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(\color{red}\frac{(1+i \sqrt{3})\alpha}{\sqrt[3]{12} \cdot t} \color{black} \color{blue}-\frac{(1-i\sqrt{3}) t}{2\sqrt[3]{18}} \color{black} \right ) + \beta\\ &= \frac{-2 \alpha^3}{3 t^3} - + \frac{\alpha^2(2(1+i\sqrt{3})-(2+\sqrt{3}i))}{2t\sqrt[3]{12}} + \frac{t^3}{18} + \beta\\ - &= \frac{-24 \alpha^3 + (3\sqrt[3]{18}t^2)(\alpha^2\sqrt{3}i) + 2t^3+36 t^3 \beta}{36t^3} + &= \frac{-12 \alpha^3 + t^6+18 t^3 \beta}{18t^3} +\end{align} + +Now continue with only the numerator +\begin{align} + 0 &\stackrel{!}{=} + - 12 \alpha^3 + + (\sqrt{3(4 \alpha^3 + 27 \beta^2)}-9\beta)^2 + + 18 (\sqrt{3(4 \alpha^3 + 27 \beta^2)} - 9 \beta) \beta\\ + &= + \color{red}- 12 \alpha^3 \color{black}+ + \left ( + 3(\color{red}4 \alpha^3\color{black} + \color{blue}27 \beta^2 \color{black}) + \color{orange}- 2 \cdot \sqrt{3(4 \alpha^3 + 27 \beta^2)} \cdot 9\beta\color{black} + + \color{blue}81 \beta^2\color{black} + \right )\\ + &\hphantom{{}=}+ 18 \beta (\color{orange}\sqrt{3(4 \alpha^3 + 27 \beta^2)}\color{black} \color{blue}- 9 \beta\color{black}) \end{align} diff --git a/documents/math-minimal-distance-to-cubic-function/quadratic-case-2.3.tex b/documents/math-minimal-distance-to-cubic-function/quadratic-case-2.3.tex index 60d4525..7c3d24e 100644 --- a/documents/math-minimal-distance-to-cubic-function/quadratic-case-2.3.tex +++ b/documents/math-minimal-distance-to-cubic-function/quadratic-case-2.3.tex @@ -2,90 +2,5 @@ One solution is \[x = \frac{(1-i \sqrt{3}) \alpha}{\sqrt[3]{12} \cdot t} -\frac{(1+i\sqrt{3}) t}{2\sqrt[3]{18}}\] -We will verify it in multiple steps. First, get $x^3$: -\begin{align} - x^3 &= \left (\frac{(1-i \sqrt{3}) \alpha}{\sqrt[3]{12} \cdot t} - \frac{(1+i\sqrt{3}) t}{2\sqrt[3]{18}} \right)^3\\ - &= \left (\frac{(2\sqrt[3]{18})(1-i \sqrt{3}) \alpha - (\sqrt[3]{12} \cdot t)(1+i\sqrt{3}) t}{\sqrt[3]{12} t \cdot 2 \sqrt[3]{18}} \right)^3\\ - &= \left (\frac{2\sqrt[3]{18}\alpha (1-i \sqrt{3}) - \sqrt[3]{12} t^2(1+i\sqrt{3})}{2t \cdot 6} \right )^3\\ - &= \bigg (\frac{\overbrace{\sqrt[3]{12} \alpha (1-i \sqrt{3}) - t^2 (1+i\sqrt{3})}^{\text{numerator}}}{\sqrt[3]{12^2} t} \bigg )^3 -\end{align} - -Now calculate numerator$^3$: -\begin{align} - \left (\sqrt[3]{12} \alpha (1-i \sqrt{3}) - t^2(1+i\sqrt{3}) \right )^3 &= - 12 \alpha^3 (1-i\sqrt{3})^3 \\ - &\hphantom{{}=}- 3 \sqrt[3]{12^2} \alpha^2(1-i\sqrt{3})^2 (t^2(1+i \sqrt{3}))\\ - &\hphantom{{}=}+ 3 \sqrt[3]{12\hphantom{^2}} \alpha\hphantom{^2} (1-i\sqrt{3}) t^4 (1+i\sqrt{3})^2 - t^6 (1+i\sqrt{3})^3\\ - &= 12 \alpha^3 \cdot (-8) \\ - &\hphantom{{}=}- 3 \sqrt[3]{12^2} \alpha^2(-2(1+i\sqrt{3}))(t^2(1+i \sqrt{3}))\\ - &\hphantom{{}=}+ 3 \sqrt[3]{12} \alpha (1-i\sqrt{3}) t^4 (-2(1-i\sqrt{3})) - t^6 (-8)\\ - &= -96 \alpha^3 + 6 \sqrt[3]{12^2} \alpha^2 t^2 (1+i \sqrt{3})^2\\ - &\hphantom{{}=}- 6 \sqrt[3]{12} \alpha t^4 (1-i\sqrt{3})^2 +8 t^6\\ - &= -96 \alpha^3 - 12 \sqrt[3]{12^2} \alpha^2 t^2 (1-i \sqrt{3})\\ - &\hphantom{{}=}+ 12 \sqrt[3]{12} \alpha t^4 (1+i \sqrt{3}) +8 t^6\\ - &= -96 \alpha^3 - 24 \sqrt[3]{18} \alpha^2 t^2 (1-i \sqrt{3})\\ - &\hphantom{{}=}+ 12 \sqrt[3]{12} \alpha t^4 (1+i \sqrt{3}) +8 t^6 -\end{align} -\goodbreak -Now back to the original equation: -\begin{align} -0 &\stackrel{!}{=} x^3 + \alpha x + \beta\\ - &= \frac{-96 \alpha^3 - 24 \sqrt[3]{18} \alpha^2 t^2 (1-i \sqrt{3}) + 12 \sqrt[3]{12} \alpha t^4 (1+i \sqrt{3}) +8 t^6}{12^2 t^3}\\ - &\hphantom{{}=}+\alpha \left (\sqrt[3]{12} \cdot \frac{\sqrt[3]{12} \alpha (1-i \sqrt{3}) - t^2(1+i\sqrt{3})}{12t} \right ) + \beta -\end{align} - -\todo[inline]{the calculation above seems to be wrong / too long. Next try} - -When you insert this in Equation~\ref{eq:simple-cubic-equation-for-quadratic-distance} -you get:\footnote{Remember, that $(1+i\sqrt{3})^2 = -2 (1-i \sqrt{3})$ and $(1-i \sqrt{3})^2 = -2 (1+i \sqrt{3})$ -and $(1 \pm i \sqrt{3})^3 = -8$} -\begin{align} - 0 &\stackrel{!}{=} \left( \frac{(1-i \sqrt{3}) \alpha}{\sqrt[3]{12} \cdot t} - -\frac{(1+i\sqrt{3}) t}{2\sqrt[3]{18}} \right)^3 - + \alpha \left ( \frac{(1-i \sqrt{3}) \alpha}{\sqrt[3]{12} \cdot t} -\frac{(1+i\sqrt{3}) t}{2\sqrt[3]{18}}\right ) - + \beta\\%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - &= \frac{(1-i \sqrt{3})^3 \alpha^3}{12 \cdot t^3} - - 3 \left (\frac{(1-i \sqrt{3}) \alpha}{\sqrt[3]{12} \cdot t} \right )^2 \frac{(1+i\sqrt{3}) t}{2\sqrt[3]{18}} - + 3 \frac{(1-i \sqrt{3}) \alpha}{\sqrt[3]{12} \cdot t} \left (\frac{(1+i\sqrt{3}) t}{2\sqrt[3]{18}} \right)^2\\ - &\hphantom{{}=} - + \frac{(1+i\sqrt{3})^3 t^3}{2^3 \cdot 18} - + \alpha \left ( \frac{(1-i \sqrt{3}) \alpha}{\sqrt[3]{12} \cdot t} -\frac{(1+i\sqrt{3}) t}{2\sqrt[3]{18}}\right ) - + \beta\\%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - &= \frac{-8 \alpha^3}{12t^3} - - 3 \frac{-2(1+i \sqrt{3}) \alpha^2}{\sqrt[3]{12^2} t^2} \frac{(1+i\sqrt{3}) t}{2\sqrt[3]{18}} - + 3 \frac{(1-i \sqrt{3}) \alpha}{\sqrt[3]{12} \cdot t} \frac{-2(1-i\sqrt{3}) t^2}{4\sqrt[3]{18^2}}\\ - &\hphantom{{}=} - + \frac{-8 t^3}{2^3 \cdot 18} - + \alpha \left ( \frac{(1-i \sqrt{3}) \alpha}{\sqrt[3]{12} \cdot t} -\frac{(1+i\sqrt{3}) t}{2\sqrt[3]{18}}\right ) - + \beta\\%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - &= \frac{-2 \alpha^3}{3t^3} - + \frac{6 \alpha^2 t (-2)(1-i \sqrt{3})}{(\sqrt[3]{12^2} t^2)(2\sqrt[3]{18})} - + \frac{12 \alpha t^2 (1+i \sqrt{3})}{(\sqrt[3]{12} \cdot t)(4\sqrt[3]{18^2)}}\\ - &\hphantom{{}=} - + \frac{- t^3}{18} - + \alpha \left ( \frac{(1-i \sqrt{3}) \alpha}{\sqrt[3]{12} \cdot t} -\frac{(1+i\sqrt{3}) t}{2\sqrt[3]{18}}\right ) - + \beta\\%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - &= \frac{-2 \alpha^3}{3t^3} - + \frac{-6 \alpha^2 (1-i \sqrt{3})}{6 \sqrt[3]{12} t} - + \frac{3 \alpha t (1+i \sqrt{3})}{6\sqrt[3]{18}} - + \frac{- t^3}{18}\\ - &\hphantom{{}=}+ \alpha \left ( \frac{(1-i \sqrt{3}) \alpha}{\sqrt[3]{12} \cdot t} -\frac{(1+i\sqrt{3}) t}{2\sqrt[3]{18}}\right ) - + \beta\\%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - &= \frac{-2 \alpha^3}{3t^3} - + \frac{-\alpha^2 (1-i \sqrt{3})}{\sqrt[3]{12} t} - + \frac{\alpha t (1+i \sqrt{3})}{2\sqrt[3]{18}} - + \frac{- t^3}{18}\\ - &\hphantom{{}=}+ \alpha \left ( \frac{(1-i \sqrt{3}) \alpha}{\sqrt[3]{12} \cdot t} -\frac{(1+i\sqrt{3}) t}{2\sqrt[3]{18}}\right ) - + \beta\\%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - &= \frac{12 \cdot (-2 \alpha^3) +(6 \sqrt[3]{18}t^2)(-\alpha^2 (1-i \sqrt{3}))+ (3 \sqrt[3]{12})(\alpha t (1+i \sqrt{3})) + (2t^3)(- t^3)}{36t^3}\\ - &\hphantom{{}=}+ \frac{(6 \sqrt[3]{18})((1-i \sqrt{3}) \alpha) - (3 \sqrt[3]{12})((1+i\sqrt{3}) t) + 36t^3 \beta}{36t^3}\\%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\end{align} - -\goodbreak -Now calculate only the numerator: -\begin{align} - 0 &\stackrel{!}{=} -12 \alpha^3 - 6 \sqrt[3]{18} t^2 \alpha^2 (1 - i \sqrt{3}) - + 3 \sqrt[3]{12} \alpha t (1+i\sqrt{3}) - 2t^6\\ - &\hphantom{{}=} + 6\sqrt[3]{18} \alpha (1- i \sqrt{3}) - - 3 \sqrt[3]{12} t (1+i \sqrt{3}) + 36 t^3 \beta -\end{align} +The verification of this case is pretty much the same as for +case 2.2. diff --git a/documents/math-minimal-distance-to-cubic-function/quadratic-functions.tex b/documents/math-minimal-distance-to-cubic-function/quadratic-functions.tex index 9e03226..88c7b9b 100644 --- a/documents/math-minimal-distance-to-cubic-function/quadratic-functions.tex +++ b/documents/math-minimal-distance-to-cubic-function/quadratic-functions.tex @@ -171,7 +171,7 @@ I will make use of the following identities: \textbf{Case 2.1:} \input{quadratic-case-2.1} - +\goodbreak \textbf{Case 2.2:} \input{quadratic-case-2.2}