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\begin{center}
\textsc{Universidad de los Andes \hfill Departamento de
Matemáticas}\newline {\bf MATE 2410 Geometría Diferencial 1 \\
Curvas Importantes}
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Prof. José Ricardo ARTEAGA %

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\begin{enumerate}

%%%%%%%%%%%%%%%%% PUNTO 1 %%%%%%%%%%%%%%%%

\item%
La \textbf{bruja de Agnesi}.
\[
W(t)=(2a \tan t, 2a \cos^2 t)
\]
\item%
\textbf{Espiral} logarítmica.
\[
\alpha (t) = (e^{bt}\cos t,e^{bt}\sin t )\qquad \kappa = \frac{1}{bs}
\]

\item%
\textbf{Cicloide}.
\[
\alpha(t)=(A+a(t-\sin t), B-a(1-\cos t))
\]
\item%
\textbf{Astroide}.
\[
\alpha (t)=(a \cos^3 t, a \sin^3 t), \qquad \alpha (t)=(a \cos^n t, a \sin^n t)
\]
\item%
\textbf{Catenaria}.
\[
y=c \cosh (x/c) \qquad \alpha (t) = (t, \cosh(\frac{t}{c}), \qquad \alpha (t) = (\cosh(\frac{t}{c},t)
\quad \kappa =\frac{-c}{c^2+s^2}
\]

\item%
\textbf{Tractriz}.
\[
\alpha (t) = a\left(\sin t,  \cos t + \ln(\tan(\frac{t}{2}))\right)
\]

\item%
\textbf{Cardiodes}.
\[
\alpha (t) = \left(2a \cos t (1 + \cos t),  2a \sin t (1 + \cos t)\right)
\]

\item%
\textbf{Lemniscatas} de Bernoulli.
\[
\alpha (t) = \left(\frac{a \cos t}{1 + \sin^2 t}, \frac{a \sin t  \cos t}{1 + \sin^2 t})\right)
\]

\item%
\textbf{Cisoide} de Diocles.
\[
\alpha (t) = \left(\frac{2at^2}{1+t^2}, \frac{2at^3}{1+t^2})\right)
\]

\item%
\textbf{Lissajous}.
\[
\alpha (t) = \left(a\sin(n t + d), b \sin t\right)
\]

\item%
\textbf{Limacon}.
\[
\alpha (t) = (2a\cos t + b)\left(\cos t, \sin t \right)
\]


\item%
Curva misteriosa
\[
\alpha (t) = \left(\frac{1}{\sqrt{2}}\cos t, \sin t, \frac{1}{\sqrt{2}}\cos t \right)
\]


\item%
\textbf{Hélice}.
\[
\alpha (t) = (a \cos t, a \sin t, bt)
\]
\item%
Curva de \textbf{Viviani}.
\[
\alpha (t) = (a(1+ \cos t), a \sin t, 2a\sin(\frac{t}{2}))
\]

\end{enumerate}

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