text:
\section{Introduction}

The objects we will focus on in this paper are the so--called \emph{$n$--pods}.
For $n=5$ they are referred as pentapods and for $n = 6$ as hexapods, which are also
known as \emph{Stewart Gough platforms}.
As described in \cite{Nawratil2014}, the geometry of this kind of mechanical manipulators is defined by the coordinates of the $n$ platform anchor points $p_i = (a_i, b_i, c_i) \in \R^3$ and of the $n$ base anchor points $P_i = (A_i, B_i, C_i) \in \R^3$ in one of their possible configurations. All pairs of points $(p_i, P_i)$ are connected by a rigid body, called \emph{leg}, so that for all possible configurations the distance $d_i = \left\| p_i - P_i \right\|$ is preserved.
\begin{figure}[!ht]
	\begin{center}
		\begin{overpic}[width=0.35\textwidth]{pentapod}
			\begin{small}
				\put(-8,4){$P_1$}
				\put(32,-5){$P_2$}
				\put(99.5,4){$P_3$}
				\put(58,21){$P_4$}
				\put(16,16){$P_5$}

				\put(-6,72){$p_1$}
				\put(41,68){$p_2$}
				\put(69.5,61){$p_3$}
				\put(96.5,81){$p_4$}
				\put(12,87){$p_5$}
			\end{small}
		\end{overpic}
	\end{center}
	\caption{Sketch of an $n$-pod for $n = 5$, a pentapod.}
\end{figure}
\begin{notation}
	We will think of an $n$--pod $\Pi$ as a triple
	\[ \Pi = \Big( (p_1,\dots,p_n),(P_1,\dots,P_n),(d_1,\dots,d_n) \Big) \]
	where $p_i$, $P_i$ and $d_i$ are defined as above.
\end{notation}
We are interested in describing the \emph{self--motions} of a given $n$--pod
$\Pi$, namely which direct isometries $\sigma$ of $\R^3$ satisfy the condition
(which is called the \emph{spherical condition})
\begin{equation}
\label{equation:spherical_condition}
	\left\| \sigma(p_i) - P_i \right\| \; = \; d_i \quad \quad \textrm{for all }
i \in \{1, \ldots, n\}
\end{equation}
In particular we want to understand what is the \emph{dimension} of the set of
these isometries, namely the \emph{mobility} of $\Pi$ (we will make these
concepts precise at the beginning of Section \ref{bond_geometric}), and what
conditions we have to impose on the base and platform points to reach a
prescribed mobility.

In order to do so we first study the group of direct isometries of $\R^3$, and
in particular we embed and compactify it in a projective space in a way
specifically tuned for $n$--pods. The key idea behind
all of this is that, if we introduce suitable coordinates in the space of
direct isometries and we consider the condition given by Equation
(\ref{equation:spherical_condition}) then the latter becomes
linear in these coordinates. These coordinates provide the desired
compactification, and Section \ref{compactification} is devoted to the study of
some of its geometric properties, which play an important role in the proofs of
the results of the subsequent section. In particular, we focus our attention on
the natural action of direct isometries on this compactification and on its
boundary. In Section \ref{bond_geometric} we use this information to establish
some results which can be framed in the so--called \emph{bond theory} (see
\cite{HegeduesSchichoSchroecker}, \cite{Nawratil2014}): the presence of some
boundary point in the projective closure of the set of self--motions of an
$n$--pod implies precise geometric constraints on base and platform points.
The set of boundary points is a $5$--dimensional complex algebraic variety, but
we give geometric interpretation of its complex points, which form a
$10$--dimensional real algebraic variety.
These results allow us to provide some necessary conditions for the mobility
of $n$--pods. Eventually we consider the case of $n$--pods with high mobility
(namely with strictly more than one degree of freedom) and we also provide
necessary conditions on the geometry of these devices.

