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\begin{document}

\section*{Computer Graphics - Excercise 2}
\begin{align*}
	f(u,v) = R \begin{pmatrix} cos(u) \\ sin(u) \\ 0 \end{pmatrix} + r \begin{pmatrix} cos(u) \times cos(v) \\ sin(u) \times cos(v) \\ sin(v) \end{pmatrix}
\end{align*}
\subsection*{2.1}
\subsection*{a)}
\begin{align*}
		\frac {\delta f }{\delta u} (u,v) &= \begin{pmatrix}  R (-sin(u)) + r (cos(v) \times -sin(u)) \\ R \times cos(u) + r(cos(v)\times cos(u)) \\ 0 \end{pmatrix}\\
		\\
		\frac {\delta f }{\delta v} (u,v) &= \begin{pmatrix}   r (cos(u) \times (-sin(v))) \\ r(-sin(v) \times sin(u)) \\ r \times cos(v) \end{pmatrix}\\
		\\
		\Rightarrow Jacobian:\\
		J_f (u,f) &=\begin{pmatrix} R \times (-sin(u)) +r (cos(v) \times -sin(u)) & r (cos(u) \times (-sin(v))) \\ sin(u) \times cos(v) & r(-sin(v) \times sin(u)) \\ 0 & r \times cos(v)\end{pmatrix}\\
\end{align*}
\subsection*{b)}
$I_{\tau}^s = \begin{pmatrix} E & F \\ F & G \end{pmatrix}$ mit $s_u=\frac {\delta f }{\delta u} (u,v), s_v=\frac {\delta f }{\delta v} (u,v)\\
\\
E = ||s_u||^2 = \\ 
\sqrt{(R (-sin(u)) + r (cos(v) \times -sin(u)))^2+(R \times cos(u) + r(cos(v)\times cos(u)))^2+0^2}^2 = \\
\sqrt{R^2 ((-sin(u))^2+cos(u)^2)+2Rrcos(v)((-sin(u))^2 +cos(u)^2)+r^2 cos(v)((-sin(u))^2+cos(u)^2)}^2 =
\sqrt{(R^2 + 2Rrcos(v) + r^2cos(v)^2}^2 = (r(cos(u))+R)^2\\
\\
F = < s_u , s_v > = 0\\
\\
G = ||s_v||^2 = \\
\sqrt{(r (cos(u) \times (-sin(v))))^2+(r(-sin(v) \times sin(u)))^2+(r \times cos(v))^2}^2 =\\
\sqrt{(r^2 \times cos(u)^2 \times (-sin(v))^2)+(r^2((-sin(v))^2 \times sin(u)^2))+(r^2 \times cos(v)^2)}^2 =\\
\sqrt{r^2(sin(v)^2 (cos(u)^2 \times sin(u)^2)+cos(v)^2}^2 =\\
\sqrt{r^2(sin(v)^2+cos(v)^2}^2 = r^2\\
$
\\
$\Rightarrow I_{\tau}^f = \begin{pmatrix} (r(cos(u))+R)^2 & 0 \\ 0 & r^2 \end{pmatrix}$

\subsection*{c)}

Aus der Vorlesung: $dA = \sqrt{det I_{\tau}^f} \,du\,dv$\\
\begin{align*}
\Rightarrow dA &= \sqrt{det \begin{pmatrix} (r(cos(u))+R)^2 & 0 \\ 0 & r^2 \end{pmatrix}} \,du\,dv\\
\Rightarrow dA &=  r(r(cos(u))+R) \,du\,dv\\
\end{align*}
\\
$\Rightarrow A = \int_{0}^{2\pi} \int_{0}^{2\pi}  r(r(cos(u))+R) \,du\,dv = 2 \pi r \times  \int_{0}^{2\pi}  r(cos(u))+R \,dv =\\\\ 2 \pi r [Rv + r(sin(v)]_{0}^{2\pi} = 2 \pi r \times (2\pi R) = \underline{\underline{4\pi^2Rr}} $
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