ortho-normal triples of vector fields)

•

rate of

rotation

of an o.f.f.

•

euclidean

spin connection

•

ℝ

3

Cartan structural equations

I, II

Variational calculus

•

calculus of variations

•

classical isoperimetric problem

•

least surface of revolution

•

brachistochrone curve

•

equation of catenary via calculus of variations

Title

classical differential geometry

Canonical name

ClassicalDifferentialGeometry

Date of creation

2013-03-22 15:17:03

Last modified on

2013-03-22 15:17:03

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

39

Author

matte (1858)

Entry type

Topic

Classification

msc 53A05

Classification

msc 53A04

Related topic

DifferentialGeometry

Generated on Fri Feb 9 22:23:10 2018 by

LaTeXML
derivation of Pappus’s centroid theorem

derivation of Pappus’s centroid theorem

I.

Let

s

denote the arc rotating about the

x

-axis (and its length) and

R

be the

y

-

coordinate

of the

centroid

of the arc.  If the arc may be given by the equation

y

=

y

⁢

(

x

)

where

a

≤

x

≤

b

, the area of the formed

surface of revolution

is

A

=

2

⁢

π

⁢

∫

a

b

y

⁢

(

x

)

⁢

1

+

[

y

′

⁢

(

x

)

]

2

⁢

𝑑

x

.

This can be concisely written

A

=

2

⁢

π

⁢

∫

s

y

⁢

𝑑

s

(1)

since differential-geometrically, the product

1

+

[

y

′

⁢

(

x

)

]

2

⁢

d

⁢

x

is the arc-element.  We rewrite (1) as

A

=

s

⋅

2

⁢

π

⋅

1

s

⁢

∫

s

y

⁢

𝑑

s

.

Here, the last factor is the

ordinate

of the centroid of the rotating arc, whence we have the result

A

=

s

⋅

2

⁢

π

⁢

R

which states the first Pappus’s centroid theorem.

II.

For deriving the second Pappus’s centroid theorem, we suppose that the region defined by

a

≤

x

≤

b

,

0

≤

y

1

⁢

(

x

)

≤

y

≤

y

2

⁢

(

x

)

,

having the area

A

and the centroid with the ordinate

R

, rotates about the

x

-axis and forms the

solid of revolution

with the volume

V

.  The centroid of the area-element between the arcs

y

=

y

1

⁢

(

x

)

and

y

=

y

2

⁢

(

x

)

is

[

y

2

⁢

(

x

)

+

y

1

⁢

(

x

)

]

/

2

when the

abscissa

is

x

; the area of this element with the width

d

⁢

x

is

[

y

2

⁢

(

x

)

-

y

1

⁢

(

x

)

]

⁢

d

⁢

x

.  Thus we get the equation

R

=

1

A

⁢

∫

a

b

y

2

⁢

(

x

)

+

y

1

⁢

(

x

)

2

⁢

[

y

2

⁢

(

x

)

-

y

1

⁢

(

x

)

]

⁢

𝑑

x

which may be written shortly

R

=

1

2

⁢

A

⁢

∫

a

b

(

y

2

2

-

y

1

2

)

⁢

𝑑

x

.

(2)

The volume of the solid of revolution is

V

=

π

⁢

∫

a

b

(

y

2

2

-

y

1

2

)

⁢

𝑑

x

=

A

⋅

2

⁢

π

⋅

1

2

⁢

A

⁢

∫

a

b

(

y

2

2

-

y

1

2

)

⁢

𝑑

x

.

By (2), this attains the form

V

=

A

⋅

2

⁢

π

⁢

R

.

