Let $f\!:M\!\rightarrow\!\mathbb{R}$ be a smooth function on a manifold and $p\!\in\!M$. Is there any way to geometrically/visually characterize the conditions
$p$ is a critical point (i.e. $D(f)_p\!=\!0$) and
$p$ is a nondegenerate critical point (i.e. $\det D^2(f)_p\!\neq\!0$)?
If $f(x)=\langle x,a\rangle$ is a height function on a surface, then $f$ is linear, so $D(f)_p\!=\!f\!:\, T_pM\rightarrow T_p\mathbb{R}\!=\!\mathbb{R}$. Thus $f$ is the zero map at those $p$ for which $T_pM$ is perpendicular to $a$, i.e. the critical points of $f$ are those points at which $T_pM=a^\bot$. But what about higher dimensions and different $f$s?
And what about nondegeneracy?