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[gnuastrocommits] master e965a70 5/5: Merged updates to Distance on a 2
From: 
Mohammad Akhlaghi 
Subject: 
[gnuastrocommits] master e965a70 5/5: Merged updates to Distance on a 2D curved space section 
Date: 
Wed, 18 Oct 2017 07:52:03 0400 (EDT) 
branch: master
commit e965a7016ca95fa1884171e25b813443abd503f8
Merge: aea662d 7500ac6
Author: Mohammad Akhlaghi <address@hidden>
Commit: Mohammad Akhlaghi <address@hidden>
Merged updates to Distance on a 2D curved space section
The changes in this section are now merged with master. Also, I noticed
that Boud had used different names `boud' in the commits of this branch,
but his previous commits were with the name `Boud Roukema' (same email
address). So the `.mailmap' file was updated for Git to correct it in its
`shortlog' output (which is used in Gnuastro to generate the list of
authors).
IMPORTANT NOTE: with this commit, it is important to rebootstrap Gnuastro
because one of the book images was renamed and thus has to be rebuilt
during bootstrapping.

.mailmap  1 +
doc/gnuastro.texi  327 +++++++++++
doc/plotsrc/Makefile  2 +
doc/plotsrc/all.tex  2 +
.../tex/{sphericalplane.tex => sphereandplane.tex}  0
5 files changed, 174 insertions(+), 158 deletions()
diff git a/.mailmap b/.mailmap
index 1da35be..bccc8e5 100644
 a/.mailmap
+++ b/.mailmap
@@ 1 +1,2 @@
+Boud Roukema <address@hidden>
<address@hidden> <address@hidden>
\ No newline at end of file
diff git a/doc/gnuastro.texi b/doc/gnuastro.texi
index daeb86e..36633bc 100644
 a/doc/gnuastro.texi
+++ b/doc/gnuastro.texi
@@ 2089,9 +2089,9 @@ Warp started on Mon Apr 6 16:51:59 953
Using 8 CPU threads.
Input: cat_convolved.fits (hdu: 1)
matrix:
 0.2000 0.0000 0.4000
 0.0000 0.2000 0.4000
 0.0000 0.0000 1.0000
+ 0.2000 0.0000 0.4000
+ 0.0000 0.2000 0.4000
+ 0.0000 0.0000 1.0000
$ ls
0_cat.fits cat_convolved_scaled.fits cat.txt
@@ 15963,8 +15963,8 @@ One line examples:
## Add noise with a standard deviation of 100 to image:
$ astmknoise sigma=100 image.fits
## Add noise to input image assuming a background magnitude (with zeropoint
## magnitude of 0) and a certain instrumental noise:
+## Add noise to input image assuming a background magnitude (with
+## zeropoint magnitude of 0) and a certain instrumental noise:
$ astmknoise background=10 z0 instrumental=20 mockimage.fits
@end example
@@ 16086,36 +16086,39 @@ interested readers can study those books.
@node Distance on a 2D curved space, Extending distance concepts to 3D,
CosmicCalculator, CosmicCalculator
@subsection Distance on a 2D curved space
The observations to date (for example the Plank 2013 results), have
not measured the presence of a significant curvature in the
universe. However to be generic (and allow its measurement if it does
in fact exist), it is very important to create a framework that allows
curvature. As 3D beings, it is impossible for us to mentally create
(visualize) a picture of the curvature of a 3D volume in a 4D
space. Hence, here we will assume a 2D surface and discuss distances
on that 2D surface when it is flat, or when the 2D surface is curved
(in a 3D space). Once the concepts have been created/visualized here,
in @ref{Extending distance concepts to 3D}, we will extend them to the
real 3D universe we live in and hope to study.

To be more understandable (actively discuss from an observer's point
of view) let's assume we have an imaginary 2D friend living on the 2D
space (which @emph{might} be curved in 3D). So here we will be working
with it in its efforts to analyze distances on its 2D universe. The
start of the analysis might seem too mundane, but since it is
impossible to imagine a 3D curved space, it is important to review all
the very basic concepts thoroughly for an easy transition to a
universe we cannot visualize any more (a curved 3D space in 4D).
