surface normal

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

If the surface $S$ is given in a parametric form

x=x(u,\\,v),\\quad y=y(u,\\,v),\\quad 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

\\displaystyle\\begin{cases}\\displaystyle{a\\frac{\\partial x}{\\partial u}+b\\frac{%\\partial y}{\\partial u}+c\\frac{\\partial z}{\\partial u}=0},\\\\\\\\\\displaystyle{a\\frac{\\partial x}{\\partial v}+b\\frac{\\partial y}{\\partial v}+c%\\frac{\\partial z}{\\partial v}=0.}\\end{cases}

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

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

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

We have the Jacobians

\\left|\\begin{matrix}\\frac{\\partial y}{\\partial u}&\\frac{\\partial z}{\\partial u%}\\\\\\frac{\\partial y}{\\partial v}&\\frac{\\partial z}{\\partial v}\\end{matrix}\\right|%=\\left|\\begin{matrix}\\sin{v}&0\\\\u\\cos{v}&c\\end{matrix}\\right|=c\\sin{v},\\;\\;\\left|\\begin{matrix}\\frac{\\partial z%}{\\partial u}&\\frac{\\partial x}{\\partial u}\\\\\\frac{\\partial z}{\\partial v}&\\frac{\\partial x}{\\partial v}\\end{matrix}\\right|%=\\left|\\begin{matrix}0&\\cos{v}\\\\c&-u\\sin{v}\\end{matrix}\\right|=-c\\cos{v},\\;\\;\\left|\\begin{matrix}\\frac{%\\partial x}{\\partial u}&\\frac{\\partial y}{\\partial u}\\\\\\frac{\\partial x}{\\partial v}&\\frac{\\partial y}{\\partial v}\\end{matrix}\\right|%=\\left|\\begin{matrix}\\cos{v}&\\sin{v}\\\\-u\\sin{v}&u\\cos{v}\\end{matrix}\\right|=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 $\\sqrt{(c\\sin{v})^{2}+(-c\\cos{v})^{2}+u^{2}}=\\sqrt{u^{2}+c^{2}}$ . 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:

\\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}}}.

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