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\begin{center}
{\bf Math 155A --- Computer Graphics --- Winter 2019 } \\
{\bf Homework \#1 ---
Due Tuesday, January 29, 9:00pm } \\
{ Hand in via Gradescope --- Use separate pages for each problem 1.-5. }
\end{center}

References to figures are mostly in draft A.2.c of the second edition. \newline
(At \hbox{http://www.math.ucsd.edu/~sbuss/CourseWeb/Math155A\_2019Winter/SecondEdDraft.pdf.})

\begin{description}
\item[1. (a)] 
Define functions
\begin{itemize}
\item $f_1 : \langle x_1, x_2 \rangle \mapsto \langle x_2, -x_1 \rangle$, and
\item $f_2 : \langle x_1, x_2 \rangle \mapsto \langle 2 x_1, -\frac 12 x_2 \rangle$, and
\item $f_3 : \langle x_1, x_2 \rangle \mapsto \langle x_1-\frac 12 x_2,x_2 \rangle$, and
\item $f_4 : \langle x_1, x_2 \rangle \mapsto \langle x_2, x_1-\frac 12 x_2 \rangle$.
\end{itemize}
Verify (but do not show your work) that each $f_i$ is linear.
Draw figures showing how these
four functions transform the ``{\tt F}''-shape.
Your four figures should be similar to the drawing on the right in Figure~II.3 on
page~22 in the first edition,
or in the drawings Figure~II.3 on page~36 (and Figures II.4 and II.5 on page~40) of the second edition draft A.2.c.
\item[~~~~(b)] Consider the functions $f_1,\ldots,f_4$ from problem~1.
Which of the four $f_i$'s are affine? Which of the four $f_i$'s are rigid? Which of the $f_i$'s are
orientation-preserving?

\item[2.] Give the $2\times 2$ matrices representing the linear transformations
in $\RR$ in parts (b), (c) and (e) of Figure~II.3 on page 36 (second edition draft A.2.c).
They are a uniform scaling $S_{\frac 12}$, a nonuniform scaling $S_{\langle \frac32, \frac12\rangle}$
and a shearing transformation.

\item[3. (a)] Let $f$ be the affine transformation of $\RR$ shown in Figure~II.5
on page 40 of draft A.2.c. Express this function as an affine function
by writing it explicitly in the form $f(\vecx) = M\vecx + \vecu$
where $M$ is a $2\times 2$ matrix.  Then give its
$3\times 3$ matrix represention.

\item[~~~~~(b)] Now express its inverse~$f^{-1}$
in the form $N\vecx + \vecv$ where $N$ is a $2\times 2$ matrix.
Second, give the $3\times 3$
matrix representing the inverse $f^{-1}$ of~$f$.

\item[4.] Let $T_{\vecu}$ and $R_{\theta,\vecu}$ be the translation and rotation
operators in $\RRR$. Give the $4\times 4$ matrix representations of
$T_{\veci-\veck}\circ R_{\pi/2,\vecj}$ and of
$R_{\pi/2,\vecj} \circ T_{\veci-\veck}$.  (You can do this either
by multiplying matrices or by visualizing. If you do it with matrices, you
are highly recommended to try visualizing afterwards.)

\item[5.] A $2\times2\times2$ cube has the eight vertices $\langle \pm1, \pm1, \pm1 \rangle$.
Show how to render the six faces of the cube with two triangle fans, by explicitly
listing the vertices used for the triangle fans.  Make sure that the usual CCW
front faces are facing outwards. (There is more than one way to do this, but please use
$\langle 1,1,1 \rangle$ as the first (central) vertex for one of the triangle fans.)
\end{description}

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