+The observations to date (for example the Planck 2015 results), have not
address@hidden observations are interpeted under the assumption of
+uniform curvature. For a relativistic alternative to dark energy (and maybe
+also some part of dark matter), nonuniform curvature may be even be more
+critical, but that is beyond the scope of this brief explanation.} the
+presence of significant curvature in the universe. However to be generic
+(and allow its measurement if it does in fact exist), it is very important
+to create a framework that allows nonzero uniform curvature. As 3D beings,
+it is difficult for us to mentally create (visualize) a picture of the
+curvature of a 3D volume embedded in a 4D space. Hence, here we will assume
+a 2D surface and discuss distances on that 2D surface when it is flat and
+when it is curved (embedded in a flat 3D space). Once the concepts have
+been created/visualized here, in @ref{Extending distance concepts to 3D},
+we will extend them to the real 3D universe we live in and hope to study.
+
+To be more understandable (actively discuss from an observer's point of
+view) let's assume there's an imaginary 2D creature living on the 2D space
+(which @emph{might} be curved in 3D). Here, we will be working with this
+creature in its efforts to analyze distances in its 2D universe. The start
+of the analysis might seem too mundane, but since it is difficult to
+imagine a 3D curved space, it is important to review all the very basic
+concepts thoroughly for an easy transition to a universe that is more
+difficult to visualize (a curved 3D space embedded in 4D).
To start, let's assume a static (not expanding or shrinking), flat 2D
surface similar to @ref{flatplane} and that our 2D friend is observing its
universe from point @mymath{A}. One of the most basic ways to parametrize
this space is through the Cartesian coordinates (@mymath{x},
+surface similar to @ref{flatplane} and that the 2D creature is observing
+its universe from point @mymath{A}. One of the most basic ways to
+parametrize this space is through the Cartesian coordinates (@mymath{x},
@mymath{y}). In @ref{flatplane}, the basic axes of these two coordinates
are plotted. An infinitesimal change in the direction of each axis is
written as @mymath{dx} and @mymath{dy}. For each point, the infinitesimal
changes are parallel with the respective axes and are not shown for
clarity. Another very useful way of parameterizing this space is through
+clarity. Another very useful way of parametrizing this space is through
polar coordinates. For each point, we define a radius (@mymath{r}) and
angle (@mymath{\phi}) from a fixed (but arbitrary) reference axis. In
@ref{flatplane} the infinitesimal changes for each polar coordinate are
@@ 16129,69 +16132,72 @@ the same radius.
plane.}
@end float
Assuming a certain position, which can be parameterized as @mymath{(x,y)},
or @mymath{(r,\phi)}, a general infinitesimal change change in its position
will place it in the coordinates @mymath{(x+dx,y+dy)} and
address@hidden(r+dr,\phi+d\phi)}. The distance (on the flat 2D surface) that is
covered by this infinitesimal change in the static universe (@mymath{ds_s},
the subscript signifies the static nature of this universe) can be written
as:
+Assuming an object is placed at a certain position, which can be
+parameterized as @mymath{(x,y)}, or @mymath{(r,\phi)}, a general
+infinitesimal change in its position will place it in the coordinates
address@hidden(x+dx,y+dy)} and @mymath{(r+dr,\phi+d\phi)}. The distance (on the
+flat 2D surface) that is covered by this infinitesimal change in the static
+universe (@mymath{ds_s}, the subscript signifies the static nature of this
+universe) can be written as:
@dispmath{ds_s=dx^2+dy^2=dr^2+r^2d\phi^2}
The main question is this: how can our 2D friend incorporate the (possible)
curvature in its universe when it is calculating distances? The universe it
lives in might equally be a locally flat but globally curved surface like
address@hidden The answer to this question but for a 3D being (us)
is the whole purpose to this discussion. So here we want to give our 2D
friend (and later, ourselves) the tools to measure distances if the space
+The main question is this: how can the 2D creature incorporate the
+(possible) curvature in its universe when it's calculating distances? The
+universe that it lives in might equally be a curved surface like
address@hidden The answer to this question but for a 3D being (us)
+is the whole purpose to this discussion. Here, we want to give the 2D
+creature (and later, ourselves) the tools to measure distances if the space
(that hosts the objects) is curved.
address@hidden assumes a spherical shell with radius @mymath{R}
as the curved 2D plane for simplicity. The spherical shell is tangent
to the 2D plane and only touches it at @mymath{A}. The result will be
generalized afterwards. The first step in measuring the distance in a
curved space is to imagine a third dimension along the @mymath{z} axis
as shown in @ref{sphericalplane}. For simplicity, the @mymath{z} axis
is assumed to pass through the center of the spherical shell. Our
imaginary 2D friend cannot visualize the third dimension or a curved
2D surface within it, so the remainder of this discussion is purely
abstract for it (similar to us being unable to visualize a 3D curved
space in 4D). But since we are 3D creatures, we have the advantage of
visualizing the following steps. Fortunately our 2D friend knows our
mathematics, so it can follow along with us.

