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\begin{center}
{\bf Math 155A --- Computer Graphics --- Winter 2019 } \\
{\bf Homework \#2 ---
Due Monday, February 4, 9:00pm } \\
{ Hand in via Gradescope --- Use separate pages for each problem 1.-3. }
\end{center}

\begin{description}
\item[1.]  Suppose $\cal C$ is a radius~1, height~2 cylinder centered at the origin, with central axis the $y$-axis.
The top face of $\cal C$ is the horizontal disk of radius one centered $\langle 0,1,0 \rangle$.
The bottom face of $\cal C$ is the horizontal disk of radius one centered $\langle 0,-1,0 \rangle$.
(``Horizontal'' means parallel to the $xz$-plane.)

Let ${\cal D}$ a skewed cylinder (also called an oblique cylinder, please ``google'' it)
which has central axis the line where $y=x$ and $z=0$,
and has perpendicular height~4, with the top face of ${\cal D}$ the horizontal radius~1 disk centered at $\langle 4,4,0\rangle$
and the bottom face of ${\cal D}$ the horizontal radius~1 disk centered at $\langle 0,0,0\rangle$.
\begin{description}
\item[a.] Draw a picture of~$D$.
\item[b.] Give a $4\times$ matrix~$M$ so that $M$ represents the affine transformation sending $\cal C$ to $\cal D$.
\end{description}

\item[2.]Let $B$ be the transformation
$\langle x, y, z \rangle \mapsto \langle \frac{\textstyle 1+x}{\textstyle 1-y} - 1,\, 0,\, \frac {\textstyle z}{\textstyle 1-y}\rangle $.
Give a $4\times 4$ matrix which represents this transformation over homogeneous coordinates.
(When $0<y<1$, $B$~give the transformation for a shadow cast from a light at $\langle -1,1,0 \rangle$
onto the plane $y=0$. You do {\bf not} need to use this fact to work the problem!)

\item[3.]
A light source is placed at $\langle -10, 0 , 0 \rangle$ and it casts shadows onto
the $yz$-plane~$P$ defined by $x=0$.
The $yz$-plane is like an infinite wall.  \newline\newline
When $\langle x,y,z\rangle$ is a point in ${\mathbb R}^3$ with $-10 {<} x{\le}0$, define
$A(\langle x,y,z\rangle)$ to be the position of the shadow of
the point on the $yz$-plane.  For example, $A(\langle-5,1,2\rangle)=\langle0,2,4\rangle$,
and $A(\langle-8,1,2\rangle)=\langle0,5,10\rangle$
\begin{description}
\item[a.] Working in ordinary coordinates (not homogeneous) give the formula
expressing the mapping
$A(\langle x,y,z \rangle) = \langle x^\prime,y^\prime, z^\prime \rangle$.
That is, give formulas for $x^\prime,y^\prime, z^\prime$ in terms of $x,y,z$.
\item[b.]
Give a $4\times 4$-matrix that represents the transformation~$A$ over
homogeneous coordinates.
\end{description}
\end{description}

\end{document}




