[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 160)
knarf at cct.lsu.edu
knarf at cct.lsu.edu
Sun Nov 6 00:20:20 CDT 2011
User: knarf
Date: 2011/11/06 12:20 AM
Modified:
/
ET.tex
/figures/
bbh-boxes.fig, bbh-boxes.pdf
Log:
a lot of small details
File Changes:
Directory: /
============
File [modified]: ET.tex
Delta lines: +93 -99
===================================================================
--- ET.tex 2011-11-06 03:59:34 UTC (rev 159)
+++ ET.tex 2011-11-06 05:20:20 UTC (rev 160)
@@ -412,7 +412,7 @@
The generic, parallel {\tt Cactus} computationa toolkit consists of modules providing
parallel drivers, coordinates, boundary conditions, interpolators,
reduction operators, and efficient I/O in different data
-formats. Generic interfaces are used, making it possible to use
+formats. Generic interfaces are used, enabling the usage of
external packages and improved modules which are immediately available
to its users. {\tt Cactus} is involved in the NSF Blue Waters consortium
for petascale computing, and has funding from NSF SDCI to
@@ -439,10 +439,10 @@
group has remained from numerical relativity. The {\tt Cactus} team has
traditionally developed and supported a set of core modules for numerical
relativity, as part of the \texttt{CactusEinstein} arrangement. Over the
-last few years however, the relevance of many of the modules has declined,
+years however, the relevance of many of the modules has declined,
and more and more of the basic infrastructure for numerical relativity
has been provided by open modules distributed by research
-groups in the community.
+groups within the community.
The Einstein Toolkit now collects the widely used parts of CactusEinstein,
combined with contributions from the community.
@@ -499,13 +499,13 @@
i.e.\ without requiring function calls in user code.
Figure \ref{fig:carpet-details} describes some details of the
Berger-Oliger time stepping algorithm. More details are described in
-\ref{Schnetter:2003rb}.
+\cite{Schnetter:2003rb}.
\begin{figure}
\centering
- \includegraphics[width=0.45\textwidth]{figures/carpet-timestepping}
- \hfill
- \includegraphics[width=0.45\textwidth]{figures/carpet-interpolation}
+ \includegraphics[width=0.3\textwidth]{figures/carpet-timestepping}
+ \hspace{3em}
+ \includegraphics[width=0.3\textwidth]{figures/carpet-interpolation}
\caption{Berger-Oliger time stepping details, showing a coarse and a
fine grid, with time advancing upwards. \textbf{Left:} Time stepping
algorithm. First the coarse grid takes a large time step, then the
@@ -522,8 +522,8 @@
used in production on up to several thousand cores. We estimate that,
in 2010, about 7,000 core years of computing time (45 million core
hours) were used via {\tt Carpet} by more than a dozen research groups
-world-wide. To date, more than 90 peer-reviewed publication and more
-than 15 student theses are based on {\tt Carpet} \cite{CarpetCode:web}.
+world-wide. To date, more than 90 peer-reviewed publications and more
+than 15 student theses are based on {\tt Carpet}~\cite{CarpetCode:web}.
\subsection{Simulation Factory\pagesdone{1 Erik}}
@@ -548,7 +548,7 @@
trail ensuring repeatable and well-documented scientific results.
Using these abstractions, most operations are much simplified, many
types of potentially disastrous user errors are avoided, allowing different
-supercomputers can be used in a uniform manner.
+supercomputers to be used in a uniform manner.
Using the Simulation Factory, we are able to offer a
tutorial for the Einstein Toolkit \cite{EinsteinToolkit:web} that lets
@@ -636,7 +636,7 @@
and conventions (see section~\ref{sec:base_modules}). Others provide
initial data (see section~\ref{sec:initial_data}), which may
be evolved using the different evolution methods described in
-section~\ref{sec:evol}.
+sections~\ref{sec:Kevol} and~\ref{sec:GRHydro}.
Simulations including
matter require their own evolution techniques (see section~\ref{sec:GRHydro})
and a description of the matter properties, encoded in equations
@@ -651,10 +651,10 @@
Modular designs have proven to be essential for distributed development
of complex software systems, and require the use of well-defined interfaces.
Low-level interoperability within the Einstein Toolkit is provided by
-the {\tt Cactus} infrastructure. One example of this is using one module
+the {\tt Cactus} infrastructure. One example of this is the usage of one module
from within another, e.g., by calling a function within another thorn
-independent of programming language used for both the calling and the
-called function. Technical challenges like this can be and are
+independent of programming language used for both, the calling and the
+called function. Solutions for technical challenges like this can be and are
provided by the underlying framework, in this case by {\tt Cactus}.
