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3.3.1 Surface Normal
Generally, the normal vector at point
on the LS of a smooth function is given by

(3.21) 
At grid points
the
th component of the normal vector can be approximated by

(3.22) 
Here
is the central difference operator as defined in (3.6).
The normal vector for a grid point close to the surface
represented by the zero LS is also a good approximation for the normal on the surface for the closest surface point. The closest surface point
of a nearby grid point
can be approximated by [135]

(3.23) 
if the grid point indices
are equal to the grid point coordinates. Here the last factor corresponds to the approximated signed distance to the surface. For the denominator the same approximation is used as in (3.22).
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Otmar Ertl: Numerical Methods for Topography Simulation