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\title{Subdivision Curve Definition and Analysis Exercises}

\section{Definition by Box-splines}

\subsection{Determine the stencils for the binary univariate box-splines of degrees 3, 4 and 5}


\subsection{Determine the stencils for the quarternary (arity=4) box spline of degree 2.}


\section{Dubuc-Deslauriers interpolating refinements}

\subsection{Determine the masks for the binary 4-pt and 6-pt schemes}



\subsection{Determine the mask for the Dubuc-Deslauriers ternary 4-pt scheme}


\section{z-transform analysis of convergence}

\firstsubsection {Making a difference mask}

The z-transform approach to analysing continuity and convergence shows how, given a subdivision mask, a derived mask can be computed, which relates differences of the subdivided polygon to those of the original.

$$S'(z) = {{(1-z)}\over{(1-z^a)}}S(z)$$

This form begs the question of whether $(1-z)S(z)$ is actually divisible without remainder by $(1-z^a)$.   If it is not, then the above expression does not give a Laurent Polynomial.

\pop Prove that if a scheme $S$ has constant precision, $(1-z)S(z)$ is exactly divisible by $(1-z^a)$

\pop Prove that if a scheme $S$ has both constant precision and (functional) linear precision, we may construct derived schemes for both first and second differences.


\section{Eigenanalysis}

\firstsubsection{Carry out the odd and even eigenanalyses for [1,4,4,1]/5} 


\subsection{Carry out the odd and even eigenanalyses for [-1,0,9,16,9,0,-1]/16} 



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