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Revise on the section 1 and 2 of the user manual

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Packages/Subdivision_surfaces_3/doc_tex/Subdivision_surfaces_3/main.tex
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@ -22,7 +22,7 @@
\ccParDims
\chapter{Subdivision Surfaces}
\chapter{Subdivision Methods}
\label{chapterSubdivision}
\ccChapterRelease{\subdivisionRev. \ \subdivisionDate}
\ccChapterAuthor{Le-Jeng Andy Shiue}
@ -41,15 +41,15 @@
% +------------------------------------------------------------------------+
\section{Introduction} \label{sectionSubIntro}
% +------------------------------------------------------------------------+
The subdivision algorithm is a simple yet powerful way to
The subdivision method is a simple yet powerful way to
generate smooth surfaces from arbitrary polyhedral meshes.
Unlike spline-based surfaces (e.g NURBS) or other numeric-based
modeling techniques, users of subdivision
surfaces do not need in-depth mathematics
knowledge on subdivision algorithms. An artist only need
knowledge on subdivision methods. Users only need
natural geometric intuitions to control the subdivision
surfaces. It is hence especially popular for character
modeling in video games and character animations.
surface; hence subdivision methods are popular choices for
character modeling in video games and character animations.
%Subdivision algorithms (see e.g.~\cite{cgal:ww-smgd-02})
%recursively refine coarse meshes and generate ever closer
@ -64,13 +64,14 @@ modeling in video games and character animations.
%Each subdivision algorithm is a refinement function parametrized
%by a set of rountines of geometric averageing.
While many subdivision schemes have been developed and new schemes
are still being researched, \ccc{Subdivision_surfaces_3}
supports the families of the four most used schemes:
Loop, Catmull-Clark, Doo-Sabin and $\sqrt{3}$ subdivisions.
\ccc{Subdivision_surfaces_3}, working on the concept of
the \ccc{CGAL::Polyhedron_3} (see Chapter~\ref{chapterPolyhedron}),
aims to be simple to use and powerful to extend.
\ccc{Subdivision_surfaces_3} aims to be simple to use and extend.
It supports four most used subdivision methods, including Loop,
Catmull-Clark, Doo-Sabin and $\sqrt{3}$ subdivisions. Variations
of these methods can be easily implemented by plugging
in the user-defined computation rules. \ccc{Subdivision_surfaces_3}
is designed to work on \ccc{CGAL::Polyhedron_3}
(see Chapter~\ref{chapterPolyhedron}),
\begin{ccHtmlOnly}
<CENTER>
@ -83,30 +84,44 @@ aims to be simple to use and powerful to extend.
\label{secSubAlgo}
% +------------------------------------------------------------------------+
In this chapter, we explain some fundamentals of
subdivision surfaces. We only focus on the topics helping you
subdivision methods. We only focus on the topics helping you
to understand the design of the package; interested readers can
still refer to \cite{cgal:ww-smgd-02} for an in-depth introduction.
Some terminologies introduced in this section will be used again
in later sections. For users just want to use Loop (or
Catmull-Clark ...etc) subdivision, Section \ref{secFirstSub}
gives a quick tutorial on using these subdivisions.
in later sections. If you are only interested in using a
specific subdivision method, Section \ref{secFirstSub}
gives a quick tutorial on Catmull-Clark subdivision.
Subdivision algorithms recursively refine coarse meshes and generate
ever closer approximations to a smooth surface.
A subdivision method recursively refines a coarse mesh and
generates an ever closer approximation to a smooth surface.
The coarse mesh can have arbitrary shapes, but it has to
be a 2-manifold. In a 2-manifold, every interior point has
a neighborhood homomorphic to a 2D disk. Subdivision methods
on non-manifolds have be developed, but are not considered
in this work.
%Subdivision algorithms (see e.g.~\cite{cgal:ww-smgd-02})
%define surfaces as the limit
%of recursive refinement of a polyhedral mesh, called
%\emph{control mesh}.
%The refinements generate mesh points
%to be smoothed to approximate the limit surface.
The chapter teaser shows a sequence of a subdivision on
a chess model.
The chapter teaser shows the process of a subdivision method on
a chess model. The coarse chess is repeatly refined by a pattern,
and more and more points are generated to approximate the
smooth chess.
Many refinement patterns are used in practice.
\ccc{Subdivision_surfaces_3} supports the four most popular
patterns, and each of them is used by Loop,
Catmull-Clark\cite{cgal:cc-rgbss-78}, Doo-Sabin
and $\sqrt{3}$ subdivision.
patterns, and each of them is used by
Catmull-Clark\cite{cgal:cc-rgbss-78}, Loop, Doo-Sabin
and $\sqrt{3}$ subdivision (left to right in the
figure). Note we name these patterns by their topological
characteristics, not the associated subdivision methods.
PQQ indicates the \emph{P}rimal \emph{Q}uadtrateral \emph{Q}uadrisection.
PTQ indicates the \emph{P}rimal \emph{T}riangle \emph{Q}uadrisection.
DQQ indicates the \emph{D}ual \emph{Q}uadtrateral \emph{Q}uadrisection.
