Height Function:

Given an immersion 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:

zf (p) := z.f (p)

For almost all directions, the critical points of a height function are isolated; the height function in such a direction is called a Morse height function.

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 tight immersions of surfaces in space.

See also:

Polar height functions

8/12/94 dpvc@geom.umn.edu -- The Geometry Center