\medskip
From the kinematic point of view this paper contains the following main
results (see Corollary \ref{corollary:mob1} and Theorem
\ref{theorem:mobility_two}):

\medskip
\noindent
{\bf Result 1} {\it
If an $n$--pod is mobile, then one of the following conditions holds:
	\begin{itemize}
	\item[(i)]
		There exists at least one pair of orthogonal projections $\pi_L$ and
$\pi_R$ such that the projections of the platform points $p_1, \ldots, p_n$ by
$\pi_L$ and of the base points $P_1, \ldots, P_n$ by $\pi_R$ differ by an
inversion or a similarity.
	\item[(ii)]
		There exists $m \leq n$ such that $p_1, \ldots, p_m$ are collinear and
$P_{m+1}, \ldots, P_n$ are collinear, up to permutation of indices.
	\end{itemize}
}

\medskip
In the following we only give one example for each of the two cases of Result 1,
as a full listing of all examples known in the literature is beyond the scope of
this paper (for some of them, see for example \cite{GeissSchreyer},
\cite{Karger} and \cite{KargerHusty}).

\smallskip
{\bf \noindent Example ad (i):}

It was proven by Bricard (cf.\ Chapter VI of \cite{bricard}) that there is exactly one type of non--trivial\footnote{The trivial motions with this property are translations with spherical trajectories, and the rotation of the moving system about a fixed axis or fixed point, respectively.} motions, where all points have spherical paths.
Moreover it is well known (cf.\ page 324--325 of \cite{bottroth}) that this motion--type is a composition of a rotation about a fixed axis and a translation parallel to this axis. Without loss of generality we can assume that this axis is the $z$--axis. We can take any number of points $p_1, \ldots, p_n$ as platform anchor points, and the centers $P_1, \ldots, P_n$ of the spheres containing their paths as base anchor points. For $n \geq 6$ and generic choice of $p_1, \ldots, p_n$, we will get an $n$--pod with mobility $1$. If we project the points $p_i$ and $P_i$ onto the $xy$--plane, then the resulting points $\pi_z(p_i)$ and $\pi_z(P_i)$ are coupled by  an inversion $\iota$ followed by a rotation $\rho$ and therefore the condition (i) is fulfilled.

Based on this example we also want to show that the condition (i) is not
sufficient for the existence of a self-motion.
To do so, we add an extra leg $(p_{n+1}, P_{n+1})$ where $P_{n+1}$ is not the
center of the sphere holding the path of $p_{n+1}$, but another point with the
same $xy$--coordinates. Then the hypothesis of (i) in Result 1
is fulfilled, but the mobility of the new $(n+1)$--pod is zero.

\smallskip
{\bf \noindent Example ad (ii):}

On the contrary, condition (ii) is  sufficient for the existence of a self--motion.
If the platform is located in a way that the carrier line of $p_1, \ldots, p_m$ coincides with the carrier line of $P_{m+1}, \ldots, P_n$, then the platform can rotate freely about this line. Therefore we get a 1--dimensional set of so--called {\it butterfly--motions}.

Note that $n$--pods which fulfill the condition (ii) can also have further
self--motions beside these butterfly--motions, even if they do not possess the
property of item (i). Good examples for this fact are the three types of
Bircard's flexible octahedra \cite{bricardoct}, as they can be interpreted as
hexapods.

\medskip
\noindent
{\bf Result 2} {\it
	Let $\Pi$ be an $n$--pod with mobility $2$ or higher. Then one of the
following holds:
	\begin{itemize}
		\item[(a)]\smallskip
			there are infinitely many pairs $(L,R)$ of elements of $S^2$ such that the
points $\pi_L(p_1), \ldots, \pi_L(p_n)$ and $\pi_R(P_1), \ldots, \pi_R(P_n)$
are equivalent by an inversion or a similarity;
		\item[(b)]
			there exists $m \leq n$ such that $p_1, \ldots, p_m$
are collinear and $P_{m+1} = \ldots = P_n$, up to permutation of indices and
interchange between base and platform;
		\item[(c)]
			there exists $m \leq n$ with $1 < m < n-1$  such that $p_1, \ldots, p_m$
lie on a line $g \subseteq \R^3$ and $p_{m+1}, \ldots, p_n$ lie on a line $g'
\subseteq \R^3$ parallel to $g$, and $P_1, \ldots, P_m$ lie on a line $G
\subseteq \R^3$ and $P_{m+1}, \ldots, P_n$ lie on a line $G' \subseteq \R^3$
parallel to $G$, up to permutation of indices.
	\end{itemize}
}

\smallskip
This last result, in particular condition (a), is the starting point of further
investigations on pentapods with mobility $2$, which are carried on in
\cite{GalletNawratilSchicho}, relying on a new technique called \emph{M\"obius
Photogrammetry}.

