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