However, certain other standards are very hard or impossible to
@@ -669,9 +669,8 @@
The Einstein Toolkit provides modules whose sole purpose is to
declare commonly used variables and define their meaning and units.
These conditions are not strictly enforced, but instead documented for the
-convenience of the user community. Three of these base modules, ADMBase, HydroBase, and TmunuBase,
-are described in more detail below, because they declare quantities that might be
-defined differently in other simulation codes.
+convenience of the user community. Three of these base modules, \codename{ADMBase}, \codename{HydroBase}, and \codename{TmunuBase},
+are described in more detail below.
In the following, we assume that the reader is familiar with the
basics of numerical relativity and GR hydrodynamics, including the underlying differential geometry and tensor analysis. Detailed
@@ -698,8 +697,8 @@
modules using different formalisms. In the $3+1$ approach, 4-dimensional spacetime is foliated into sequences of spacelike
3-dimensional hypersurfaces (slices) connected by timelike normal vectors. The $3+1$ split introduces 4
gauge degrees of freedom, the lapse function $\alpha$ that describes
-the advance of proper time with respect to coordinate for a normal
-observer\footnote{A normal observer follows a wordline tangent to the
+the advance of proper time with respect to coordinate time for a normal
+observer\footnote{A normal observer follows a worldline tangent to the
unit normal on the 3-hypersurface.} and the shift vector $\beta^i$
that describes how spatial coordinates change from one slice to the
next.
@@ -770,7 +769,7 @@
function is to store variables which are common to most if not all
hydrodynamics codes solving the Euler equations, the so-called primitive
variables. These are also the variables which are needed to couple to a
-spacetime solver often by analysis thorns as well. As with
+spacetime solver, and often by analysis thorns as well. As with
ADMBase, the usage of a common set of variables by different hydrodynamics
codes creates the possibility of sharing parts of the code, e.g., initial data
solvers or analysis routines.
@@ -804,11 +803,6 @@
$n^\mu$.
\end{itemize}
-Note that while this includes definitions for variables used to store information
-about magnetic fields, the implementation of an evolution system for magnetic fields
-within the Einstein Toolkit is currently still in development and
-not yet officially released.
-
HydroBase also sets up scheduling blocks that organize the main functions that modules of a
hydrodynamics code may need. All of those scheduling blocks are optional, but when used
they simplify existing codes and make them more interoperable. HydroBase itself does
@@ -828,7 +822,7 @@
a central access point for analysis thorns.
\subsubsection{TmunuBase}
-In the Einstein Toolkit, we typically choose the stress energy tensor $T^{\mu\nu}$ to be that of an ideal relativistic fluid,
+In the Einstein Toolkit the stress energy tensor $T^{\mu\nu}$ is often chosen to be that of an ideal relativistic fluid,
\begin{equation}
T^{\mu\nu} = \rho h u^\mu u^\nu - g^{\mu\nu} P\,\,,
\end{equation}
@@ -843,7 +837,7 @@
interdependence. The resulting stress-energy tensor can then be used
by the spacetime evolution thorn (again without explicit dependence on any
matter thorns). When thorn \codename{MoL} is used for time evolution this
-provides a high order spacetime-matter coupling.
+provides a high-order spacetime-matter coupling.
The grid functions provided by \codename{TmunuBase} are:
\begin{itemize}
@@ -851,7 +845,7 @@
\item The mixed components $T_{0i}$: {\tt eTtx}, {\tt eTty}, {\tt eTtz}
\item The spatial components $T_{ij}$: {\tt eTxx}, {\tt eTxy}, {\tt eTxz}, {\tt eTyy}, {\tt eTyz}, {\tt eTzz}
\end{itemize}
-In addition, the grid scalar {\tt stress\_energy\_state} has the value 1 if storage is on for the stress-energy tensor and 0 if not.
+In addition, the grid scalar {\tt stress\_energy\_state} has the value 1 if storage is provided for the stress-energy tensor and 0 if not.
Thorn \codename{ADMCoupling} provides a similar (but older) interface between
space-time and matter thorns. However, since it is based on an include file
@@ -1077,8 +1071,9 @@
handling with some care the potentially singular terms that appear at the origin.
To facilitate the construction of stars in more complicated dynamical configurations,
-one may also apply a uniform velocity to the NS, though this does not affect
-the ODE solution nor the resulting density profile.
