title: Pinch point (mathematics)
text: # 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
  
      $f(u,v,w)=u^{2}-vw^{2}+[4]\,$
  
  where [4] denotes terms of degree 4 or more and $v$ is not a square in the ring of functions.
  
  For example the surface $1-2x+x^{2}-yz^{2}=0$ near the point $(1,0,0)$, meaning in coordinates vanishing at that point, has the form above. In fact, if $u=1-x,v=y$ and $w=z$ then {$u,v,w$} is a system of coordinates vanishing at $(1,0,0)$ then $1-2x+x^{2}-yz^{2}=(1-x)^{2}-yz^{2}=u^{2}-vw^{2}$ is written in the canonical form.
  
  The simplest example of a pinch point is the hypersurface defined by the equation $u^{2}-vw^{2}=0$ called Whitney umbrella.
  
  The pinch point (in this case the origin) is a limit of normal crossings singular points (the $v$-axis in this case). These singular points are intimately related in the sense that in order to resolve the pinch point singularity one must blow-up the whole $v$-axis and not only the pinch point.
  
  