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\def\Reals{{\mathbb R}}
\def\RR{{\Reals^2}}
\def\RRR{{\Reals^3}}
\def\veca{{\mathbf a}}
\def\vecb{{\mathbf b}}
\def\vecc{{\mathbf c}}
\def\vecd{{\mathbf d}}
\def\vece{{\mathbf e}}
\def\vecf{{\mathbf f}}
\def\veci{{\mathbf i}}
\def\vecj{{\mathbf j}}
\def\veck{{\mathbf k}}
\def\vecm{{\mathbf m}}
\def\vecn{{\mathbf n}}
\def\vecp{{\mathbf p}}
\def\vecq{{\mathbf q}}
\def\vecu{{\mathbf u}}
\def\vecv{{\mathbf v}}
\def\vecw{{\mathbf w}}
\def\vecx{{\mathbf x}}
\def\vecy{{\mathbf y}}
\def\vecz{{\mathbf z}}
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\begin{document}

\begin{center}
{\bf Math 155A --- Computer Graphics --- Winter 2019 } \\
{\bf Homework \#6 ---
Due Thursday, March 14, 9:00pm } \\
{ Hand in via Gradescope --- Use separate pages for each problem. }
\end{center}

\begin{description}
\item[1.] Suppose a degree three B\'ezier curve $\vecq(u)$ in $\RR$
has control points $\vecp_0 = \langle -1,0\rangle$,
$\vecp_1 = \langle 0,-2 \rangle$,
$\vecp_2 = \langle 3,0 \rangle$, and
$\vecp_2 = \langle 1,3 \rangle$.
Draw a graph showing these three points,
the control polygon, and a sketch of
$\vecq(u)$ for $0\le u \le 1$. Your sketch should
show the starting and ending positions and tangencies
clearly.
What are $\vecq'(0)$
and $\vecq'(1)$ equal to?

\item[2.] Suppose a
particle needs to be animated so that at time $t=0$ it is at
$\langle 0,0\rangle$ with velocity $\langle 0,3 \rangle$, and at
time $t=1$ it is at $\langle 0,3 \rangle$ with velocity
$\langle -2,0 \rangle$.  Express the position of the
particle, as a function of time~$t$, in terms of
a degree three B\'ezier curve. What are the control points for
the B\'ezier curve?

\item[3.]  Repeat problem 2 with the same position and velocity
at time $t=0$ and the same position at time $t=1$, but with the
ending velocity at time $t=1$ equal to $\veczero = \langle 0,0 \rangle$.


\end{description}

\end{document}