\section{Compactification of $\SE$}
\label{compactification}

We start our discussion in Subsection \ref{compactification:new} by introducing
a new compactification of the group of direct isometries of $\R^3$ in a
projective space. Then in Subsection \ref{compactification:action} we study the
natural action of isometries on this compactification, and we prove that it is
given by linear changes of coordinates. At last in Subsection
\ref{compactification:boundary} we describe the boundary of the
compactification.

\subsection{A new compactification}
\label{compactification:new}

We study the 6-dimensional algebraic group $\SE$ of direct isometries of affine
$3$--space $\R^3$ into itself.
One can embed $\SE$ as an open subset of a quadric hypersurface in $\p^7_{\R}$,
called the \emph{Study quadric}. This compactification of $\SE$ turns out to be
extremely useful in the study of mobility properties of objects coming from
robotics and kinematics (see for example \cite{HegeduesSchichoSchroecker},
\cite{Nawratil2014}). However, in our situation we will see that a different
compactification will lead us to a better comprehension of the phenomena which
can arise.

Any isometry of $\R^3$ can be written as a pair $(M,y)$, where $M \in \SO$ is
the linear contribution and $y \in \R^3$ is the image of the origin
$o \in \R^3$. We define $x := -M^t y = -M^{-1} y$ and $r := \langle
x,x \rangle = \langle y,y \rangle$, where $\langle \cdot, \cdot \rangle$ is the
Euclidean scalar product. The isometry $(M, y)$ is considered as a point in
$\p^{16}_{\R}$ with coordinates
\begin{itemize}
	\item[$\cdot$] $m_{11}, \dots, m_{33}$ (the entries of the matrix),
	\item[$\cdot$] $x_1,\dots,x_3$ (the coordinates of $x$),
	\item[$\cdot$] $y_1,\dots,y_3$ (the coordinates of $y$),
	\item[$\cdot$] $r$ and $h$ (a homogenization coordinate; for group elements
we can assume that it is equal to $1$).
\end{itemize}
The group $\SE$ is defined by the inequality $h \ne 0$ and equations
\[ M M^t = M^t M = h^2 \cdot \mathrm{id}_{\R^3}, \;\; \det(M)=h^3, \]
\[ M^t y + h x = 0, \;\; M x + h y = 0 , \]
\[ \langle x,x \rangle = \langle y,y \rangle = r h \]
(not all equations are needed, for instance $M^t y + h x = 0$ is a consequence
of the other equations and the inequality).
We define $X_{\R}$ as the Zariski closure of $\SE$, i.e. the zero set of the set
of all equations vanishing at $\SE$.
Using computer algebra, a Gr\"obner basis for this set of equations can be
computed\footnote{This can be done, for example, by adjoining a temporary
variable $u$, computing a Gr\"obner basis of the equations above and equation
$hu-1$ with an elimination order that eliminates $u$, and then taking the subset
of the basis of elements that have degree $0$ in $u$.}. The degree of $X_{\R}$
can also be computed using computer algebra by the leading monomials of the
Gr\"obner basis: it is $40$.
\begin{remark}
	In the first version of this paper we constructed a projectively isomorphic
compactification of $\SE$ using Study parameters. However it turned out that
the construction above is more direct, computationally simpler and easier to
generalize to higher dimensions.
\end{remark}
We consider the spherical conditions $\left\| \sigma(p_i) - P_i \right\| = d_i$
we want to impose to rigid motions in $\R^3$; if we set $h = 1$, this can
be expressed by:
\begin{equation}
\label{equation:sphere_condition}
\begin{aligned}
	d_i^2 & = \langle M p_i + y - P_i,M p_i + y - P_i \rangle \\
	& = \langle M p_i,M p_i\rangle + 2\langle M p_i, y \rangle + r +\langle
P_i,P_i\rangle
	-2\langle M p_i, P_i \rangle - 2 \langle y, P_i\rangle \\
	& = \langle p_i, p_i \rangle -\langle P_i, P_i\rangle + r + 2\langle p_i,M^t
y \rangle -2\langle M p_i, P_i \rangle - 2 \langle y, P_i \rangle \\
	& = \langle p_i, p_i\rangle -\langle P_i, P_i \rangle + r - 2\langle
p_i, x\rangle - 2 \langle y, P_i \rangle -2\langle M p_i, P_i\rangle.
\end{aligned}
\end{equation}