With the third axis added, a generic infinitesimal change over
address@hidden full} 3D space corresponds to the distance:
address@hidden is very
important to recognize that this change of distance is for @emph{any}
point in the 3D space, not just those changes that occur on the 2D
spherical shell of @ref{sphericalplane}. Recall that our 2D friend can
only do measurements in the 2D spherical shell, not the full 3D
space. So we have to constrain this general change to any change on
the 2D spherical shell. To do that, let's look at the arbitrary point
address@hidden on the 2D spherical shell. Its image (@mymath{P'}) on the
flat plain is also displayed. From the dark triangle, we see that

address@hidden Figure,sphericalplane
address@hidden@image{gnuastrofigures/sphericalplane, 10cm, , }

address@hidden spherical plane (centered on @mymath{O}) and flat plane
(gray) tangent to it at point @mymath{A}.}
address@hidden assumes a spherical shell with radius @mymath{R} as
+the curved 2D plane for simplicity. The 2D plane is tangent to the
+spherical shell and only touches it at @mymath{A}. This idea will be
+generalized later. The first step in measuring the distance in a curved
+space is to imagine a third dimension along the @mymath{z} axis as shown in
address@hidden For simplicity, the @mymath{z} axis is assumed to
+pass through the center of the spherical shell. Our imaginary 2D creature
+cannot visualize the third dimension or a curved 2D surface within it, so
+the remainder of this discussion is purely abstract for it (similar to us
+having difficulty in visualizing a 3D curved space in 4D). But since we are
+3D creatures, we have the advantage of visualizing the following
+steps. Fortunately the 2D creature is already familiar with our
+mathematical constructs, so it can follow our reasoning.
+
+With the third axis added, a generic infinitesimal change over @emph{the
+full} 3D space corresponds to the distance:
+
address@hidden
+
address@hidden Figure,sphereandplane
address@hidden@image{gnuastrofigures/sphereandplane, 10cm, , }
+
address@hidden spherical shell (centered on @mymath{O}) and flat plane (light
+gray) tangent to it at point @mymath{A}.}
@end float
+It is very important to recognize that this change of distance is for
address@hidden point in the 3D space, not just those changes that occur on the
+2D spherical shell of @ref{sphereandplane}. Recall that our 2D friend can
+only do measurements on the 2D surfaces, not the full 3D space. So we have
+to constrain this general change to any change on the 2D spherical
+shell. To do that, let's look at the arbitrary point @mymath{P} on the 2D
+spherical shell. Its image (@mymath{P'}) on the flat plain is also
+displayed. From the dark gray triangle, we see that
+
@dispmath{\sin\theta={r\over R},\quad\cos\theta={Rz\over R}.}These
relations allow our 2D friend to find the value of @mymath{z} (an
abstract dimension for it) as a function of r (distance on a flat 2D
plane, which it can visualize) and thus eliminate @mymath{z}. From
address@hidden, we get @mymath{z^22Rz+r^2=0}
and solving for @mymath{z}, we find:
address@hidden(1\pm\sqrt{1{r^2\over R^2}}\right).}The
address@hidden can be understood from @ref{sphericalplane}: For each
address@hidden, there are two points on the sphere, one in the upper
hemisphere and one in the lower hemisphere. An infinitesimal change in
address@hidden, will create the following infinitesimal change in
address@hidden:
+relations allow the 2D creature to find the value of @mymath{z} (an
+abstract dimension for it) as a function of r (distance on a flat 2D plane,
+which it can visualize) and thus eliminate @mymath{z}. From
address@hidden, we get @mymath{z^22Rz+r^2=0} and
+solving for @mymath{z}, we find:
+
address@hidden(1\pm\sqrt{1{r^2\over R^2}}\right).}
+
+The @mymath{\pm} can be understood from @ref{sphereandplane}: For each
address@hidden, there are two points on the sphere, one in the upper hemisphere
+and one in the lower hemisphere. An infinitesimal change in @mymath{r},
+will create the following infinitesimal change in @mymath{z}:
@dispmath{dz={\mp r\over R}\left(1\over
\sqrt{1{r^2/R^2}}\right)dr.}Using the positive signed equation
@@ 16199,45 +16205,48 @@ instead of @mymath{dz} in the @mymath{ds_s^2}
equation above, we get:
@dispmath{ds_s^2={dr^2\over 1r^2/R^2}+r^2d\phi^2.}
The derivation above was done for a spherical shell of radius
address@hidden as a curved 2D surface. To generalize it to any surface, we
can define @mymath{K=1/R^2} as the curvature parameter. Then the
general infinitesimal change in a static universe can be written as:
address@hidden 1Kr^2}+r^2d\phi^2.}Therefore, we see that
a positive @mymath{K} represents a real @mymath{R} which signifies a
closed 2D spherical shell like @ref{sphericalplane}. When
address@hidden, we have a flat plane (@ref{flatplane}) and a negative
address@hidden will correspond to an imaginary @mymath{R}. The latter two
cases are open universes (where @mymath{r} can extend to infinity).
