Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2012 / 9 / b / iv

Differentiating the projected parameterization givesIt vanishes exactly when $s=-1$. The image is the tangency half-line from part (iii), $x=t-1$, $y=-t$, $t\ge 0$. This is a fold caustic: the parameterized surface itself remains regular, but its projection ceases to be locally invertible, so it cannot there be expressed as a differentiable single-valued graph over the plane.