PlanetMath: surface normal

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surface normal

(Definition)

Let be a smooth surface in $\mathbb{R}^3$ . The surface normal of at a point of is the line passing through and perpendicular to the tangent plane $\tau$ of at the point , i.e. perpendicular to all lines in $\tau$ .

If the surface is given in a parametric form

$\displaystyle x = x(u,\,v),\quad y = y(u,\,v),\quad z = z(u,\,v),$

it is useful to interpret the parameters

and

as the rectangular coordinates of a point in a plane, the so-called parameter plane. We can consider on

the so-called parameter curves, namely the

-curves which correspond the lines parallel to the

-axis and the

-curves which correspond the lines parallel to the

-axis in the parameter plane. One

-curve and one

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

and

in a point of

are the Gaussian coordinates of this point). The surface normal at any point of

is perpendicular to both parameter curves, and thus its direction cosines

satisfy the equations

$\begin{align*}\begin{cases}\displaystyle{a\frac{\partial x}{\partial u}+b\frac{\... ...rtial y}{\partial v}+c\frac{\partial z}{\partial v}= 0.} \end{cases}\end{align*}$

This homogeneous pair of linear equations determines the ratio of the direction cosines

$\displaystyle a:b:c = \frac{\partial(y,\,z)}{\partial(u,\,v)}:\frac{\partial(z,\,x)}{\partial(u,\,v)}:\frac{\partial(x,\,y)}{\partial(u,\,v)}$

via the Jacobians.

Example. Determine the direction cosines of the normal of the helicoid

$\displaystyle x = u\cos{v},\quad y = u\sin{v},\quad z = cv.$

We have the Jacobians

$\displaystyle \left\vert \begin{matrix} \frac{\partial y}{\partial u} & \frac{\... ...in{matrix}\cos{v} & \sin{v}\ -u\sin{v} & u\cos{v}\end{matrix}\right\vert = u.$

These are the components of the normal vector of the helicoid surface in the point with the Gaussian coordinates

and

. The length of the vector is $\sqrt{(c\sin{v})^2+(-c\cos{v})^2+u^2} = \sqrt{u^2+c^2}$ . If we divide the vector by its length, we obtain a unit vector, the components of which are the direction cosines of the surface normal:

$\displaystyle \frac{c\sin{v}}{\sqrt{u^2+c^2}},\;\;-\frac{c\cos{v}}{\sqrt{u^2+c^2}},\;\;\frac{u}{\sqrt{u^2+c^2}}.$

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"surface normal" is owned by pahio. [ full author list (2) ]

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See Also: normal line, equation of plane

Other names:

surface normal line, normal of surface

Also defines:

parameter plane, parameter curve, Gaussian coordinates

This object's parent.

Attachments:: normal section (Definition) by pahio

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Cross-references: unit vector, vector, length, normal vector, components, helicoid, normal, Jacobians, ratio, linear equations, homogeneous, equations, direction cosines, parallel, plane, rectangular coordinates, parameters, parametric form, tangent plane, perpendicular, passing through, line, point, surface, smooth
There are 5 references to this entry.

This is version 9 of surface normal, born on 2007-07-08, modified 2007-08-17.
Object id is 9753, canonical name is SurfaceNormal.
Accessed 1353 times total.

Classification:

AMS MSC:	26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems)
	26B05 (Real functions :: Functions of several variables :: Continuity and differentiation questions)

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