From 236db8ec14cc68709ea3a82138f364db41945768 Mon Sep 17 00:00:00 2001 From: bobarna <barnabas.borcsok@gmail.com> Date: Wed, 9 Dec 2020 14:18:24 +0100 Subject: [PATCH] algorithm overview partly done --- docs/documentation.aux | 19 +-- docs/documentation.log | 304 +++++++++++++++++++++-------------------- docs/documentation.pdf | Bin 476184 -> 572023 bytes docs/documentation.tex | 108 +++++++++++++-- 4 files changed, 262 insertions(+), 169 deletions(-) diff --git a/docs/documentation.aux b/docs/documentation.aux index 0aaf0d2..9204b4f 100644 --- a/docs/documentation.aux +++ b/docs/documentation.aux @@ -35,13 +35,10 @@ \@writefile{toc}{\contentsline {subsection}{\numberline {2.3}Follow-the-Leader (FTL)}{2}{subsection.2.3}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\numberline {3}Simulation of a Single Hair Strand}{2}{section.3}\protected@file@percent } \@writefile{toc}{\contentsline {subsection}{\numberline {3.1}A Particle}{2}{subsection.3.1}\protected@file@percent } +\citation{MullerPBD} \citation{PixarVolumetricHair} \citation{FTLHair} \citation{RappRealTime} -\citation{UmenhofferSimulation} -\bibstyle{ACM-Reference-Format} -\bibdata{references} -\bibcite{PixarPaper}{{1}{2013}{{Iben et~al\mbox {.}}}{{Iben, Meyer, Petrovic, Soares, Anderson, and Witkin}}} \@writefile{loa}{\contentsline {algocf}{\numberline {1}{\ignorespaces pseudo code for the PBD simulation\relax }}{3}{algocf.1}\protected@file@percent } \newlabel{alg:pseudoPBD}{{1}{3}{A Particle}{algocf.1}{}} \@writefile{toc}{\contentsline {section}{\numberline {4}Putting Hair on the Head}{3}{section.4}\protected@file@percent } @@ -52,9 +49,10 @@ \@writefile{toc}{\contentsline {subsection}{\numberline {7.2}Refinement and proper customizability of the propagation of hair strands on the head object.}{3}{subsection.7.2}\protected@file@percent } \@writefile{toc}{\contentsline {subsection}{\numberline {7.3}Better customizability}{3}{subsection.7.3}\protected@file@percent } \@writefile{toc}{\contentsline {subsection}{\numberline {7.4}Halton sequence for randomizing hair positions}{3}{subsection.7.4}\protected@file@percent } -\@writefile{toc}{\contentsline {subsection}{\numberline {7.5}Rendering}{3}{subsection.7.5}\protected@file@percent } -\@writefile{toc}{\contentsline {subsection}{\numberline {7.6}Moving to the GPU}{3}{subsection.7.6}\protected@file@percent } -\@writefile{toc}{\contentsline {section}{Acknowledgments}{3}{section*.8}\protected@file@percent } +\citation{UmenhofferSimulation} +\bibstyle{ACM-Reference-Format} +\bibdata{references} +\bibcite{PixarPaper}{{1}{2013}{{Iben et~al\mbox {.}}}{{Iben, Meyer, Petrovic, Soares, Anderson, and Witkin}}} \bibcite{MullerPBD}{{2}{2007}{{M\"{u}ller et~al\mbox {.}}}{{M\"{u}ller, Heidelberger, Hennix, and Ratcliff}}} \bibcite{FTLHair}{{3}{2012}{{Müller et~al\mbox {.}}}{{Müller, Kim, and Chentanez}}} \bibcite{PixarVolumetricHair}{{4}{2006}{{Petrovic et~al\mbox {.}}}{{Petrovic, Henne, and Pixar}}} @@ -62,9 +60,12 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LaTeX Info: Redefining \tempurl on input line 104. @@ -1114,16 +1125,9 @@ Underfull \hbox (badness 10000) in paragraph at lines 111--118 [] ) -Overfull \hbox (11.38634pt too wide) in paragraph at lines 418--421 -\T1/LinuxLibertineT-TLF/m/n/10 (-20) at 24 FPS, with the nam-ing con-ven-tion r -en-derXXXX[.bmp/.png], - [] - - -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. 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-\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; -- GitLab