geometric meaning of complex cubic polynomial coefficients

A complex cubic polynomial arranged as $f(z)=z^3-3az^2+3b^2z-c^3$ with coefficients $a,b,c\in\mathbb{C}$ can be represented as the product of (unknown) factors $f(z)=(z-p)(z-q)(z-r)=z^3-3\left(\frac{p+q+r}{3}\right)z^2+3\left(\sqrt \frac{pq+qr+rp}{3}\right)^2z-\left(\sqrt[3]{pqr}\right)^3$ with roots $p,q,r\in\mathbb{C}$.

Thus we can read the centroid $a=\frac{p+q+r}{3}$ of the Steiner ellipse of the triangle $p,q,r\in\mathbb{C}$ right from the coefficients (e.g. reasoning with Marden’s theorem).

Do the coefficients $b=\sqrt \frac{pq+qr+rp}{3}$ and $c=\sqrt[3]{pqr}$ have similar nice intuitive geometric interpretation, e.g. describe characteristic points or features of the triangle or its Steiner ellipse?

Solution 1:

Vieta's Formulae

\begin{align} f(z) &= z^3-3az^2+3b^2z-c^3 \\ &= (z-p)(z-q)(z-r) \\ a &= \frac{p+q+r}{3} \\ b^2 &= \frac{pq+qr+rp}{3} \\ c^3 &= pqr \end{align}

Foci of Steiner ellipse

\begin{align} f'(\lambda) &=0 \\ \lambda &= a\pm \sqrt{a^2-b^2} \\ &= \frac{p+q+r \pm \sqrt{(p+q+r)^2-3(pq+qr+rp)}}{3} \end{align}

Resolvents \begin{align} u &= \sqrt[3]{\frac{c^3+3ab^2-2a^3}{2}+ \sqrt{(b^2-a^2)^3+ \left( \frac{c^3+3ab^2-2a^3}{2} \right)^2}} \\ v &= \sqrt[3]{\frac{c^3+3ab^2-2a^3}{2}- \sqrt{(b^2-a^2)^3+ \left( \frac{c^3+3ab^2-2a^3}{2} \right)^2}} \\ 0 &= f(a+u\, \omega^n+v\, \omega^{2n}) \tag{$\omega=e^{2\pi i/3}$} \end{align}

• Centroid $$a=\frac{p+q+r}{3}$$

• Linear eccentricity of Steiner ellipse $$\sqrt{|a^2-b^2|}=\sqrt{|uv|}$$

• Semi-major axis of Steiner ellipse $$\frac{|u|+|v|}{2}$$

• Semi-minor axis of Steiner ellipse $$\frac{||u|-|v||}{2}$$