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\begin{document}

\begin{center}
{\bf Math 155A --- Computer Graphics --- Winter 2019 } \\
{\bf Homework \#3 ---
Due Wednesday, February 20, 9:00pm } \\
{ Hand in via Gradescope --- Use separate pages for each problem. }
\end{center}

\begin{description}
\item[1.]  Let $h(r) = 1 - r^2$ define a surface of revolution~$\vecf(r,\theta)$ by revolving
the graph of~$h$ around the $y$-axis. Give the formula
for a point $\vecf(\theta, r)$ on the surface of rotation.
Also give a formula for a normal vector (not necessarily a
unit vector) at that point on the surface of rotation. Your normal
vector should be pointing generally upward.

\item[2.] Let $S$ be the paraboloid surface
\begin{equation}\label{eq:ExerParabNormal}
S ~=~ \{ \langle x, y, z \rangle : y = 1-x^2 -z^2 \}.
\end{equation}
Use the gradient method to give a formula $\vecn = \vecn(x, y, z)$ for a
vector normal to~$S$ at a point $\langle x, y, z \rangle$
on~$S$. The vector~$\vecn$ should be pointing generally
upward.

Does your answer agree with the answer to the previous problem?
If so, how? If not, why not?

\item[3.]
A smooth surface $S$ in $\RRR$ is transformed by the linear
transformation $f:\RRR\rightarrow\RRR$ represented by the matrix
\[
N ~=~ \left( \begin{array}{ccc}
2 & 0 & 0 \\ 0 & 0 & \frac12 \\  0 & 4 & 0
\end{array} \right)
\]
to form the surface $f(S)$.
Give a $3\times3$ matrix~$M$
such that whenever $\vecn$ is normal to~$S$ at a point~$\vecx$ on~$S$,
then the vector $\vecm = M\vecn$ is normal to the
point $f(\vecx)$ on the surface $f(S)$.
[Hint: It is not difficult to invert~$N$.]

\item[4.]
The upper half of a hyperboloid~$\cal H$ is specified by
the equation $y = \sqrt{1 + x^2 + 2z^2}$. Equivalently,
${\cal H} = \{\langle x,y,z \rangle : y^2 = 1+x^2+2z^2, \, y{\ge}0\}$.
\begin{itemize}
\item[\rm (a)] Draw a rough sketch of the hyperboloid~$\cal H$.
\item[\rm (b)] Suppose $\langle x, y, z \rangle$ is
a point on~$\cal H$.  Give a formula for a vector normal
to~$\cal H$ at the point $\langle x, y, z \rangle$,
using crossproducts of partial derivatives).  Your
vector need not be a unit vector, but it should point downward
from the surface (so as to be outward facing for a viewer placed at the
origin).
\end{itemize}

\end{description}

\end{document}




