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fruitstaa 2018-12-10 13:18:40 +01:00
parent a635a50205
commit 130c08f054
2 changed files with 13 additions and 2 deletions
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exercise2/Theorie.pdf
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exercise2/Theorie.tex
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@ -11,6 +11,9 @@
\usepackage{enumerate} \usepackage{enumerate}
\usepackage{geometry}
\geometry{a4paper, top=25mm, left=18mm, right=15mm, bottom=30mm,
headsep=10mm, footskip=12mm}
\begin{document} \begin{document}
@ -32,11 +35,19 @@
\subsection*{b)} \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)\\ $I_{\tau}^s = \begin{pmatrix} E & F \\ F & G \end{pmatrix}$ mit $s_u=\frac {\delta f }{\delta u} (u,v), s_v=\frac {\delta f }{\delta v} (u,v)\\
\\ \\
E = ||s_u||^2 = (r(cos(u))+R)^2\\ E = ||s_u||^2 = \\
\sqrt{(R (-sin(u)) + r (cos(v) \times -sin(u)))^2+(R \times cos(u) + r(cos(v)\times cos(u)))^2+0^2}^2 = \\
\sqrt{R^2 ((-sin(u))^2+cos(u)^2)+2Rrcos(v)((-sin(u))^2 +cos(u)^2)+r^2 cos(v)((-sin(u))^2+cos(u)^2)}^2 =
\sqrt{(R^2 + 2Rrcos(v) + r^2cos(v)^2}^2 = (r(cos(u))+R)^2\\
\\ \\
F = < s_u , s_v > = 0\\ F = < s_u , s_v > = 0\\
\\ \\
G = ||s_v||^2 = r^2\\$ 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\\
$
\\ \\
$\Rightarrow I_{\tau}^f = \begin{pmatrix} (r(cos(u))+R)^2 & 0 \\ 0 & r^2 \end{pmatrix}$ $\Rightarrow I_{\tau}^f = \begin{pmatrix} (r(cos(u))+R)^2 & 0 \\ 0 & r^2 \end{pmatrix}$