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@ -351,9 +351,21 @@ For all other points $P = (0, w)$, there are exactly two minima $x_{1,2} = \pm \
The solution of Equation~\ref{eq:simple-cubic-equation-for-quadratic-distance} The solution of Equation~\ref{eq:simple-cubic-equation-for-quadratic-distance}
is is
\[t := \sqrt[3]{\sqrt{3 \cdot (4a^3 + 27 b^2)} -9b}\] \[t := \sqrt[3]{\sqrt{3 \cdot (4 \alpha^3 + 27 \beta^2)} -9\beta}\]
\[x = \frac{t}{\sqrt[3]{18}} - \frac{\sqrt[3]{\frac{2}{3}} a }{t}\] \[x = \frac{t}{\sqrt[3]{18}} - \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t}\]
When you insert is in Equation~\ref{eq:simple-cubic-equation-for-quadratic-distance}
you get:
\begin{align}
0 &= \left (\frac{t}{\sqrt[3]{18}} - \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t} \right )^3 + \alpha \left (\frac{t}{\sqrt[3]{18}} - \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t} \right ) + \beta\\
&= (\frac{t}{\sqrt[3]{18}})^3 - 3 (\frac{t}{\sqrt[3]{18}})^2 \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t} + 3 (\frac{t}{\sqrt[3]{18}})(\frac{\sqrt[3]{\frac{2}{3}} \alpha }{t})^2 + (\frac{\sqrt[3]{\frac{2}{3}} \alpha }{t})^3 + \alpha \left (\frac{t}{\sqrt[3]{18}} - \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t} \right ) + \beta\\
&= \frac{t^3}{18} - \frac{3t^2}{\sqrt[3]{18^2}} \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t} + \frac{3t}{\sqrt[3]{18}} \frac{\sqrt[3]{\frac{4}{9}} \alpha^2 }{t^2} + \frac{\frac{2}{3} \alpha^3 }{t^3} + \alpha \left (\frac{t}{\sqrt[3]{18}} - \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t} \right ) + \beta\\
&= \frac{t^3}{18} - \frac{\sqrt[3]{18} t \alpha}{\sqrt[3]{18^2}} + \frac{\sqrt[3]{12} \alpha^2}{\sqrt[3]{18} t} + \frac{\frac{2}{3} \alpha^3 }{t^3} + \alpha \left (\frac{t}{\sqrt[3]{18}} - \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t} \right ) + \beta\\
&= \frac{t^3}{18} - \frac{t \alpha}{\sqrt[3]{18}} + \frac{\sqrt[3]{2} \alpha^2}{\sqrt[3]{3} t} + \frac{\frac{2}{3} \alpha^3 }{t^3} + \alpha \left (\frac{t}{\sqrt[3]{18}} - \frac{\sqrt[3]{\frac{2}{3}} \alpha }{t} \right ) + \beta\\
&= \frac{t^3}{18} - \frac{t \alpha}{\sqrt[3]{18}} + \frac{\frac{2}{3} \alpha^3 }{t^3} + \frac{\alpha t}{\sqrt[3]{18}} + \beta\\
&= \frac{t^3}{18} + \frac{\frac{2}{3} \alpha^3 }{t^3} + \beta\\
\end{align}
\todo[inline]{verify this solution} \todo[inline]{verify this solution}