 Singular projection of an implicit curve

For two constraints $F(x,y,z)=G(x,y,z)=0$, the implicit function theorem gives differentiable functions $y(x),z(x)$ when the Jacobian determinant $\partial (F,G)/\partial (y,z)$ is nonzero. At its zeros this particular projection can fail even when a geometric curve remains. A nonzero tangent vector with zero $x$ component gives a vertical tangent in that projection. Singular constraint representations require further care: a determinant of zero alone does not prove an infinite derivative or even a regular intersection.