Pinch point (mathematics)
In geometry, a pinch point or cuspidal point is a type of singular point on an algebraic surface.
The equation for the surface near a pinch point may be put in the form
where denotes terms of degree 4 or more and is not a square in the ring of functions.
For example the surface near the point, meaning in coordinates vanishing at that point, has the form above. In fact, if and then is a system of coordinates vanishing at then is written in the canonical form.
The simplest example of a pinch point is the hypersurface defined by the equation called Whitney umbrella.
The pinch point is a limit of normal crossings singular points. These singular points are intimately related in the sense that in order to resolve the pinch point singularity one must blow-up the whole -axis and not only the pinch point.