+one may also apply a uniform velocity to the NS, though this does neither affect
+the ODE solution, the resulting density profile nor does it fulfill the spacetime
+constraint equations.
\subsection{Spacetime Curvature Evolution}
\label{sec:Kevol}
@@ -1248,8 +1243,8 @@
\subsubsection{Initial Conditions}
-Initial conditions from the ADM variables $g_{ij}$, $K_{ij}$, lapse
-$\alpha$, and shift $\beta^i$, as provided by the initial data
+Initial conditions for the ADM variables $g_{ij}$, $K_{ij}$, lapse
+$\alpha$, and shift $\beta^i$, are provided by the initial data
discussed in Sec.~\ref{sec:initial_data}. From these the BSSN
quantities are calculated via their definition, setting $B^i=0$, and
using cubic extrapolation for $\tilde\Gamma^i$ at the outer
@@ -1369,8 +1364,7 @@
variables are not smooth and may not be finite differenced accurately.
All other source terms in the evolution equations may contain only the
hydrodynamic variables themselves as well as the metric variables and
-derivatives of the latter, since they must formally be smooth and thus
-differentiable using finite differencing techniques. Summarizing
+derivatives of the latter. Summarizing
these methods briefly, the following stages occur every timestep:
\begin{itemize}
\item The primitive variables are ``reconstructed'' at cell faces
@@ -1380,7 +1374,7 @@
\item A Riemann problem is solved at each cell face using an
approximate solver. Currently implemented versions include HLLE
(Harten-Lax-van Leer-Einfeldt), Roe, and Marquina solvers.
-\item The conserved variables are advanced a timestep, and used to
+\item The conserved variables are advanced one timestep, and used to
recalculate the new values of the primitive variables.
\end{itemize}
We discuss the GRHD formalism, the stages within a timestep, and the
@@ -1635,7 +1629,7 @@
needed to close the system of GR hydrodynamics equations. The module
\codename{EOS\_Omni} provides a unified general equation of state
(EOS) interface and backend for simple analytic and complex
-microphysical EOS.
+microphysical EOSs.
The polytropic EOS,
\begin{equation}
@@ -1724,7 +1718,7 @@
functionality provided by the Einstein Toolkit. In most cases, the analysis modules
work on the variables stored in the base modules discussed in
Sec.~\ref{sec:base_modules}, \codename{ADMBase}, \codename{TmunuBase},
-and \codename{HydroBase}, to create as portable a tool as possible.
+and \codename{HydroBase}, to be as portable as possible.
\subsubsection{Horizons}
When spacetimes contain a BH, localizing its horizon is necessary
@@ -1769,7 +1763,7 @@
the approximate position of the anticipated event horizon to further
narrow the region containing the event horizon.
-Unfortunately, event horizons can only be found after the full spacetime
+However, event horizons can only be found after the full spacetime
has been evolved. It is often useful to know the positions
and shapes of any BH on a given hypersurface for purposes such as
excision, accretion, and local measures of its mass and spin. The
@@ -1817,8 +1811,8 @@
The \codename{AHFinderDirect} module~\cite{Thornburg:2003sf} is a
faster alternative to \codename{AHFinder}. Its approach is to view
-Eq.\ref{eq:ah_theta} as an elliptic PDE for $h(\theta,\phi)$ on $S^2$
-using standard finite differencing methods. Rewriting Eq.~\ref{eq:ah_theta}
+eq.~\ref{eq:ah_theta} as an elliptic PDE for $h(\theta,\phi)$ on $S^2$
+using standard finite differencing methods. Rewriting eq.~\ref{eq:ah_theta}
in the form
\begin{equation}
\Theta \equiv \Theta\left(h,\partial_u h,\partial_{uv}h;
@@ -1828,7 +1822,8 @@
using a Newton-Raphson method to solve $\bf{J}\cdot\delta h=-\Theta$
where $\bf{J}$ is the Jacobian matrix.
The drawback of this method is that it is not guaranteed to give the
-outermost marginally trapped surface.
+outermost marginally trapped surface. In practice however, this limitation
+can be easily overcome by either a good, or multiple initial guesses.
\subsubsection{Masses and Momenta}
Two distinct measures of mass and momenta are available in relativistic
@@ -1857,7 +1852,7 @@
that the integrals of $\tilde{D}^i e^\phi$ and
$e^{6\phi} \epsilon_{ij}{}^k x^j \tilde{A}^\ell{}_k$ over the boundaries of
the computational domain vanish. The ADM mass and angular momentum
-can also be calculated rom the
+can also be calculated from the
variables stored in the base modules using the \codename{Extract} module,
as surface integrals~\cite{Bowen:1980yu}
\begin{eqnarray}
@@ -1869,7 +1864,7 @@
the domain since these quantities are only properly defined when
calculated at infinity.
