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\begin{document}

\begin{center}
{\bf Math 155A --- Computer Graphics --- Winter 2019 } \\
{\bf Homework \#7 ---
Due Wednesday, March 20, 9:00pm } \\
{ Hand in via Gradescope --- Use separate pages for each problem. }
\end{center}

\begin{description}
\item[1.] A particle moving in $\Bbb R$ (on a line) has position 
given by a cubic function $q(t)$.  At time $t=0$ it is
at position $q(0)=1$ and has initial velocity $q^\prime(0)=3$.
At time $t=1$ it is at $q(1) = 8$ and has velocity $q^\prime(1) = -6$.
Express $q(u)$ as a B\'ezier ``curve'' in~$\Bbb R$ by giving its four
control points.  

 

The convex hull property can be used to bound the position
$q(t)$ for $t\in[0,1]$. What bounds does this give? (Do not find the
actual maximum, just give the upper bound that comes from the convex hull
property.)

Express $q(u)$ as a degree three polynomial of~$t$.

\item[2] A degree three B\'ezier curve in $\RR$ has control
points 
$\vecp_0 = \langle 0,0\rangle$,
$\vecp_1 = \langle 1,-2 \rangle$,
$\vecp_2 = \langle 1,3 \rangle$, and
$\vecp_2 = \langle 3,0 \rangle$.

Draw a graph of the curve, and the results of (one round of)
recursive subdivision
with $u=\frac 12$. Show your work clearly in your sketch of the
curve. What is $q(\frac12)$?  (This is essentially the same as using
the de Casteljau algorithm to find $q(\frac12)$.)

The second half of the curve $q(u)$ where $u\in[\frac12,1]$ is
also a degree three B\'ezier curve.  What are its control points?

\item[3.] (Catmull-Rom interpolation.)
Let
$\vecp_0 = \langle 0,-2 \rangle$.
$\vecp_1 = \langle 2,0 \rangle$,
$\vecp_2 = \langle 2,1 \rangle$,
$\vecp_3 = \langle 4,1 \rangle$, and
$\vecp_4 = \langle 4,0 \rangle$.

Sketch the Catmull-Rom curve $\vecq(u)$ specified by these
points.  What is its starting point? What is its ending point?
What is the slope of $\vecq(u)$ at each interpolated point~$\vecp_i$?

The subcurve of~$\vecq$ that joins $\vecp_1$ and $\vecp_2$
is a degree B\'ezier curve.
What are its control points?


\end{description}

\end{document}