\begin{remark}
	After homogenization, Equation~(\ref{equation:sphere_condition}) becomes
linear in the projective coordinates of $\p^{16}_{\R}$.
\end{remark}

\smallskip
By introducing this compactification of $\SE$ we reduced the problem of dealing
with Equation (\ref{equation:spherical_condition}) to the problem of
understanding linear equations on $X_{\R} \subseteq \p^{16}_{\R}$. In order to
fully use the techniques from algebraic geometry and to be able to set up bond
theory, we need to extend our ground field to the complex numbers.

\begin{definition}
	From now on we work with the complexification of $X_{\R}$, denoted by $X$. In
	order to do this we simply take the equations defining $X$ (which have real
	coefficients), and we think of them as polynomials over $\C$. Hence what we
	get is a projective variety in $\p^{16}_{\C}$ of complex dimension $6$ and
	degree $40$ whose real points are in bijection with the points of $X_{\R}$.
	Inside $X$ we can consider the complexification of $\SE$, which we will denote
	by $\SEC$, hence we have an injective map $\Phi: \SE \hookrightarrow \SEC
	\subseteq X$.
\end{definition}

\subsection{Action of $\SE$ on $X$}
\label{compactification:action}

In this subsection we want to extend the natural actions of $\SE$ on itself,
given by composition on the left and on the right, to actions of $\SE$ on
$\p^{16}_{\C}$ which restrict to actions on $X$. This will be useful
in the proofs of Section \ref{bond_geometric}, because it will allow us to
exploit the symmetries of $X$.

Let $\sigma_1: v \mapsto M_1 v + y_1$ and $\sigma_2: v \mapsto M_2 v + y_2$ be
isometries. Then the product $\sigma_{12} = \sigma_1 \sigma_2$ maps
\[ v \; \mapsto \; (M_1 M_2) v + (M_1 y_2 + y_1). \]
We set $M_{12} := M_1 M_2$ and $y_{12} := M_1 y_2 + y_1$. The remaining affine
coordinates of $\sigma_{12}$ are
\[
	\begin{spreadlines}{0.8em}
		\begin{aligned}
			x_{12} & = -M_{12}^t y_{12} = -M_2^t M_1^t M_1 y_2 - M_2^t M_1^t y_1 \\
			& = -M_2^{t} y_2 - M_2^t M_1^t y_1 = x_2 + M_2^t x_1, \\
			r_{12} & = \langle y_{12}, y_{12}\rangle = \langle y_1,y_1\rangle +
		\langle{M_1 y_2, M_1 y_2}\rangle + 2 \langle M_1 y_2, y_1\rangle \\
			& = r_1 + r_2 - 2 \langle x_1,y_2 \rangle.
		\end{aligned}
	\end{spreadlines}
\]
This product becomes bilinear after homogenization, and the projective coordinates of $\sigma_{12}$ are
\begin{equation}
\label{equation:product}
	\big( \underbrace{h_1 h_2}_{h_{12}}: \underbrace{M_1
	M_2}_{M_{12}}:
	\underbrace{M_2^t x_1 + h_1 x_2}_{x_{12}}: \underbrace{h_2 y_1
	+ M_1 y_2}_{y_{12}}: \underbrace{h_2 r_1 + h_1 r_2 - 2\langle
	x_1,y_2\rangle}_{r_{12}} \big).
\end{equation}
(The matrices and vectors appearing in the above coordinates should be replaced
by their entries.)