-There are the quasi-local measures of mass and angular momentum. From
+There are also the quasi-local measures of mass and angular momentum, from
any \ahz{s} found during the spacetime. Both
\codename{AHFinderDirect} and \codename{AHFinder} output the
corresponding mass derived from the area of the horizon $m_H =
@@ -1886,14 +1881,14 @@
(see, e.g., \cite{Lovelace:2009dg}), and are therefore
an indispensable tool in numerical relativity.
\codename{QuasiLocalMeasures} takes as input a horizon surface, or any
-other surface that the user specifies,like a large coordinate
+other surface that the user specifies, like a large coordinate
sphere, and
can calculate useful quantities such as the Weyl or Ricci scalars
or the three-volume element of the horizon world tube,
-in addition to physical observables such as mass and momenta,
+in addition to physical observables such as mass and momenta.
Finally, the module \codename{HydroAnalysis} additionally locates the
-center of mass as well as the point of maximum rest mass density of a
+(coordinate) center of mass as well as the point of maximum rest mass density of a
matter field.
\subsubsection{Gravitational Waves}
@@ -1903,7 +1898,7 @@
data from the various gravitational wave detectors around the
globe. The Einstein Toolkit includes modules for extracting
gravitational waves via either the Moncrief formalism of a perturbation
-on a Schwarzschild background or calculation of the Weyl scalar $\Psi_4$.
+on a Schwarzschild background or the calculation of the Weyl scalar $\Psi_4$.
The module \codename{Extract} uses the Moncrief formalism~\cite{
Moncrief:1974am} to extract gauge-invariant wavefunctions $Q_{\ell m}^\times$ and $Q_{\ell
@@ -1994,6 +1989,34 @@
\end{equation}
of a projection onto spin-weighted spherical harmonics ${}_s Y_{\ell m}$.
+\todo{CCE}
+
+\subsubsection{Object tracking}
+\label{sec:object-tracking}
+We provide a module (\codename{PunctureTracker}) for tracking BH
+positions evolved with moving puncture techniques. It can be used
+with (\codename{CarpetTracker}) to have the mesh refinement regions follow the
+BHs as they move across the grid. The BH position is
+stored as the centroid of a spherical surface (even though there is no surface)
+provided by \codename{SphericalSurface}.
+
+Since the punctures only move due to the shift advection terms in
+the BSSN equations, the puncture location is evolved very simply as
+\begin{equation}
+ \frac{d x^i}{d t} = -\beta^i, \label{eq:puncturetracking}
+\end{equation}
+where $x^i$ is the puncture location and $\beta^i$ is the shift. Since the
+puncture location usually does not coincide with gridpoints, the shift is
+interpolated to the location of the puncture.
+Equation~(\ref{eq:puncturetracking}) is implemented with a simple first order
+Euler scheme, which seems to be accurate enough for controlling the location
+of the mesh refinement hierarchy.
+
+Another class of objects which often needs to be tracked are neutron stars.
+Here is it usually sufficient to locate the position of the maximum density,
+and to adapt AMR resolution in these regions accordingly, coupled with the
+condition that this location can only move at a specifyable maximum speed.
+
\subsubsection{Other analysis modules}
The remaining analysis capabilities of the Einstein Toolkit span a
variety of primarily vacuum-based functions.
@@ -2021,37 +2044,6 @@
$\mathcal{R}$, as well as the 3-metric and extrinsic curvature converted to
spherical coordinates.
-
-\subsection{Relativity Tools\pages{3 Peter}}
-
-\subsubsection*{Object tracking}
-\label{sec:object-tracking}
-We provide a module (\codename{PunctureTracker}) for tracking BH
-positions evolved with moving puncture techniques. It can be used
-with (\codename{CarpetTracker}) to have the mesh refinement regions follow the
-BHs as they move across the grid. The BH position is
-stored as the centroid of a spherical surface (even though there is no surface)
-provided by \codename{SphericalSurface}.
-
-Since the punctures only move due to the shift advection terms in
-the BSSN equations, the puncture location is evolved very simply as
-\begin{equation}
- \frac{d x^i}{d t} = -\beta^i, \label{eq:puncturetracking}
-\end{equation}
-where $x^i$ is the puncture location and $\beta^i$ is the shift. Since the
-puncture location usually does not coincide with gridpoints, the shift is
-interpolated to the location of the puncture.
