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index 0aaf0d2f43d641d951dd31323b2b85e984d02cd5..9204b4f93cbc94f0dcda6cd05a75cc49a1c88764 100644
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\citation{RappRealTime}
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\bibcite{MullerPBD}{{2}{2007}{{M\"{u}ller et~al\mbox {.}}}{{M\"{u}ller, Heidelberger, Hennix, and Ratcliff}}}
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\bibcite{PixarVolumetricHair}{{4}{2006}{{Petrovic et~al\mbox {.}}}{{Petrovic, Henne, and Pixar}}}
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index 14924faebef54dced24218cec755469603384ef0..5f1c3b4484043cfeb19e5571b60684c604177c56 100644
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Package microtype Info: Loading generic protrusion settings for font family
(microtype) `zi4' (encoding: T1).
(microtype) For optimal results, create family-specific settings.
(microtype) See the microtype manual for details.
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+[3.3] Processing 'acks' comment.
\openout3 = `comment.cut'.
Comment 'acks' writing to comment.cut.
Straight input of comment.cut. (./comment.cut) (./documentation.bbl
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@@ -1114,16 +1125,9 @@ Underfull \hbox (badness 10000) in paragraph at lines 111--118
[]
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-Overfull \hbox (3.58836pt too wide) in paragraph at lines 423--427
-\T1/zi4/m/n/10 make_video.sh \T1/LinuxLibertineT-TLF/m/n/10 (-20) script to as-
-sem-ble the \T1/zi4/m/n/10 output.mp4 \T1/LinuxLibertineT-TLF/m/n/10 (-20) and
-delete
+Overfull \hbox (18.1797pt too wide) in paragraph at lines 493--496
+[]\T1/LinuxLibertineT-TLF/m/n/8 (-20) For han-dling the ex-port of the ren-dere
+d im-age into files, the stb_image_write.h[][][][][]
[]
@@ -1131,47 +1135,51 @@ Package balance Warning: You have called \balance in second column
(balance) Columns might not be balanced.
[4.4]
-Package atveryend Info: Empty hook `BeforeClearDocument' on input line 429.
-Package atveryend Info: Empty hook `AfterLastShipout' on input line 429.
+Package atveryend Info: Empty hook `BeforeClearDocument' on input line 509.
+Package atveryend Info: Empty hook `AfterLastShipout' on input line 509.
(./documentation.aux)
-Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 429.
-Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 429.
+Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 509.
+Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 509.
Package rerunfilecheck Info: File `documentation.out' has not changed.
(rerunfilecheck) Checksum: 86A066FEE8792F691364A237681A9047;4299.
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diff --git a/docs/documentation.pdf b/docs/documentation.pdf
index 2b0f0a0f7463aec6bb2994b27783be1479b60fcf..ee518687ac384c5960291e82c5838564949c7866 100644
Binary files a/docs/documentation.pdf and b/docs/documentation.pdf differ
diff --git a/docs/documentation.tex b/docs/documentation.tex
index 1077027df5db5414d0a7179b35aea583fcac538d..3a46daf845ce9943d9104a0ff59199c1ca89645c 100644
--- a/docs/documentation.tex
+++ b/docs/documentation.tex
@@ -17,12 +17,14 @@
\newcommand{\specialcell}[2][t]{%
\begin{tabular}[#1]{@{}l@{}}#2\end{tabular}}
-\usepackage[]{algorithm2e}
-\usepackage{algorithmic}
+ \usepackage[ruled, linesnumbered]{algorithm2e}
+% \usepackage{algorithmic}
+ \renewcommand{\b}{\boldsymbol}
%% end of the preamble, start of the body of the document source.
\begin{document}
+% \renewcommand{\@algocf@capt@plain}{top} % formerly {bottom}
%%
%% The "title" command has an optional parameter,
%% allowing the author to define a "short title" to be used in page headers.
@@ -204,17 +206,17 @@ not further investigate if there could have been improvements made to my
implementation to eliminate the stretching in the presence of greater forces.
-\section{Simulation of a Single Hair Strand}
+\section{Position Based Simulation}
-\subsection{A Particle}
+\subsection Algorithm Overview
-We model an $S$ hair strand as a chain of $P$ particles and a set of $C$
+We model an $S$ hair strand as a chain of $N$ particles and a set of $M$
constraints. Each particle $p \in [1,\ldots,N]$ has three atributes: mass ($m_p$), position
($\boldsymbol{x_p}$) and velocity ($\boldsymbol{v_p}$).
-A constraint $c \in [1,\ldots,C]$ with cardinality $n_c$ is a function $C_c
+A constraint $c \in [1,\ldots,M]$ with cardinality $n_c$ is a function $C_c
: \mathbb{R} ^{3n_c} \mapsto \mathbb{R}$. It operates on a set of indices $\{i_1,\ldots
-i_{n_c}\}, i_k \in [1,...,P]$. The constraint funtion also has a stiffnes
+i_{n_c}\}, i_k \in [1,...,N]$. The constraint funtion also has a stiffnes
parameter $k_c \in [0...1]$ and a type of either \emph{equality} or
\emph{inequality}.
