Fold caustic ($\det D_{(s,t)}(x,y)=0$)

A characteristic surface may remain smooth while its projection to the independent-variable plane fails to be locally invertible. At a generic fold, smooth changes of parameter and plane coordinates reduce this projection to $(u,v)\mapsto (u,v^2)$. There are two sheets on one side of the image of $v=0$, no sheet on the other, and one merged sheet on the fold itself. The image is a fold caustic. A characteristic graph can terminate there even though the space curves defining the surface continue smoothly. Vanishing of the Jacobian determinant detects a candidate singular locus; verifying the fold requires a nonzero quadratic term in its null direction, as in $v^2$.