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 [4] denotes terms of degree 4 or more and ${\displaystyle v}$ is not a square in the ring of functions.

For example the surface ${\displaystyle 1-2x+x^{2}-yz^{2}=0}$ near the point ${\displaystyle (1,0,0)}$, meaning in coordinates vanishing at that point, has the form above. In fact, if ${\displaystyle u=1-x,v=y}$ and ${\displaystyle w=z}$ then {${\displaystyle u,v,w}$} is a system of coordinates vanishing at ${\displaystyle (1,0,0)}$ then ${\displaystyle 1-2x+x^{2}-yz^{2}=(1-x)^{2}-yz^{2}=u^{2}-vw^{2}}$ is written in the canonical form.

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