surface normal
Let be a smooth surface in . The surface normal of at a point of is the line passing through and perpendicular to the tangent plane of at the point , i.e. perpendicular to all lines in .
If the surface is given in a parametric form
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
This homogeneous pair of linear equations determines the ratio of the direction cosines
via the Jacobians.
Example. Determine the direction cosines of the normal of the helicoid
We have the Jacobians
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 . 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:
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 |