Cuspidal cubic
= Cuspidal cubic
The projective <algebraic curve> $D=\{y^2z=x^3\}\subset\mathbb P^2$ is an irreducible cubic with a <cusp> at $[0:0:1]$. In the chart $z=1$, its equation is $y^2=x^3$, with parametrization $(x,y)=(t^2,t^3)$. Its affine version $y^2=z^3$, used below, differs only by renaming a coordinate. At the point $[0:1:0]$, the chart $y=1$ gives $z=x^3$, which is smooth and exhibits its <unique smooth flex of a cuspidal cubic>.