Title

derivation of Pappus’s centroid theorem

Canonical name

DerivationOfPappussCentroidTheorem

Date of creation

2013-03-22 19:36:11

Last modified on

2013-03-22 19:36:11

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Derivation

Classification

msc 53A05

Generated on Fri Feb 9 22:23:13 2018 by

LaTeXML
Gauss-Bonnet theorem

Gauss-Bonnet theorem

(Carl Friedrich Gauss and Pierre Ossian Bonnet) Given a two-dimensional

compact

Riemannian manifold

M

with

boundary

,

Gaussian curvature

of points

G

and

geodesic curvature

of points

g

x

on the boundary

∂

⁡

M

, it is the
case that

∫

M

G

⁢

𝑑

A

+

∫

∂

⁡

M

g

x

⁢

𝑑

s

=

2

⁢

π

⁢

χ

⁢

(

M

)

,

where

χ

⁢

(

M

)

is the

Euler characteristic

of the

manifold

,

d

⁢

A

denotes the measure with respect to area, and

d

⁢

s

denotes the measure with respect to arclength on the boundary. This theorem expresses a

topological invariant

in
terms of geometrical information.

Title

Gauss-Bonnet theorem

Canonical name

GaussBonnetTheorem

Date of creation

2013-03-22 16:36:37

Last modified on

2013-03-22 16:36:37

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 53A05

Generated on Fri Feb 9 22:23:15 2018 by

LaTeXML
Gauss-Bonnet theorem for surfaces without boundary

Gauss-Bonnet theorem for surfaces without boundary

If

S

is a compact,

orientable surface

without boundary, then

∫

S

K

=

2

⁢

π

⁢

χ

⁢

(

S

)

,

where

K

is the

Gaussian curvature

of

S

and

χ

⁢

(

S

)

its
Euler-Poincaré characteristic. (

http://planetmath.org/EulerrCharacteristic

)

Title

Gauss-Bonnet theorem for surfaces without boundary

Canonical name

GaussBonnetTheoremForSurfacesWithoutBoundary

Date of creation

2013-03-22 16:37:30

Last modified on

2013-03-22 16:37:30

Owner

Simone (5904)

Last modified by

Simone (5904)

Numerical id

8

Author

Simone (5904)

Entry type

Theorem

Classification

msc 53A05

Generated on Fri Feb 9 22:23:17 2018 by

LaTeXML
Gaussian curvature

Gaussian curvature

The

Gaussian curvature

of a surface at a point

p

is the product

K

=

κ

1

⁢

κ

2

of the two

principal curvatures

of the surface at

p

.

The

arithmetic mean

of the principal curvatures at a point

p

H

=

κ

1

+

κ

2

2

is called the

mean curvature

of the surface at

p

.

Title

Gaussian curvature

Canonical name

GaussianCurvature

Date of creation

2013-03-22 17:00:56

Last modified on

2013-03-22 17:00:56

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

6

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 53A05

Synonym

total curvature

Synonym

total normal curvature

Related topic

MeanCurvatureAtSurfacePoint

Defines

mean curvature

Generated on Fri Feb 9 22:23:19 2018 by

LaTeXML
level curve

level curve

The

level curves

(in German

Niveaukurve

, in French

ligne de niveau

) of a

surface

z

=

f

⁢

(

x

,

y

)

(1)

in

ℝ

3

are the intersection curves of the surface and the planes

z

=

constant

. Thus the

projections

of the level curves on the

x

⁢

y

-plane have equations of the form

f

⁢

(

x

,

y

)

=

c

(2)

where

c

is a constant.

For example, the level curves of the

hyperbolic paraboloid

(

http://planetmath.org/RuledSurface

)

z

=

x

⁢

y

are the

rectangular hyperbolas

x

⁢

y

=

c

(cf. this entry (

http://planetmath.org/GraphOfEquationXyConstant

)).

The

gradient

f

x

′

⁢

(

x

,

y

)

⁢

i

→

+

f

y

′

⁢

(

x

,

y

)

⁢

j

→

of the function

f

in any point of the surface (1) is

perpendicular

to the level curve (2), since the slope of the gradient is

f

y

′

f

x

′

and the slope of the level curve is

-

f

x

′

f

y

′

, whence the slopes are opposite

inverses

.

Analogically one can define the

level surfaces

(or

contour surfaces

)

F

⁢

(

x

,

y

,

z

)

=

c

(3)

for a function

F

of three variables

x

,

y

,

z

. The gradient of

F

in a point

(

x

,

y

,

z

)

is

parallel

to the

surface normal

of the level surface

passing through

this point.

