surface normal

surface normal

Let

S

be a smooth surface in

ℝ

3

. The

surface normal

of

S

at a point

P

of

S

is the line

passing through

P

and

perpendicular

to the

tangent plane

τ

of

S

at the point

P

, i.e. perpendicular to all lines in

τ

.

If the surface

S

is given in a parametric form

x

=

x

⁢

(

u

,

v

)

,

y

=

y

⁢

(

u

,

v

)

,

z

=

z

⁢

(

u

,

v

)

,

it is useful to interpret the parameters

u

and

v

as the

rectangular coordinates

of a point in a plane, the so-called

parameter plane

. We can consider on

S

the so-called

parameter curves

, namely the

u

-

curves

which correspond the lines

parallel

to the

u

-axis and the

v

-

curves

which correspond the lines parallel to the

v

-axis in the parameter plane. One

u

-curve and one

v

-curve passes through every point on the surface (the values of

u

and

v

in a point of

S

are the

Gaussian coordinates

of this point). The surface normal at any point of

S

is perpendicular to both parameter curves, and thus its

direction cosines

a

,

b

,

c

satisfy the equations

{

a

⁢

∂

⁡

x

∂

⁡

u

+

b

⁢

∂

⁡

y

∂

⁡

u

+

c

⁢

∂

⁡

z

∂

⁡

u

=

0

,

a

⁢

∂

⁡

x

∂

⁡

v

+

b

⁢

∂

⁡

y

∂

⁡

v

+

c

⁢

∂

⁡

z

∂

⁡

v

=

0

.

This

homogeneous

pair of

linear equations

determines the ratio of the direction cosines

a

:

b

:

c

=

∂

⁡

(

y

,

z

)

∂

⁡

(

u

,

v

)

:

∂

⁡

(

z

,

x

)

∂

⁡

(

u

,

v

)

:

∂

⁡

(

x

,

y

)

∂

⁡

(

u

,

v

)

via the

Jacobians

.

Example.

Determine the direction cosines of the normal of the

helicoid

x

=

u

⁢

cos

⁡

v

,

y

=

u

⁢

sin

⁡

v

,

z

=

c

⁢

v

.

We have the Jacobians

|

∂

⁡

y

∂

⁡

u

∂

⁡

z

∂

⁡

u

∂

⁡

y

∂

⁡

v

∂

⁡

z

∂

⁡

v

|

=

|

sin

⁡

v

0

u

⁢

cos

⁡

v

c

|

=

c

⁢

sin

⁡

v

,

|

∂

⁡

z

∂

⁡

u

∂

⁡

x

∂

⁡

u

∂

⁡

z

∂

⁡

v

∂

⁡

x

∂

⁡

v

|

=

|

0

cos

⁡

v

c

-

u

⁢

sin

⁡

v

|

=

-

c

⁢

cos

⁡

v

,

|

∂

⁡

x

∂

⁡

u

∂

⁡

y

∂

⁡

u

∂

⁡

x

∂

⁡

v

∂

⁡

y

∂

⁡

v

|

=

|

cos

⁡

v

sin

⁡

v

-

u

⁢

sin

⁡

v

u

⁢

cos

⁡

v

|

=

u

.

These are the

components

of the

normal vector

of the helicoid surface in the point with the Gaussian coordinates

u

and

v

.  The length of the vector is

(

c

⁢

sin

⁡

v

)

2

+

(

-

c

⁢

cos

⁡

v

)

2

+

u

2

=

u

2

+

c

2

.  If we divide (

http://planetmath.org/Division

) the vector by its length, we obtain a

unit vector

, the components of which are the direction cosines of the surface normal:

c

⁢

sin

⁡

v

u

2

+

c

2

,

-

c

⁢

cos

⁡

v

u

2

+

c

2

,

u

u

2

+

c

2

.

Title

surface normal

Canonical name

SurfaceNormal

Date of creation

2013-03-22 17:23:10

Last modified on

2013-03-22 17:23:10

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

16

Author

pahio (2872)

Entry type

Definition

Classification

msc 26B05

Classification

msc 26A24

Classification

msc 53A04

Classification

msc 53A05

Synonym

surface normal line

Synonym

normal of surface

Related topic

NormalLine

Related topic

EquationOfPlane

Related topic

Parameter

Defines

parametre plane

Defines

parameter plane

Defines

parametre curve

Defines

parameter curve

Defines

Gaussian coordinates

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