\input header
\input epsf

\title{Subdivision Surfaces: exercises}

\section{Biquadratic subdivision}

\firstsubsection{Apply two steps of biquadratic subdivision to a cube.}

You will need some coefficients for the triangle face.  Use the stencil
$$\matrix{1\cr4&1} /6$$
and its rotations.

\subsection{Apply two steps of biquadratic subdivision to a coarse toroidal shape with 16 vertices.}

\section{Box-splines}

\firstsubsection{ Expand out the generating function
$$(1+x)^2(1+y)^2(1+xy)^2$$
to give a mask.  What are the stencils ?}

\subsection{ Expand out the generating function
$$(1+x+x^2)^2(1+y+y^2)^2(1+xy+x^2y^2)^2$$
to give a mask.  What are the stencils ?}

\section{Others}

\firstsubsection{ Determine the mask which results when 
$$\matrix{
&&-1&&-1\cr
&0&&0&&0\cr
-1&&10&&10&&-1\cr
&0&&32&&0\cr
-1&&10&&10&&-1\cr
&0&&0&&0\cr
&&-1&&-1}\ /32$$
is applied twice.}

Remember that this is a $\sqrt2$ scheme, and so the configuration needs to be rotated through 45 degrees between the two steps.

\subsection{Determine the mask which results when 
$$\matrix{
&&1\cr
&2&&2\cr
1&&4&&1\cr
&2&&2\cr
&&1}\ /8$$
is applied twice.}

Remember that this is a $\sqrt2$ scheme, and so the configuration needs to be rotated through 45 degrees between the two steps.

\section{Generation of all subdivision schemes}

\pop Sketch one example of each of the eight categories possible within the combinations of quad, tri and hex grids and primal, dual, both and neither types.

Draw the new grid in one colour and the old one in another.

\pop Sketch one example on a quad grid, one on a tri-grid and one on a hex-grid each of which has a complex arity.

\end

