Let $f(x)$ be a smooth real-valued function of $n$ real variables. For what follows, $n=2$ and our variables are $y$ and $z$. We will also use a parameter $u$.

A function’s 1-dimensional height ridge generalizes the concept of local maxima by relaxing one ‘orthogonality’ condition and one ‘curvature’ condition implicit in a characterization of local maxima. We will not go into the definition of a 1-dimensional height ridge here.

Similarly, a function’s 1-dimensional relative critical set generalizes the concept of critical points by relaxing one ‘orthogonality’ condition implicit in a characterization of critical point. We will not go into the definition of a 1-dimensional relative critical set here.

What we will do here is give three examples of families of functions whose relative critical sets exhibit non-generic behavior at the origin when the parameter vanishes. The geometric structure of the relative critical sets are well understood, so these functions can serve as ‘unit tests’ for algorithms that aim to extract a function’s relative critical set.

Each family of functions is named for the degeneracy it exhibits when its parameter $u$ vanishes: the Gauss Type function, the Gauss-Hessian Type function, and the Degenerate Critical Point funtion. They are, in that order:

\[z+uy-\frac{1}{2}y^2+\frac{1}{2}z^2-y^2z\] \[z+uy^2+z^2-zy^45-yz^2\] \[uy+\frac{1}{2}z^2+y^3+2yz^2\]