Title

level curve

Canonical name

LevelCurve

Date of creation

2013-03-22 17:35:27

Last modified on

2013-03-22 17:35:27

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Definition

Classification

msc 53A05

Classification

msc 53A04

Classification

msc 51M04

Synonym

contour curve

Synonym

isopleth

Related topic

LevelSet

Related topic

ConvexAngle

Defines

level surface

Defines

contour surface

Generated on Fri Feb 9 22:23:21 2018 by

LaTeXML
line of curvature

line of curvature

A line

γ

on a surface

S

is a

line of curvature

of

S

, if in every point of

γ

one of the

principal sections

has common

tangent

with

γ

.

By the parent entry (

http://planetmath.org/NormalCurvatures

), a surface

F

⁢

(

x

,

y

,

z

)

=

0

,  where

F

has

continuous

first and

partial derivatives

, has two distinct families of lines of curvature, which families are

orthogonal

(

http://planetmath.org/ConvexAngle

) to each other.

For example, the

meridian curves

and the circles of latitude are the two families of the lines of curvature on a

surface of revolution

.

On a

developable surface

, the other family of its curvature lines consists of the

generatrices

of the surface.

A necessary and

sufficient

condition for that the

surface normals

of a surface

S

set along a curve

c

on

S

would form a developable surface, is that

c

is a line of curvature of

S

.

Title

line of curvature

Canonical name

LineOfCurvature

Date of creation

2013-03-22 18:08:44

Last modified on

2013-03-22 18:08:44

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Definition

Classification

msc 53A05

Classification

msc 26B05

Classification

msc 26A24

Synonym

curvature line

Related topic

TiltCurve

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

LaTeXML
loxodrome

loxodrome

Think in

ℝ

3

a sphere with radius

r

and two

antipodal points

of it wich we call the

North pole

and the

South pole

.

Meridians

are

great circles

passing through

the .  A

loxodrome

is a curve on the sphere intersecting all meridians at the same angle.

Let

{

x

=

r

⁢

sin

⁡

u

⁢

cos

⁡

v

y

=

r

⁢

sin

⁡

u

⁢

sin

⁡

v

z

=

r

⁢

cos

⁡

u

be a

parametric presentation

of the sphere (cf. the

spherical coordinates

).  We will show that

b

⁢

v

=

ln

⁡

tan

⁡

u

2

+

c

,

(1)

where

b

and

c

are constants, is an equation of loxodromes in the

Gaussian coordinates

u

,

v

.

We denote

1

b

:=

a

,  whence the equation of the family (1) in the parameter plane reads

v

=

a

⁢

ln

⁡

tan

⁡

u

2

+

a

⁢

c

:=

v

⁢

(

u

)

.

(2)

When we denote also the position vector of a point of the sphere by

k

→

=

k

→

⁢

(

u

,

v

)

:=

(

r

⁢

sin

⁡

u

⁢

cos

⁡

v

,

r

⁢

sin

⁡

u

⁢

sin

⁡

v

,

r

⁢

cos

⁡

u

)

,

we have the

tangent vector

of a curve (1) on the sphere:

t

→

:=

d

d

⁢

u

⁢

k

→

⁢

(

u

,

v

⁢

(

u

)

)

=

k

→

u

′

+

k

→

v

′

⋅

v

′

⁢

(

u

)

.

Since

k

→

u

′

=

(

r

⁢

cos

⁡

u

⁢

cos

⁡

v

,

r

⁢

cos

⁡

u

⁢

sin

⁡

v

,

-

r

⁢

sin

⁡

u

)

,

k

→

v

′

=

(

-

r

⁢

sin

⁡

u

⁢

sin

⁡

v

,

r

⁢

sin

⁡

u

⁢

cos

⁡

v

,

0

)

and since

v

′

⁢

(

u

)

=

a

tan

⁡

u

2

⋅

1

cos

2

⁡

u

2

⋅

1

2

=

a

2

⁢

sin

⁡

u

2

⁢

cos

⁡

u

2

=

a

sin

⁡

u

,

we can write the tangent vector of the curve as

t

→

=

r

⋅

(

cos

⁡

u

⁢

cos

⁡

v

-

a

⁢

sin

⁡

v

,

cos

⁡

u

⁢

sin

⁡

v

+

a

⁢

cos

⁡

v

,

-

sin

⁡

u

)

.

