Argument

Calculus texts define curvature and provide methods for its computation.  Suppose an ellipse with semi-major axis [latex]a[/latex] and semi-minor axis [latex]b[/latex] has the
parametric equations

\begin{equation}
x=a \cos{t} \ \ \ \ \ \     y=b \sin {t}.
\end{equation}

A formula for the curvature from [2] (see exercises) is

\begin{equation}
\kappa = \frac{|x’y^{\prime\prime} – y’x^{\prime\prime}|}{[(x’)^2 + (y’)^2]^\frac{3}{2}}
\end{equation}

The prime indicates differentiation with respect to the parameter.

Routine calculation shows that the critical points for the curvature are the
endpoints of the major and minor axis.  At the endpoints of the major axis, the curvature is [latex]a/b^2[/latex] and at the endpoints of the minor axis it is [latex]b/a^2[/latex].  The latter is the smaller value.

Over the closed segments between successive critical points the curvature is strictly monotone [4].  It follows that maxima occur at the endpoints of the major axis and minima occur at the endpoints of the minor axis.
An ellipse has four vertices.