However, when @mymath{K>0}, we have a closed universe, where
address@hidden cannot become larger than @mymath{R} as in
address@hidden
+The derivation above was done for a spherical shell of radius @mymath{R} as
+a curved 2D surface. To generalize it to any surface, we can define
address@hidden/R^2} as the curvature parameter. Then the general infinitesimal
+change in a static universe can be written as:
+
address@hidden 1Kr^2}+r^2d\phi^2.}
+
+Therefore, when @mymath{K>0} (and curvature is the same everywhere), we
+have a finite universe, where @mymath{r} cannot become larger than
address@hidden as in @ref{sphereandplane}. When @mymath{K=0}, we have a flat
+plane (@ref{flatplane}) and a negative @mymath{K} will correspond to an
+imaginary @mymath{R}. The latter two cases may be infinite in area (which
+is not a simple concept, but mathematically can be modelled with @mymath{r}
+extending infinitely), or finitearea (like a cylinder is flat everywhere
+with @mymath{ds_s^2={dx^2 + dy^2}}, but finite in one direction in size).
@cindex Proper distance
A very important issue that can be discussed now (while we are still
in 2D and can actually visualize things) is that
address@hidden is tangent to the curved space at the
observer's position. In other words, it is on the gray flat surface of
address@hidden, even when the universe if curved:
address@hidden'A}. Therefore for the point @mymath{P}
on a curved space, the raw coordinate @mymath{r} is the distance to
address@hidden'}, not @mymath{P}. The distance to the point @mymath{P} (at
a specific coordinate @mymath{r} on the flat plane) on the curved
surface (thick line in @ref{sphericalplane}) is called the
address@hidden distance} and is displayed with @mymath{l}. For the
specific example of @ref{sphericalplane}, the proper distance can be
calculated with: @mymath{l=R\theta} (@mymath{\theta} is in
radians). using the @mymath{\sin\theta} relation found above, we can
find @mymath{l} as a function of @mymath{r}:
+A very important issue that can be discussed now (while we are still in 2D
+and can actually visualize things) is that @mymath{\overrightarrow{r}} is
+tangent to the curved space at the observer's position. In other words, it
+is on the gray flat surface of @ref{sphereandplane}, even when the universe
+if curved: @mymath{\overrightarrow{r}=P'A}. Therefore for the point
address@hidden on a curved space, the raw coordinate @mymath{r} is the distance
+to @mymath{P'}, not @mymath{P}. The distance to the point @mymath{P} (at a
+specific coordinate @mymath{r} on the flat plane) over the curved surface
+(thick line in @ref{sphereandplane}) is called the @emph{proper distance}
+and is displayed with @mymath{l}. For the specific example of
address@hidden, the proper distance can be calculated with:
address@hidden (@mymath{\theta} is in radians). using the
address@hidden relation found above, we can find @mymath{l} as a
+function of @mymath{r}:
@dispmath{\theta=\sin^{1}\left({r\over R}\right)\quad\rightarrow\quad
l(r)=R\sin^{1}\left({r\over R}\right)address@hidden is just an arbitrary
constant and can be directly found from @mymath{K}, so for cleaner
equations, it is common practice to set @mymath{R=1}, which gives:
address@hidden(r)=\sin^{1}r}. Also note that if @mymath{R=1}, then
address@hidden Generally, depending on the the curvature, in a
address@hidden universe the proper distance can be written as a
function of the coordinate @mymath{r} as (from now on we are assuming
+l(r)=R\sin^{1}\left({r\over R}\right)}
+
+
address@hidden is just an arbitrary constant and can be directly found from
address@hidden, so for cleaner equations, it is common practice to set
address@hidden, which gives: @mymath{l(r)=\sin^{1}r}. Also note that when
address@hidden, then @mymath{l=\theta}. Generally, depending on the the
+curvature, in a @emph{static} universe the proper distance can be written
+as a function of the coordinate @mymath{r} as (from now on we are assuming
@mymath{R=1}):
@dispmath{l(r)=\sin^{1}(r)\quad(K>0),\quad\quad