For a tangent vector of a meridian, the

partial derivative

k

→

u

′

may be taken.
Thus we obtain the value

cos

⁡

(

t

→

,

k

→

u

′

)

=

t

→

⋅

k

→

u

′

|

t

→

|

⁢

|

k

→

u

′

|

=

1

1

+

a

2

=

b

b

2

+

1

,

which is a constant.  It means that the angle

α

between the curve (1) and a meridian is constant.

Pictures in

http://hu.wikipedia.org/wiki/Loxodroma

Wiki

Title

loxodrome

Canonical

name

Loxodrome

Date of creation

2013-03-22 19:11:02

Last modified on

2013-03-22 19:11:02

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Definition

Classification

msc 53A05

Classification

msc 53A04

Classification

msc 26B05

Classification

msc 26A24

Defines

meridian

Generated on Fri Feb 9 22:23:34 2018 by

LaTeXML
mean curvature at surface point

mean curvature at surface point

Let

P

be a point on the

surface

F

⁢

(

x

,

y

,

z

)

=

0

where the function

F

is twice

continuously differentiable

on a neighbourhood of

P

. Then the

normal curvature

ϰ

θ

at

P

is, by Euler’s theorem, via the

principal curvatures

ϰ

1

and

ϰ

2

as

ϰ

θ

=

ϰ

1

⁢

cos

2

⁡

θ

+

ϰ

2

⁢

sin

2

⁡

θ

,

(1)

where

θ

is the angle between (

http://planetmath.org/AngleBetweenTwoPlanes

) the

normal section

plane corresponding

ϰ

1

and the normal section plane corresponding

ϰ

θ

. When we apply (1) by taking instead

θ

the angle

θ

+

π

2

, we may write

ϰ

θ

+

π

2

=

ϰ

1

⁢

sin

2

⁡

θ

+

ϰ

2

⁢

cos

2

⁡

θ

.

Adding this equation to (1) then yields

ϰ

θ

+

ϰ

θ

+

π

2

2

=

ϰ

1

+

ϰ

2

2

.

The contents of this result is the

Theorem.

The

arithmetic mean

of the

curvatures

(

http://planetmath.org/CurvaturePlaneCurve

) of two

perpendicular

normal sections has a value, which is equal to the arithmetic mean of the principal curvatures. This mean is called the

mean curvature

at the point in question.

References

1

Ernst Lindelöf

:

Differentiali- ja integralilasku
ja sen sovellutukset II

.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1932).

Title

mean curvature at surface point

Canonical name

MeanCurvatureAtSurfacePoint

Date of creation

2013-03-22 17:26:56

Last modified on

2013-03-22 17:26:56

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Theorem

Classification

msc 53A05

Classification

msc 26B05

Classification

msc 26A24

Related topic

AdditionFormulasForSineAndCosine

Related topic

GaussianCurvature

Related topic

MeanCurvature

Defines

mean curvature

Generated on Fri Feb 9 22:23:36 2018 by

LaTeXML
Meusnier’s theorem

Meusnier’s theorem

Let

P

be a point of a

surface

F

⁢

(

x

,

y

,

z

)

=

0

where

F

is twice

continuously differentiable

in a neighbourhood of

P

.  Set at

P

a

tangent

of the surface. At the point

P

, set through this tangent both the

normal plane

and a skew plane forming the angle (

http://planetmath.org/AngleBetweenTwoPlanes

)

ω

with the normal plane. Let

ϱ

be the

radius of curvature

of the

normal section

and

ϱ

ω

the radius of curvature of the

inclined section

.

Meusnier proved in 1779 that the equation

ϱ

ω

=

ϱ

⁢

cos

⁡

ω

between these radii of

curvature

is valid.

One can obtain an illustrative interpretation for the Meusnier’s theorem, if one thinks the sphere with radius the radius

ϱ

of curvature of the normal section and with centre the corresponding centre of curvature. Then the equation utters that the circle, which is intersected from the sphere by the inclined plane, is the

circle of curvature

of the intersection curve of this plane and the surface

F

⁢

(

x

,

y

,

z

)

=

0

.