\begin{proposition}
\label{proposition:product}
	Let $(h_1:M_1:x_1:y_1:r_1),(h_2:M_2:x_2:y_2:r_2) \in X$. Then the above
product	is defined if at least one of $h_1$, $h_2$ is not equal to zero.
\end{proposition}

\begin{proof}
	Assume that the product is undefined, which means that all entries are zero.
	In particular, $h_1 h_2 = 0$. We assume $h_1 = 0$ (the other case $h_2 = 0$
	can be treated analogously). Assume, indirectly, that $h_2\ne 0$. Since
	$\det(M_2) = h_2^3$, it follows that $M_2$ is invertible.
	Since $M_1 M_2 = 0$, it follows that $M_1=0$. Since $M_2^t x_1 + h_1 x_2=0$,
	it follows that $x_1 = 0$. Since $h_2 r_1 + h_1 r_2 - 2 \langle x_1,
	y_2 \rangle = 0$, it follows that $r_1 = 0$. Then all coordinates of the first
	element vanish,	a contradiction.
\end{proof}

\begin{remark}
	Since the $h$--coordinate of any element $\sigma \in \Phi(\SE)$ is always
different from zero, then Proposition \ref{proposition:product} ensures that
left and right multiplication by $\sigma$, which a priori are maps from
$\Phi(\SE)$ to itself, extend to linear maps $X \longrightarrow X$.
\end{remark}

\begin{remark}
\label{remark:action}
	We specialize Equation \eqref{equation:product} to the cases of left and
right multiplication by translations or rotations along the origin. We fix
$\sigma \in X$ and we suppose that it has coordinates $\sigma = (h: M: x: y:
r)$.
\begin{adjustwidth}{0.4cm}{0.4cm}
\begin{itemize}
	\item[a)]
		Given a vector $s \in \R^3$, the translation by $s$ gives the following
		element	$\sigma' \in \Phi(\SE)$:
		\[ \sigma' = \big( 1: \mathrm{id}: -s: s: \langle s, s \rangle \big). \]
		Then left multiplication by $\sigma'$ provides
		\[
			\sigma' \tth \sigma \; = \; \big( h: M: -M^t s + x: hs + y: h \tth
			\langle s, s \rangle + r + 2 \langle s, y \rangle \big),
		\]
		while right multiplication by $\sigma'$ provides
		\[
			\sigma \tth \sigma' \; = \; \big( h: M: x - hs: y + Ms: r + h \tth \langle
			s, s \rangle - 2 \langle x, s \rangle \big).
		\]
	\item[b)]
		Given an orthogonal matrix $M' \in \SO$, the rotation around the origin by
		$M'$ gives the following element $\sigma' \in \Phi(\SE)$:
		\[ \sigma' \; = \; (1: M: 0: 0: 0). \]
		Then left multiplication by $\sigma'$ provides
		\[
			\sigma' \tth \sigma \; = \; \big( h: M'M: x: M'y: r \big)
		\]
		while right multiplication by $\sigma'$ provides
		\[
			\sigma \tth \sigma' \; = \; \big( h: M M': M'x: y: r \big)
		\]
\end{itemize}
\end{adjustwidth}
\end{remark}

\subsection{Boundary of $X$.}
\label{compactification:boundary}

\begin{definition}
The \emph{boundary} of $X$ is defined as $B := X \setminus \SEC$. It is the
closed subset of $X$ cut out by the linear equation $h = 0$.
\end{definition}

For any point $(h: M: x: y: r) \in B$, we have
\[ M M^t \; = \; M^t M \; = \; 0, \quad \quad M^t y \; = \; M x \; = \; 0, \]
\[ \langle x,x \rangle \; = \; \langle y,y \rangle \; = \; 0 . \]
The first equation shows that $\mathrm{rank}(M) \le 1$. Hence $M = v w^t$ for
two suitable vectors $v, w \in \C^3$. It should be noted that $v$ and $w$ are
not unique: one may multiply $v$ by a nonzero complex number and $w$ by its
inverse. We have two cases:
\begin{itemize}
	\item[i.]
		if $M = 0$, we can take both $v$ and $w$ to be zero;
	\item[ii.]
		if $M \neq 0$, then again by the same equation, it follows that
$\langle v,v \rangle = \langle w,w \rangle = 0$. The second equation implies
$\langle x,w \rangle = \langle y,v \rangle = 0$. Then the subspace spanned by
$x$ and $w$ is totally isotropic with respect to $\langle \cdot, \cdot \rangle$,
and this implies it has dimension $1$, so $x$ and $w$ are linearly dependent.
Similarly, also $y$ and $v$ are linearly dependent.
\end{itemize}