$\sqrt{3}$ has a long story of its naming; simply put, it
indicates the speed of the topological converging.
\begin{ccTexOnly}
\begin{center}
@ -122,12 +137,14 @@ and $\sqrt{3}$ subdivision.
</CENTER>
\end{ccHtmlOnly}
Four refinement patterns are demonstrated on a valence-n mesh.
The refined mesh is shown below the input
mesh (called \emph{control mesh}).
The figure demonstrates these four refinement patterns on
the 1-disk of a valence-5 vertex/facet.
Refined meshes are shown below the source meshes.
%We call a source mesh \emph{control mesh} if it is
%the origin of a squence of refinements.
Points on the refined mesh are generated by averaging
neighbor points on the input mesh. A graph, called \emph{stencil},
determines the input submesh whose points contribute to the
neighbor points on the source mesh. A graph, called \emph{stencil},
determines the source neighthood whose points contribute to the
position of a refined point. A refinement pattern usually has
several specific sets of stencils.
%Stencils are defined at the time the refinement pattern is chosen.
@ -137,7 +154,10 @@ For example, the PQQ
refinement used by Catmull-Clark subdivision has a vertex-node stencil,
which defines the 1-ring of an input vertex; an edge-node stencil,
which defines the 1-ring of an input edge; and a facet-node stencil,
which defines an input facet.
which defines an input facet. The stecils of the PQQ refinement are
shown in the following figure. The blue neighborhoods in the
top row indicate the corresponding stencils of the refined nodes
in red.
\begin{ccTexOnly}
\begin{center}
@ -154,14 +174,16 @@ which defines an input facet.
\end{ccHtmlOnly}
The blue neighborhoods in the top row indicate the corresponding
stencils of the refined nodes in red.
%while the DQQ scheme has only a corner-node stencil, which
%relates the facet of a corner to a target node.
Stencils with weights are called \emph{geometry masks}.
%One practical set of geometry masks of the PQQ scheme is
The geometric masks of Catmull-Clark subdivision are shown below.
A subdivision method defines the masks for the stencils, and
generates new points by summarizing source points weighted by the mask.
Geometry masks are usually carefully choosed to meet requirements of
certain surface smoothness and shape quality.
The geometry masks of Catmull-Clark subdivision are shown
below (masks for extraordinary vertices are not shown).
\begin{ccTexOnly}
\begin{center}
@ -178,19 +200,17 @@ The geometric masks of Catmull-Clark subdivision are shown below.
\end{ccHtmlOnly}
The weights shown here are unnormalized, and $n$ is the valence
of the vertex. The subdivided point, in red, is computed by a summation
of the weighted points on its stencil (i.e.~an input submesh).
For example, a Catmull-Clark facet-node is computed by the summation
of $1/4$ of each stencil node.
of the vertex. The generated point, in red, is computed by a summation
of the weighted points. For example, a Catmull-Clark facet-node is
computed by the summation of $1/4$ of each points on its stencil.
A stencil may have several specific sets of geometry masks. In other
words, a facet-node of Catmull-Clark refinement may be computed by
the summation of $1/5$ of each stencil node instead of $1/4$.
Although it is legal in \ccc{Subdivision_surfaces_3} to have
any kind of geometry mask, the result surfaces may be odd,
not smooth, or not even exist.
To learn how to design a good mask, both mathematically and esthetically,
readers can refer to \cite{cgal:ww-smgd-02}.
not smooth, or not even exist. \cite{cgal:ww-smgd-02} explains the
details on designing masks of quality subdivision surfaces.
%The averaging process can typically be factored into
%simpler steps \cite{Oswald-2003-CSS}.
@ -224,15 +244,14 @@ readers can refer to \cite{cgal:ww-smgd-02}.
% +-------------------------------------------------------------+
\section{Your First CGAL::Subdivision}
\label{secFirstSub}
Assuming that you already have a \emph{normal} CGAL polyhedral mesh,
say the default one, you can use the \ccc{Subdivision_surfaces_3}
without much effort.
With a default CGAL polyhedral mesh, you can
use the \ccc{Subdivision_surfaces_3} without much effort.
\ccIncludeExampleCode{Subdivision_surfaces_3/CatmullClark_subdivision.C}
This example demonstrates the use of Catmull-Clark subdivision
on a read-in CGAL polyhedral mesh. There is only one line worth some
time to explain:
on a read-in CGAL polyhedral mesh. There is only one line deserving
explanation:
\begin{ccExampleCode}
Subdivision_surfaces_3<Polyhedron>::CatmullClark_subdivision(P,d);
\end{ccExampleCode}
@ -253,8 +272,8 @@ This example shows how to apply the subdivision algorithm
on a simple \ccc{CGAL::Polyhedron_3}. When specializing the polyhedron,
there are two more things required to be kept in mind.
First, the internal storage of your
polyhedron has to be sequential ordered, such as the vector
or the linked-list. A sequential ordered container always inserts new
polyhedron has to be sequentially ordered, such as the vector
or the linked-list. A sequentially ordered container always inserts new
entries at the end. Secondly, typename \ccc{Point_3} is
required to be defined within your polyhedron, otherwise our
subdivisions will not know how to compute and where to store the