Title

Meusnier’s theorem

Canonical name

MeusniersTheorem

Date of creation

2013-03-22 17:28:39

Last modified on

2013-03-22 17:28:39

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Theorem

Classification

msc 53A05

Classification

msc 26B05

Classification

msc 26A24

Synonym

theorem of Meusnier

Related topic

EulersTheorem2

Related topic

ProjectionOfPoint

Related topic

NormalCurvatures

Defines

inclined section

Generated on Fri Feb 9 22:23:38 2018 by

LaTeXML
minimal surface

minimal surface

Among the

surfaces

F

⁢

(

x

,

y

,

z

)

=

0

,  with

F

twice

continuously differentiable

, a

minimal surface

is such that in every of its points, the

mean curvature

vanishes

.  Because the mean curvature is the

arithmetic mean

of the

principal curvatures

ϰ

1

and

ϰ

2

, the equation

ϰ

2

=

-

ϰ

1

is valid in each point of a minimal surface.

A minimal surface has also the property that every sufficiently little portion of it has smaller area than any other

regular surface

with the same boundary curve.

Trivially, a plane is a minimal surface.  The

catenoid

is the only

surface of revolution

which is also a minimal surface.

Title

minimal surface

Canonical name

MinimalSurface

Date of creation

2013-03-22 18:08:56

Last modified on

2013-03-22 18:08:56

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Definition

Classification

msc 53A05

Classification

msc 26B05

Classification

msc 26A24

Related topic

PlateausProblem

Related topic

LeastSurfaceOfRevolution

Generated on Fri Feb 9 22:23:39 2018 by

LaTeXML
non orientable surface

non orientable surface

Non

orientable

phenomena are a consequence about the consideration of the tangent bundles regarding an

embedding

. One asks if

e

:

A

→

B

is an embedding then how the tangent bundles

T

⁢

A

and

T

⁢

B

relate?

For example: we could consider the core (simple close curve) of an cylinder

S

1

×

I

or in a

Mobius band

M

⁢

o

¨

. First we can observe that if

C

1

=

S

1

×

{

1

2

}

has as a

regular

neighborhood

whose

boundary

is two

component

disconnected

curve (in fact two disjoint circles), while the boundary of a regular neighborhood

N

of the core curve

C

⁢

o

¨

is a single circle:

∂

⁡

M

⁢

o

¨

.

In terms of tangent bundles we see that we can choose along the cylinder core a

consistent

normal in the sense that if this curve is traveled then at the end we have the same basis. In contrast with happens in

C

⁢

o

¨

which after a full turn we are going to find a

reflexion

of the normal axe.

Now employing the standard

classification

of closed

surfaces

we will construct another kind.

These are the only types of orientable surfaces:

O

0

the sphere;

O

1

the two torus;

O

2

=

O

1

⁢

#

⁢

O

1

the bitoro;

O

3

=

O

1

⁢

#

⁢

O

1

⁢

#

⁢

O

1

the tritoro,… etc,
i.e.

O

g

=

O

1

⁢

#

⁢

⋯

⁢

#

⁢

O

1

So, with the

connected sum

device we have:

The

projective plane

ℝ

⁢

P

2

=

(

O

0

∖

int

⁢

D

)

∪

∂

M

⁢

o

¨

=

D

∪

∂

M

⁢

o

¨

The

Klein bottle

ℝ

⁢

P

2

⁢

#

⁢

ℝ

⁢

P

2

=

[

O

0

∖

(

int

⁢

D

1

⊔

int

⁢

D

2

)

]

∪

∂

[

(

M

⁢

o

¨

)

1

⊔

(

M

⁢

o

¨

)

2

]

=

(

M

⁢

o

¨

)

1

∪

∂

(

M

⁢

o

¨

)

2

If we standarize as

N

1

=

ℝ

⁢

P

2

and

N

2

=

ℝ

⁢

P

2

⁢

#

⁢

ℝ

⁢

P

2

, then the genus three non orientable surface is

N

3

=

ℝ

⁢

P

2

⁢

#

⁢

ℝ

⁢

P

2

⁢

#

⁢

ℝ

⁢

P

2

=

N

2

⁢

#

⁢

ℝ

⁢

P

2

=

O

1

⁢

#

⁢

ℝ

⁢

P

2

=

(

[

O

0

∖

(

int

D

1

⊔

int

D

2

⊔

int

D

3

)

