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\begin{document}
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\shadowbox{\Large Lijepljenje torusa}\\[2pt]
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Torus je ploha koja ima implicitnu jednad\v{z}bu
$$\big(x^2+y^2+z^2+R^2-r^2\big)^2-4R^2\big(x^2+y^2\big)=0.$$
Za razli\v{c}ite odabire konstanti $R$ i $r$ dobivamo druk\v{c}ije toruse.\\[5pt]
Dvostruki torus nastaje lijepljenjem dva torusa. Proces lijepljenja odgovara produktu
implicitnih jednad\v{z}bi torusa umanjenom za neku konstantu. Konkretnije, implicitna
jednad\v{z}ba dvostrukog torusa glasi
$$f_1(x,y,z)\cdot f_2(x,y,z)=c,$$
pri \v{c}emu je
\begin{align*}
f_1(x,y,z)&=\big((x+a)^2+y^2+z^2+R^2-r^2\big)^2-4R^2\big((x+a)^2+y^2\big)\\
f_2(x,y,z)&=\big((x+b)^2+y^2+z^2+R^2-r^2\big)^2-4R^2\big((x+b)^2+y^2\big)
\end{align*}
za neke odabrane konstante $a,b,c,r,R$.\\[5pt]
Trostruki torus nastaje lijepljenjem tri torusa. Mo\v{z}emo ga vizualizirati preko
implicitne jednad\v{z}be
$$g_1(x,y,z)\cdot g_2(x,y,z)\cdot g_3(x,y,z)=c,$$
pri \v{c}emu je
\begin{align*}
g_1(x,y,z)&=\big((x+a)^2+y^2+z^2+R^2-r^2\big)^2-4R^2\big((x+a)^2+y^2\big)\\
g_2(x,y,z)&=\big((x+b_1)^2+(y+b_2)^2+z^2+R^2-r^2\big)^2-4R^2\big((x+b_1)^2+(y+b_2)^2\big)\\
g_3(x,y,z)&=\big((x+c_1)^2+(y+c_2)^2+z^2+R^2-r^2\big)^2-4R^2\big((x+c_1)^2+(y+c_2)^2\big)
\end{align*}
i vrijedi $b_1=a\cos{\frac{2}{3}\pi}$, $b_2=a\sin{\frac{2}{3}\pi}$, $c_1=a\cos{\frac{4}{3}\pi}$,
$c_2=a\sin{\frac{4}{3}\pi}$ za neke odabrane\vspace*{2pt} konstante $a,c,r,R$. Ako promijenimo konstante $b_2$ i $c_2$ tako da stavimo $b_2=0$ i $c_2=0$, dobivamo modificirani trostruki torus.
\begin{figure}[!h]
\centering
\subcaptionbox{dvostruki torus}{\includegraphics[scale=0.1]{dvostrukiTorus.png}}\hspace*{0.5cm}
\subcaptionbox{trostruki torus}{\includegraphics[scale=0.1]{trostrukiTorus.png}}\hspace*{0.5cm}
\subcaptionbox{modificirani trostruki torus}{\includegraphics[scale=0.1]{smijesniModel.png}}
%\caption{Dvostruki i trostruki torus}
\end{figure}
 
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