-Equation~(\ref{eq:puncturetracking}) is implemented with a simple first order
-Euler scheme, which seems to be accurate enough for controlling the location
-of the mesh refinement hierarchy.
-\todo{ES: I think this paragraph provides too much detail for this paper.}
-\todo{ES: Can also track objects via their maximum density or centre
- of mass; I believe GRHydro does something like this}
-\todo{We mention NS Tracker later, but not here?
-Also, should this be incorporated into the Analysis section immediately above, even though these
-reside in a different arrangement, since they really are analysis modules?}
-
-
\subsection{Simulation Domain, Symmetries, Boundaries\pages{3 Roland}}
\subsubsection{Domains and Coordinates.}
{\tt Cactus} distinguishes between
@@ -2122,7 +2114,7 @@
In applying symmetries to populate ghost zones, the
transformation properties of tensorial quantities (including tensor
densities and non-tensors such as Christoffel symbols) are correctly
-taken into account, just as they are in the interpolation routines present in {\tt Cactus}
+taken into account, just as they are in the interpolation routines present in {\tt Cactus}.
Thus, symmetries are handled transparently
from the point of view of user modules (see Figure~\ref{fig:faces} for an
illustration).
@@ -2180,7 +2172,7 @@
\caption{Nested boxes following the individual BHs in binary
BH merger simulation (see Section~\ref{sec:bbh-example}),
with the location of the individual BHs found by
- \codename{PunctureTracker}. The innermost three of the none
+ \codename{PunctureTracker}. The innermost three of the nine
levels of mesh refinement used in this simulation are shown. Notice the use of
\codename{RotatingSymmetry180} to reduce the computational domain.}
\label{fig:bbh-boxes}
@@ -2311,7 +2303,7 @@
is non-zero due to truncation error, but shows fourth order convergence to
zero with resolution (this mode is not present in the initial data and is not
excited during the evolution). Other modes are zero to roundoff due to
-symmetries at all resolution.
+symmetries at all resolutions.
Since there is non-trivial gravitational wave content in the initial data
the mass of the BH changes when evolved. In Figure~\ref{fig:ah_mass}
@@ -2368,10 +2360,10 @@
right plot of Figure~\ref{fig:ah_mass_spin} by the fact that the pink
(dotted) curve (the high resolution result scaled by a factor of $1.78$ for
second order convergence to the resolution of the medium resolution) and the
-green (long dashed) curve are on top of each other. The reason that we only
-see second order convergence in the spin is due to the fact that thorn
-\codename{QuasiLocalMeasures} still have some parts of the algorithm that are
-only second order accurate. The increase of about 0.22\% in the mass of the
+green (long dashed) curve are on top of each other. The reason for the
+second order convergence in the spin lies in the fact that thorn
+\codename{QuasiLocalMeasures} uses an algorithm which is
+only second order accurate overall. The increase of about 0.22\% in the mass of the
BH is caused solely by the increase in the irreducible mass.
\subsection{BHB\pages{2 Bruno}}
@@ -2472,7 +2464,7 @@
%at this extraction region ($\sim 6.25 \times 10^{-2}$).
In order to evaluate the convergence of the numerical
-solution, we ran five different simulations with different
+solution, we ran five simulations with different
resolutions, and focus our analysis on the convergence
of the Weyl scalar $\Psi_4$ phase and amplitude.
The mesh spacings adopted for the coarser grid in the
@@ -2580,6 +2572,7 @@
\end{figure}
\subsection{Linear oscillations of TOV stars\pages{2 Frank}}
+\label{sec:tov_oscilations}
The examples in the previous subsections did not include the evolution of
matter within a relativistic spacetime. One interesting test of a coupled
matter-spacetime evolution is to measure the eigenfrequencies of a stable TOV
@@ -2621,8 +2614,7 @@
an initial spike, produced by the interpolation of the one-dimensional equilibrium
solution onto the three-dimensional evolution grid. The remainder of the evolution
however, the central density evolution is dominated by continuous excitations coming
- from the interaction of the stellar surface with the artificial atmosphere. Shown
- are three different resolutions.}
+ from the interaction of the stellar surface with the artificial atmosphere.}
\end{figure}
In figure~\ref{fig:tov_mode_spectrum} we show the power spectral density (PSD)
@@ -2661,7 +2653,7 @@
surface.