]

∪

∂

[

(

M

o

¨

)

1

⊔

(

M

o

¨

)

2

⊔

(

M

o

¨

)

3

]

=

(

O

1

∖

int

⁢

D

)

∪

∂

M

⁢

o

¨

=

(

N

2

∖

int

⁢

D

)

∪

∂

M

⁢

o

¨

∙

⁣

∙

⁣

∙

{xy}

(0,10)*+

R

^2=”f”;
(13,10)*+TM
¨

o

=”e”;
(15,0)*+M
¨

o

=”m”;

\ar

@. ”f”;”e”?*!/_2mm/⊂;

\ar

”e”;”m”?*!/_3mm/p;

Title

non orientable surface

Canonical name

NonOrientableSurface

Date of creation

2013-03-22 19:02:29

Last modified on

2013-03-22 19:02:29

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

33

Author

juanman (12619)

Entry type

Definition

Classification

msc 53A05

Classification

msc 57M20

Classification

msc 14J29

Related topic

Surface

Generated on Fri Feb 9 22:23:43 2018 by

LaTeXML
normal section

normal section

Normal sections

Let

P

be a point of a surface

F

⁢

(

x

,

y

,

z

)

=

0

,

(1)

where

F

has the

continuous

first and

partial derivatives

in a neighbourhood of

P

.  If one intersects the surface with a plane containing the

surface normal

at

P

, the intersection curve is called a

normal section

.

Normal curvatures

When the direction of the intersecting plane is varied, one gets different normal sections, and their

curvatures

(

http://planetmath.org/CurvaturePlaneCurve

) at

P

, the so-called

normal curvatures

, vary having a

minimum

value

ϰ

1

and a

maximum

value

ϰ

2

.  The

arithmetic mean

of

ϰ

1

and

ϰ

2

is called the

mean curvature

of the surface at

P

.

By the suppositions on the function

F

, examining the normal curvatures can without loss of generality be to the following:  Examine the curvature of the normal sections through the origin, the surface given in the form

z

=

z

⁢

(

x

,

y

)

,

(2)

where

z

⁢

(

x

,

y

)

has the continuous first and partial derivatives in a neighbourhood of the origin and

z

⁢

(

0

,

0

)

=

z

x

′

⁢

(

0

,

0

)

=

z

y

′

⁢

(

0

,

0

)

=

0

.

Indeed, one can take a new

rectangular coordinate

system with

P

the new origin and the normal at

P

the new

z

-axis; then the new

x

⁢

y

-plane coincides with the

tangent plane

of the surface (1) at

P

. The equation (1) defines the function of (2).

Title

normal section

Canonical name

NormalSection

Date of creation

2013-03-22 17:26:25

Last modified on

2013-03-22 17:26:25

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Definition

Classification

msc 53A05

Classification

msc 26A24

Classification

msc 26B05

Related topic

SecondFundamentalForm

Related topic

DihedralAngle

Defines

normal curvature

Generated on Fri Feb 9 22:23:49 2018 by

LaTeXML
shape operator

shape operator

The

shape operator

S

of a surface

Σ

in

ℝ

3

is the

derivative

of the sphere map

N

:

Σ

→

S

2

given by

N

⁢

(

p

)

=

unit normal

vector field

at

p

. So at each

p

,

S

⁢

(

p

)

=

d

p

⁢

N

and it is the

linear transformation

S

⁢

(

p

)

:

T

p

⁢

Σ

→

T

N

⁢

(

p

)

⁢

S

2

. This is important, because the

determinant

defines the

Gaussian curvature

at

p

in

Σ

.

Title

shape operator

Canonical

name

ShapeOperator

Date of creation

2013-03-22 16:04:31

Last modified on

2013-03-22 16:04:31

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

6

Author

juanman (12619)

Entry type

Definition

Classification

msc 53A05

Related topic

SecondFundamentalForm

Generated on Fri Feb 9 22:24:00 2018 by

LaTeXML