
\hyphenation{Bool-ean com-plete other-wise manu-script}

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\def\vecp{{\mathbf p}}
\def\vecq{{\mathbf q}}
\def\vecr{{\mathbf r}}
\def\vecu{{\mathbf u}}
\def\vecv{{\mathbf v}}
\def\vecw{{\mathbf w}}
\def\vecx{{\mathbf x}}
\def\vecy{{\mathbf y}}
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\def\veczero{{\mathbf 0}}

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\begin{document}

\begin{center}
{\bf Math 155B --- Computer Graphics --- Spring 2018 } \\
{\bf Homework \#1 ---
Due 9:00pm, Monday, April 23 } \\
{\bf Hand in via GradeScope }
\end{center}


\noindent
{\bf 1.} Let $\vecq(u)$ be a degree 2 Bezier curve with
control points $\langle 0,0,0 \rangle$,
$\langle 3, 6,0 \rangle$, 
and $\langle 6, -6, 0 \rangle$.
Express $\vecq (u)$ as a degree three
Bezier curve (by giving its control points).
Also express $\vecq(u)$ as a degree four
Bezier curve.

\noindent
{\bf 2.} Suppose a degree $3\times 2$
Bezier patch with control points $\vecr_{i,j}$ is defined by
\[
\vecq(u) ~=~ \sum_{i=0}^3 \sum_{j=0}^2 B^3_i(u) B^2_j(u) \vecr_{i,j}.
\]
Express this patch as a degree $3\times 3$ Bezier patch with
control points $p_{i,j}$. Your answer should give formulas
for the $\vecp_{i,j}$'s in terms of the $\vecr_{i,j}$.
Give a brief proof of the correctness of your answer. (Hint: use
Theorem VII.8.)

\noindent
{\bf 3.} Express the ``right half'' of  the unit sphere, 
\[
\{ \langle x, y, z\rangle : x\ge 0 \hbox{~and~} x^2+y^2+z^2 = 1 \} ,
\]
as a single rational Bezier patch.

\noindent
{\bf 4.} Exercise VII.14, page 184.

\medskip

\noindent
{\bf 5.} Exercise VII.20, page 186. (You may find it easier to first
think about exercise VII.19 instead.)

\medskip

\noindent
{\bf 6.} Exercise VII.24, page 191.

\medskip

\noindent
{\bf 7.} (Linear interpolation in homogeneous coordinates.)
Suppose $w, v > 0$. Let $\langle w\vecx, w \rangle$ and $\langle v\vecy, v \rangle$
be homogeneous representations of two points $\vecx$ and $\vecy$ in~$\Reals^n$. For $0\le u\le 1$,
define $\vecq(u)$ to be the point in $\Reals^n$ represented
in homogeneous coordinates by the $(n{+}1)$-tuple

\[
\lerp(u,\,\langle w\vecx,w \rangle,\,\langle v\vecy, v \rangle ) .
\]
Evaluate the
first derivative~$\vecq^\prime(u)$ of~$\vecq(u)$. What are $\vecq^\prime(0)$ and
$\vecq^\prime(1)$?   Can you express your answer in terms of $w$, $v$, and $\vecy{-}\vecx$?
Does your answer agree with the case of ordinary linear
interpolation where $v=w= 1$?



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