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

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\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}}
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\begin{document}

\begin{center}
{\bf Math 155B --- Computer Graphics --- Spring 2017 } \\
{\bf Homework \#2 ---
Due 9:00pm, Friday, May 5 } \\
{\bf Hand in via GradeScope }
\end{center}


\noindent
Please show work and label answers clearly.

\noindent
{\bf 1.} Consider a degree 3 (order 4) B-spline curve
$\vecq(u)$ defined with the knot vector
$[ 0,0, 1, 2, 3, 3, 4, 5,5,5, 6, 7, 8, 9, 10]$.
\begin{enumerate}
\setlength{\itemsep}{0pt}
\item[a.] How many control points $\vecp_0, \vecp_1, \ldots$ are
needed to define the curve~$\vecq (u)$?
\item[a.] What is the domain of $\vecq(u)$?
\item[b.] What is the domain of $N_{0,4}(u)$?
\item[c.] What is the domain of $N_{1,4}(u)$?
\item[d.] What is the domain of $N_{2,4}(u)$?
\item[e.] What is the domain of $N_{6,4}(u)$?
\item[f.] At $u=2$, $\vecq(u)$ must be $C^\ell$-continuous, for what value~$\ell$?
\item[g.] At $u=3$, $\vecq(u)$ must be $C^\ell$-continuous, for what value~$\ell$?
\item[h.] At $u=5$, $\vecq(u)$ must be $C^\ell$-continuous, for what value~$\ell$?
\item[i.] At $u=5.5$, $\vecq(u)$ must be $C^\ell$-continuous, for what value~$\ell$?
\end{enumerate}

\medskip

\noindent
{\bf 2.} [Similar to Exercise VIII.6, page 211. See
Figure VIII.9 also on page 211.]
Consider a degree 2, order 3 B-spline curve defined
with the knot vector
$[0,0,0,1,2,3,4,4,4]$. How many control points does this
curve need to be well-defined?  Give the
formulas for the following functions:
\begin{enumerate}
\item[a.] The functions $N_{0,2}$, $N_{1,2}$, $N_{2,2}$, $N_{3,2}$.
[Hint: They are piece-wise linear].
\item[b.] Only on the interval $[0,1]$:
the functions $N_{0,3}$, $N_{1,3}$, $N_{2,3}$.
\end{enumerate}
Verify that $N_{0,3}(u)+N_{1,3}(u)+N_{2,3}(u)$
is equal to 1.
\medskip

\noindent
{\bf 3.} Page 231 expresses a system of linear equations
using a tridiagonal matrix.  The algorithm on that page solves
that system to obtain values for the $\vecp_i$'s
from values for the $\vecq_i$'s.  A tridiagonal matrix
is also called a ``band matrix with $k_1=k_2 =1 $''.
The value $k_1$ is the number of non-zero entries allowed immediately before
the diagonal entry, the value $k_2$ is the number
of non-zero entries allowed immediately after the diagonal entry.

This can be generalized to a ``band matrix with $k_1=1$ and $k_2=2$''
by allowing one more non-zero entry~$\delta_i$ on each $i$-th row $(1\le i \le n{-}1)$.
Assume the first row is unchanged and still $1~0~0 \cdots$; the last row is
likewise unchanged; the $i$-th row has
``$\alpha_i~\beta_i~\gamma_i~0$'' replaced with
``$\alpha_i~\beta_i~\gamma_i~\delta_i$'' and otherwise all zero.
You may assume (without checking) that a divide-be-zero condition never
occurs.

Problem: Rewrite the algorithm on page 231 to work with matrices of this form.
\medskip


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