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

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\begin{enumerate}

\item%
Parametrización de \textbf{Monge} para la superficie $z=f(x,y)$.
\[
X(u,v)=(u,v,f(u,v))
\]
%---------------------------------------------
\item%
\textbf{Catenoide}.
\[
X(u,v)=\left[u,\cosh(u) \cos(v), \cosh(u) \sin(v)\right]
\]
%---------------------------------------------
\item%
\textbf{Helicoide}.
\[
  X(u,v)=\left[ u \cos v,u \sin v, v \right]
\]
%---------------------------------------------
\item%
Superficie de \textbf{Enneper}.
\[
X(u,v)=\left[u-\frac{1}{3}u^3+uv^2,v-\frac{1}{3}v^3+vu^2,u^2-v^2\right]
\]
Para graficar puede tomar $-2\leq u,\quad v\leq 2$ .
%---------------------------------------------
\item%
\textbf{Scherk}.
\[
X(u,v)= \left[ u, v, \ln\left(\frac{\cos u}{\cos
v}\right)\right]
\]
%---------------------------------------------
\item%
\textbf{Catalan}.
\[
X(u,v)= \left[%
u-\sin u \cosh v,%
1-\cos u \cosh v,%
4\sin(u/2)\sinh(v/2)%
\right]
\]
%---------------------------------------------
\item%
\textbf{Henneberg}.
\[
X(u,v)= \left[%
2\sinh u \cos v-\frac{2}{3}\sinh 3u \cos 3v,%
2\sinh u \sin v+\frac{2}{3}\sinh 3u \sin 3v,%
2\cosh 2u \cos 2v%
\right]
\]

%---------------------------------------------
\item%
Superficie ``\textbf{rara}''
\[
X(u,v)=\left[
\frac{cu\pm \sin(u) \cosh(v)}{\sqrt{1-c^2}},
\frac{v\pm \cos(u) \sinh(v)}{\sqrt{1-c^2}},
\cos(u) \cosh(v)\right]
\]
Para graficar puede tomar $-3\leq u,\quad v\leq 3$ .
%---------------------------------------------
\item%
\textbf{Cilindro} vertical.
\[
\frac{x^2}{a^2} +  \frac{y^2}{a^2} =1
\]
%---------------------------------------------
\item%
\textbf{Silla} de montar (paraboloide hiperbólico).
\[
z=xy \qquad \qquad  z=x^2-y^2
\]
Use parametrización de Monge, para hallar $X=X(u,v)$
%---------------------------------------------
\item%
\textbf{Toro}.
\[
X(u,v)=\left[(R+r\cos u)\cos v, (R+r\cos u)\sin v, r\sin u\right]
\]
\end{enumerate}
%---------------------------------------------
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