\medskip
We can partition the boundary in five subsets. The nomenclature of the various
types of points will become clear in Section \ref{bond_geometric}.

\subsubsection{Vertex.}
\label{compactification:boundary:vertex}

For any real point in $B$, we have $v = w = x = y = 0$. The only nonzero
coordinate is $r$, so we have a unique real point $v_0 = (0: \ldots : 0 : 1)$ in
$B$, called the \emph{vertex}. A computer algebra computation shows that this is
a point of multiplicity $20$ on $X$, but we do not need this fact.

\subsubsection{Inversion Points.}
\label{compactification:boundary:inversion}

Consider the matrix $N := r M + 2y x^t$ (it will be clear later in the
discussion why we choose this expression for $N$). A boundary point $\beta$ with
$M \ne 0$ and $N \ne 0$ is called an \emph{inversion point}. In this case we
have $x = \lambda w$ and $y = \mu v$ with $\lambda, \mu \in \C$. Hence the
coordinates of an inversion point can be written as $(0:v w^t:\lambda w: \mu
v:r)$. Since $v$
and $w$ satisfy the quadratic equation $\langle v,v \rangle = \langle w,w
\rangle = 0$ (called the equation of the \emph{absolute conic} in $\p^2_{\C}$),
the complex dimension of the set of inversion points is 5 (one for $v$, one for
$w$, one for $\lambda$, one for $\mu$, one for $r$). One can show that these are
smooth points of the boundary, but we do not need this fact.

In order to compute normal forms, we first apply rotations. Multiplication from
the right  by a rotation of matrix $M'$ gives (see Remark
\ref{remark:action})
\[ (0: v w^t M': \lambda \tth M' w: \mu \tth v: r), \]
so it leaves $v$ fixed. Being $M'$ an orthogonal
matrix, it is in particular unitary, so it preserves both the scalar product
and the Hermitian norm of $w$, and the action is transitive on vectors with
$\langle w, w \rangle = 0$ and of the same Hermitian norm. Hence $w$ can be
taken to a vector of the form $\delta \tth (1, i, 0)^t$, where $\delta \in
\C^{*}$. Multiplication from the left acts analogously on $v$. Hence by suitable
rotations from both sides we obtain $v = \gamma \tth (1,i,0)^t$ and $w =
\delta \tth (1,i,0)^t$ with both $\gamma$ and $\delta$ different from zero
since $M \neq 0$. Then projectively we can suppose that $M = \left(
\begin{smallmatrix}
	1 & i & 0 \\
	i & -1 & 0 \\
	0 & 0 & 0
\end{smallmatrix}
\right)$.

