f of a surface and a direction z in
space, the height function, zf, in this direction is the
projection of the immersed surface onto the directed line in the
direction of z. That is the value of the height function at
some point p of M is given by the dot product of
z with the value of f at p:
For almost all directions, the
of a height function are isolated; the height function in such a
direction is called a Morse height function.
- zf (p) := z.f (p)
For smooth surfaces, a Morse height function will have critical points
where the tangent planes to the surface are perpendicular to the
direction z, namely local minima, local maxima and saddles.
Polyhedral surfaces will have critical points at the corresponding
piecewise linear structures.
Height functions play an important role in describing
immersions of surfaces in space.
Polar height functions
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The Geometry Center