Figure~\ref{fig:tov_ham_conv} shows the order of convergence of the Hamiltonian
-constraint violation, using the two highest-resolution runs, at the stellar
+constraint violation, using the three highest-resolution runs, at the stellar
center and a coordinate radius of $r=5\mathrm{M}$ which is about half-way between the
center and the surface. The observed convergence rate for most of the
simulation time lies between $1.4$ and $1.5$ at the center, and between $1.6$ and
@@ -2741,7 +2733,7 @@
disconnecting the evolution in the interior from the outside spacetime.}
\end{figure}
\begin{figure}
- \label{fig:tov_collapse_H_convergence_at0.pdf}
+ \label{fig:tov_collapse_H_convergence_at0}
\includegraphics[width=0.9\textwidth]{examples/collapse/H_convergence_at0}
\caption{Convergence factor for the Hamiltonian constraint evaluated at the
center of the collapsing star. We plot convergence factors computed using
@@ -2798,7 +2790,7 @@
\end{eqnarray}
\end{widetext}
-In Figure\ref{fig:kasner}, we show the full evolution of the $t=1$ slice
+In Figure~\ref{fig:kasner}, we show the full evolution of the $t=1$ slice
of spacetime~\ref{eq:kasner}, along with the associated error for a sequence of
time resolutions.
@@ -2815,7 +2807,7 @@
\section{Future Work\pagesdone{1 Frank}}
-In this paper we illustrated the current state of the ``Einstein Toolkit'',
+In this paper we illustrated the state of the ``Einstein Toolkit'' release ``Curie'',
a collection of freely available and easy to use computational codes
for numerical relativity. However, there is room for improvement on both
the underlying infrastructure and the included physics.
@@ -2831,17 +2823,19 @@
example is the improvement of current wave extraction techniques within the
Einstein Toolkit to include Cauchy characteristic wave extraction techniques,
as recently studied in~\cite{todo}. The authors of these research codes
-recently agreed to make their work available to the whole community within
-the Einstein toolkit. This is expected to happen for the second release of
-the toolkit in 2011.
+agreed to make their work available to the whole community within
+the Einstein toolkit, which happened in the recently released version ``Maxwell'',
+but which is not described here.
A second example for an improvement of an existing method is extending the
current implementation of ideal magneto hydrodynamics to the case of non-zero
resistivity. Such implementations exist
-and are described in the literature~\cite{todo}, but none are provided freely
-within the Einstein Toolkit yet. Resistive MHD is important for a number of
-astrophysical scenarios, one of which is the merger of two magnetized NSs,
-a candidate for short gamma-ray bursts.
+and are described in the literature~\cite{todo}, but none have been provided freely
+within the Einstein Toolkit until the recently released ``Maxwell'' version.
+Resistive MHD is important for a number of astrophysical scenarios, one of
+which is the merger of two magnetized NSs, a candidate for short gamma-ray
+bursts.
+\todo{reword - make clear that we describe Curie}
Yet another important goal is to increase the scalability of the {\tt Carpet} AMR
infrastructure. It has been shown that good scaling is limited to less than
Directory: /figures/
====================
File [modified]: bbh-boxes.fig
Delta lines: +8 -6
===================================================================
--- figures/bbh-boxes.fig 2011-11-06 03:59:34 UTC (rev 159)
+++ figures/bbh-boxes.fig 2011-11-06 05:20:20 UTC (rev 160)
@@ -8,17 +8,19 @@
0
1200 2
# polyline
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- 2848 9757 2848 9757 3856 9757 3856 7384 2848 7384
+2 1 0 1 0 0 954 0 -1 4.000 0 0 0 0 0 4
+ 2848 9757 3856 9757 3856 7384 2848 7384
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+2 1 0 1 0 0 952 0 -1 4.000 0 0 0 0 0 4
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# polyline
-2 1 0 1 0 0 951 0 -1 4.000 0 0 0 0 0 5
- 2848 8482 2848 8482 3485 8482 3485 7740 2848 7740
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# polyline
2 1 0 1 0 0 949 0 -1 4.000 0 0 0 0 0 5
2929 8303 3300 8303 3300 7933 2929 7933 2929 8303
2 1 1 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 2
2850 7200 2850 10050
+2 1 1 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 2
+ 2850 8575 4425 8575
4 1 0 50 -1 0 12 0.0000 4 135 315 2850 10200 x=0\001
File [modified]: bbh-boxes.pdf
Delta lines: +0 -0
===================================================================
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