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Parametrized surfaces.

Example 3.7   Let $x=\cos u \cos v$, $y=\cos u \sin v$ and $z=\sin u$, for $0 \leq u,v, \leq 2 \pi$. This is a parametrization of the unit sphere; we have:
\begin{align*}x^2+y^2+z^2 &=\cos^2 u \cos^2 v +\cos^2 u \sin^2 v + \sin^2 u \\
...
...cos^2 v + \sin^2 v) + \sin^2 u \\
\quad & =\cos^2 u + \sin^2 u =1.
\end{align*}
As $0 \leq u,v, \leq 2 \pi$, we get the whole sphere.


  
Figure 7: The unit sphere.
\begin{figure}
\mbox{\epsfig{file=unitsphere.eps,height=6cm} }
\end{figure}

Example 3.8   Let x=(2+cos(u)) cos(v) , y=(2+cos(u)) sin(v) and z= sin(u), for $0 \leq u,v, \leq 2 \pi$. We have a torus (cf Figure  8).
  
Figure 8: A torus.
\begin{figure}
\mbox{\epsfig{file=torus.eps,height=8cm} }
\end{figure}

Example 3.9   Let x=2+cos(u)) cos(v) , y= sin(u) sin(v) and z= sin(v), for $0 \leq u,v, \leq 2 \pi$. We have a twisted torus (cf Figure  9).
  
Figure 9: A twisted torus.
\begin{figure}
\mbox{\epsfig{file=twistedtorus.eps,height=8cm} }
\end{figure}



Noah Dana-Picard
2001-05-30