@@ 16248,47 +16257,53 @@ more simpler and abstract form of
@dispmath{ds_s^2=dl^2+r^2d\phi^2.}
@cindex Comoving distance
Until now, we had assumed a static universe (not changing with
time). But our observations so far appear to indicate that the
universe is expanding (isn't static). Since there is no reason to
expect the observed expansion is unique to our particular position of
the universe, we expect the universe to be expanding at all points
with the same rate at the same time. Therefore, to add a time
dependence to our distance measurements, we can simply add a
multiplicative scaling factor, which is a function of time:
+Until now, we had assumed a static universe (not changing with time). But
+our observations so far appear to indicate that the universe is expanding
+(it isn't static). Since there is no reason to expect the observed
+expansion is unique to our particular position of the universe, we expect
+the universe to be expanding at all points with the same rate at the same
+time. Therefore, to add a time dependence to our distance measurements, we
+can include a multiplicative scaling factor, which is a function of time:
@mymath{a(t)}. The functional form of @mymath{a(t)} comes from the
cosmology and the physics we assume for it: general relativity.

With this scaling factor, the proper distance will also depend on
time. As the universe expands (moves), the distance will also move to
larger values. We thus define a distance measure, or coordinate, that
is independent of time and thus doesn't `move' which we call the
address@hidden distance} and display with @mymath{\chi} such that:
address@hidden(r,t)=\chi(r)a(t)}. We thus shift the @mymath{r} dependence
of the proper distance we derived above for a static universe to the
comoving distance:
+cosmology, the physics we assume for it: general relativity, and the choice
+of whether the universe is uniform (`homogeneous') in density and curvature
+or inhomogeneous. In this section, the functional form of @mymath{a(t)} is
+irrelevant, so we can aviod these issues.
+
+With this scaling factor, the proper distance will also depend on time. As
+the universe expands, the distance between two given points will shift to
+larger values. We thus define a distance measure, or coordinate, that is
+independent of time and thus doesnâ€™t `move'. We call it the @emph{comoving
+distance} and display with @mymath{\chi} such that:
address@hidden(r,t)=\chi(r)a(t)}. We have therefore, shifted the @mymath{r}
+dependence of the proper distance we derived above for a static universe to
+the comoving distance:
@dispmath{\chi(r)=\sin^{1}(r)\quad(K>0),\quad\quad
\chi(r)=r\quad(K=0),\quad\quad \chi(r)=\sinh^{1}(r)\quad(K<0).}
Therefore @mymath{\chi(r)} is the proper distance of an object at a
specific reference time: @mymath{t=t_r} (the @mymath{r} subscript
signifies ``reference'') when @mymath{a(t_r)=1}. At any arbitrary
moment (@mymath{t\neq{t_r}}) before or after @mymath{t_r}, the proper
distance to the object can simply be scaled with
address@hidden(t)}. Measuring the change of distance in a timedependent
(expanding) universe will also involve the speed of the object
changing positions. Hence, let's assume that we are only thinking
about the change in distance caused by something (light) moving at the
speed of light. This speed is postulated as the only constant and
frameofreferenceindependent speed in the universe, making our
calculations easier, light is also the major source of information we
receive from the universe, so this is a reasonable assumption for most
extragalactic studies. We can thus parametrize the change in distance
as

address@hidden(t)ds_s^2 =
c^2dt^2a^2(t)(d\chi^2+r^2d\phi^2).}
+Therefore, @mymath{\chi(r)} is the proper distance to an object at a
+specific reference time: @mymath{t=t_r} (the @mymath{r} subscript signifies
+``reference'') when @mymath{a(t_r)=1}. At any arbitrary moment
+(@mymath{t\neq{t_r}}) before or after @mymath{t_r}, the proper distance to
+the object can be scaled with @mymath{a(t)}.