@@ -226,16 +228,98 @@ strength of the constraint in a range from zero to one.
Given these notations, the algorithm works in the following way:
+\TODO{decide about syntax: array of particles (OOP) or array of v, x and m?}
\begin{algorithm}
\caption{pseudo code for the PBD simulation}
\label{alg:pseudoPBD}
- \begin{algorithmic}
- \IF {$i \geq maxval$}
- \STATE asd
- \ENDIF
- \end{algorithmic}
+
+ \SetCommentSty{scriptsize}
+ \SetNlSty{texttt}{(}{)}
+
+ % to turn off vertical lines:
+ % \SetAlgoNoLine
+
+ \SetKwBlock{Loop}{loop}{end}
+
+ \SetKwFunction{dampVelocities}{dampVelocities}
+ \SetKwFunction{genCollCon}{generateCollisionConstraints}
+ \SetKwFunction{projCon}{projectConstraints}
+ \SetKwFunction{velUpdate}{velocityUpdate}
+
+ \SetKwProg{loopTimes}{loop}{ times}{end loop}
+
+ \ForEach{p in particles}{
+ initialize $\b{p_x} = \b{x_p^0}, \b{p_v} = \b{v_p^0}, p_w = 1/m_p$
+ }
+
+ \Loop{
+ \ForEach{p in particles}{
+ $\b{p_v} \gets \b{p_v} + \Delta t \cdot p_w \cdot \b{f_{external}}$
+ }
+ \dampVelocities{$p_1, \ldots, p_N$}
+ \\
+ \ForEach{p in particles}{
+ $\b{p_{tmp}} \gets \b{p_x} + \Delta t \cdot \b{p_v}$
+
+ }
+ \ForEach{p in particles}{
+ \genCollCon{$p_x \to p_{tmp}$}
+ }
+ \loopTimes{numberOfIterations}{
+ \projCon{$C_1,\ldots, C_{M+M_{coll}}, particles$}\\
+ \tcc*[f]{only the temporary positions of $particles$ are needed}
+ }
+ \ForEach{p in particles}{
+ $\b{p_v} \gets (\b{p_{tmp}} - \b{p_x})/\Delta t$\\
+ $\b{p_x} \gets \b{p_{tmp}}$
+ }
+ \velUpdate{$particles$}\\
+ \tcc*[f]{velocities of colliding particles are modified according to
+ friction and restitution coefficients.}
+ }
\end{algorithm}
+\newcommand{\lineRef}[2]{\footnotesize{\texttt{(#1)}-\texttt{(#2)}}}
+\newcommand{\lineRefSingle}[1]{\footnotesize{\texttt{(#1)}}}
+
+Lines \lineRef{1}{3} initialize the particles. We store the inverse masses $w$
+in order to be able to store infinitely heavy particles that are stationary
+during the simulation. ($\frac{1}{\infty} \approx 0$)
+
+The core idea of position based dynamics is shown in lines \lineRef{9}{11},
+\lineRef{15}{17} and \lineRef{18}{21}. In line \lineRefSingle{10}, estimated
+positions for each particle $\b{p}$ are calculated using and explicit forward
+Euler integration step. In line \lineRefSingle{16} the iterative solver manipulates
+these temporary position estimates such that they satisfy the constraints. It
+does this by repeatedly projecting each constraint in a Gauss-Seidel type
+fashion \TODO{See section ...}. Once the solver finishes with the iteartions, in
+lines \lineRef{19}{20} each particle is moved to the calculated (once temporary)
+positions, and the velocity of each particle is updated accordingly.
+
+Velocities are manipulated in lines \lineRefSingle{6}, \lineRefSingle{8} and
+\lineRefSingle{19}.
+
+Line \lineRefSingle{6} allows to account for external forces
+in the simulation if some of the forces cannot be converted to positional
+constraints. \emph{(For example the pulling force between subsequent particles
+are modelled as distance constraints instead of external forces, and so are the
+forces generated by the particles colliding with other objects.} We use it to
+add gravity, and to generate wind effects for demonstration purposes. If only
+the gravitational force is present, then the line becomes $\b{p_v} \gets \b{p_v}
++ \Delta t \cdot \b{g}$, where $\b{g}$ is the gravitational acceleration.
+
+In line \lineRefSingle{8} the velocities can be damped, if necessary. \emph{Section
+3.5} of the \citet{MullerPBD} paper gives some more sophisticated methods for
+damping, although the paper also points out that any form of damping can be
+used. In our case, simply applying a $k_s$ damping coefficient with value
+between $98\%$ to $99.9\%$ seemed to be a good enough solution.
+
+The given constraints $C_1, \ldots, C_M$ are fixed throughout the simulation. In
+addition to these initial constraints, line \lineRef{13} generates additional
+$M_coll$ collisition constrants. These change from time step to time step. The
+projection step in line \lineRef{16} considers both the fixed and the collision
+constraints. Our implementation
+
\begin{verbatim}
class Particle {
vec3 pos;