# Critical Point:

Given a mapping f of a surface into space and a unit vector z, the critical points of the mapping f in the direction z are the singularities of the height function induced by f in the direction z.

For smooth surfaces, these are the points where the tangent plane to the surface is perpendicular to z.

Generically, the critical points of a surface in space are isolated points. A level set containing a critical point is called a critical level of the surface in the direction z.

In general, the critical points are the points where the topology of the level sets (for the given direction) change, usually be a change in the number of components in the level set (a component appears or disappears, two components join at a critical point to form one component, or one component breaks into two). Because of this, the topology of the surface is completely determined by the changes that occur at the critical points. An important observation is that the Euler characteristic of a surface can be computed by adding the number of maxima and minima, and subtracting the number of saddles.

See also:

Maxima, minima, and saddles

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