+
+Measuring the change of distance in a timedependent (expanding) universe
+only makes sense if we can add up space and address@hidden other words,
+making our spacetime consistent with Minkowski spacetime geometry. In this
+geometry, different observers at a given point (event) in spacetime split
+up spacetime into `space' and `time' in different ways, just like people at
+the same spatial position can make different choices of splitting up a map
+into `leftright' and `updown'. This model is well supported by
+twentieth and twentyfirst century observations.}. But we can only add bits
+of space and time together if we measure them in the same units: with a
+conversion constant (similar to how 1000 is used to convert a kilometer
+into meters). Experimentally, we find strong support for the hypothesis
+that this conversion constant can be the speed of light in a vacuum. It is
+almost always written either as @mymath{c}, or in `natural units', as 1. We
+can thus parametrize the change in distance on an expanding 2D surface as
+
address@hidden(t)ds_s^2 = c^2dt^2a^2(t)(d\chi^2+r^2d\phi^2).}
@node Extending distance concepts to 3D, Invoking astcosmiccal, Distance on a
2D curved space, CosmicCalculator
@@ 16298,7 +16313,7 @@ The concepts of @ref{Distance on a 2D curved space} are
here extended
to a 3D space that @emph{might} be curved in a 4D space. We can start
with the generic infinitesimal distance in a static 3D universe, but
this time not in spherical coordinates instead of polar coordinates.
address@hidden is shown in @ref{sphericalplane}, but here we are 3D
address@hidden is shown in @ref{sphereandplane}, but here we are 3D
beings, positioned on @mymath{O} (the center of the sphere) and the
point @mymath{O} is tangent to a 4Dsphere. In our 3D space, a generic
infinitesimal displacement will have the distance:
diff git a/doc/plotsrc/Makefile b/doc/plotsrc/Makefile
index 3759b9c..f377300 100644
 a/doc/plotsrc/Makefile
+++ b/doc/plotsrc/Makefile
@@ 42,7 +42,7 @@ all.pdf: all.tex ./tex/*.tex ./conversions.sh
cp tikz/allfigure0.eps ../gnuastrofigures/iandtime.eps
cp tikz/allfigure1.eps ../gnuastrofigures/samplingfreq.eps
cp tikz/allfigure2.eps ../gnuastrofigures/flatplane.eps
 cp tikz/allfigure3.eps ../gnuastrofigures/sphericalplane.eps
+ cp tikz/allfigure3.eps ../gnuastrofigures/sphereandplane.eps
# Make all the conversions:
./conversions.sh ../gnuastrofigures/
diff git a/doc/plotsrc/all.tex b/doc/plotsrc/all.tex
index 8d6351d..1700910 100644
 a/doc/plotsrc/all.tex
+++ b/doc/plotsrc/all.tex
@@ 148,6 +148,6 @@ appropriate directory.
\input{tex/flatplane.tex}
\input{tex/sphericalplane.tex}
+\input{tex/sphereandplane.tex}
\end{document}
diff git a/doc/plotsrc/tex/sphericalplane.tex
b/doc/plotsrc/tex/sphereandplane.tex
similarity index 100%
rename from doc/plotsrc/tex/sphericalplane.tex
rename to doc/plotsrc/tex/sphereandplane.tex
 [gnuastrocommits] master updated (aea662d > e965a70), Mohammad Akhlaghi, 2017/10/18
 [gnuastrocommits] master 00cbca4 3/5: Tidy up 2D curvature gnuastro.texi fixes  s/sphericalplane/sphereandplane/, Mohammad Akhlaghi, 2017/10/18
 [gnuastrocommits] master 22df182 1/5: gnuastro.texi: 2D curved space: main change in Minkowski spacetime ending, Mohammad Akhlaghi, 2017/10/18
 [gnuastrocommits] master 56cc992 2/5: 2D curvature doc: deanthropomorphisation, s/sphericalplane/sphereandplane/, Mohammad Akhlaghi, 2017/10/18
 [gnuastrocommits] master e965a70 5/5: Merged updates to Distance on a 2D curved space section,
Mohammad Akhlaghi <=
 [gnuastrocommits] master 7500ac6 4/5: Edits for recent updates to the 2D curved space section, Mohammad Akhlaghi, 2017/10/18