surface normal

Let S be a smooth surface in 3. The surface normal of S at a point P of S is the line passing through P and perpendicularMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the tangent planeMathworldPlanetmath 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


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 parallelMathworldPlanetmathPlanetmath 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 cosinesMathworldPlanetmath a, b, c satisfy the equations


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


via the JacobiansMathworldPlanetmathPlanetmath.

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


We have the Jacobians


These are the componentsPlanetmathPlanetmathPlanetmath of the normal vector of the helicoid surface in the point with the Gaussian coordinates u and v.  The length of the vector is  (csinv)2+(-ccosv)2+u2=u2+c2.  If we divide ( the vector by its length, we obtain a unit vectorMathworldPlanetmath, the components of which are the direction cosines of the surface normal:

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