The action of left multiplication by a translation with vector $s \in \R^3$ on
the boundary point $\beta$ gives (see Remark \ref{remark:action})
\[ \big(0:vw^t:(-\langle v,s \rangle + \lambda) \tth w: \mu \tth v: r + 2\mu
\tth \langle v,s\rangle
\big) , \]
and similarly the action by right multiplication with vector $t \in \R^3$ gives
\[ \big(0:vw^t: \lambda \tth w: (\langle w,t \rangle + \mu) \tth v: r - 2\lambda
\tth \langle w,t \rangle \big) . \]
This shows that we can achieve $\lambda = \mu = 0$ by multiplication by
translations from both sides (for example, since we reduced to the situation $v
= (1, i,0)^t$, one can take $s_1 = \mathrm{Re} \lambda$, $s_2 = \mathrm{Im}
\lambda$ and $s_3$ to be arbitrary, where $s = (s_1, s_2, s_3)^{t}$, and
similarly for $t$). It also shows that the matrix $N$ is invariant under
translations (this was the reason why we chose $N$ in this way). So by
translations from both sides, we obtain $x = y = 0$. The value of $r$ cannot be
changed by any rotation that fixes $x = y = 0$, but we still can apply a
rotation of the form $\left(
\begin{smallmatrix}
	c & d & 0 \\
	-d & c & 0 \\
	0 & 0 & 1
\end{smallmatrix}
\right)$, with $c^2 + d^2 = 1$, from the left. The effect on $M$ is
multiplication by $(c+id)$, and we have no effect on $r$. Projectively, this
is the same as leaving $M$ untouched and multiplying $r$ by $(c+id)^{-1}$.
Hence we can reach the situation where $r \in \R_{>0}$. We notice that $r$
cannot be zero, otherwise we would have $N = 0$. So inversion points have the
following normal forms:
\[ \beta \; = \; (0: \underbrace{1:i:0:i:-1:0:0:0:0}_M:
\underbrace{0:0:0}_{x}: \underbrace{0:0:0}_{y}: r), \]
with $r\in\R_{>0}$.

\subsubsection{Butterfly Points.}
\label{compactification:boundary:butterfly}

A boundary point $\beta$ with $M \ne 0$ and $N = 0$ is called a \emph{butterfly
point}. The complex dimension of the set of butterfly points is $4$: as before,
we can choose $v$ and $w$ on the absolute conic curve, and $\lambda, \mu \in
\C^\ast$.
The normal form is constructed similarly as above. In this case, when we obtain
$x = y = 0$, the fact that $M \neq 0$ and $N = 0$ forces $r$ to be zero. In this
case, we have only a single normal form, namely
\[ \beta = (0: \underbrace{1:i:0:i:-1:0:0:0:0}_M: \underbrace{0:0:0}_{x}:
\underbrace{0:0:0}_{y}: 0). \]

\subsubsection{Similarity Points.}
\label{compactification:boundary:similarity}

The points $\beta = (0: M: x: y: r) \in B$ such that $M = 0$, $x \ne 0$ and $y
\ne 0$ are called \emph{similarity points}. Since $x$ and $y$ are on the
absolute conic, the complex dimension of the set of similarity points is $4$.

\smallskip
To compute normal forms of similarity points, we first apply rotations. As
we saw in Subsection \ref{compactification:boundary:inversion}, right
multiplication fixes $y$ and $r$ and can transform $x$ to $\gamma
\tth (1,i,0)^t$, and left multiplication fixes $x$ and $r$ and can transform
$y$ to $\delta \tth (1, i, 0)^t$, with both $\gamma$ and $\delta$ in $\C^{*}$.
Hence projectively we can always suppose that $\delta = 1$, so we can
reduce any similarity point to one such that $x = \gamma \tth (1,i,0)^t$ and $y
=
(1,i,0)^t$. Then translations act transitively on $r$, thus we may
get to the situation with $r = 0$. Eventually, as we have already seen in
Subsection \ref{compactification:boundary:inversion}, we can perform rotations
so that we can ensure that $\gamma$ is a real positive number. So we get normal
forms of the following kind
\[ \beta = (0: \underbrace{0:0:0:0:0:0:0:0:0}_M:
\underbrace{\gamma:i\gamma:0}_{x}: \underbrace{1:i:0}_{y}: 0), \]
with $\gamma \in \R_{>0}$.

\subsubsection{Collinearity Points.}
\label{compactification:boundary:collinearity}

For the last group of points $\beta$ in $B$ we have $M = 0$ and either $x = 0$,
$y \ne 0$ or $x \ne 0$, $y = 0$. These points are called \emph{collinearity
points}. There are two subsets of collinearity points, one with $x = 0$ and one
with $y = 0$. Both subsets have complex dimension $2$ (since there is still a
free value for $r$ to choose).

By rotations, we can achieve either $x = (1:i:0)^t$ or $y = (1:i:0)^t$.
Translations act transitively on $r$, so we get two normal forms, namely
\[ \beta = (0: \underbrace{0:0:0:0:0:0:0:0:0}_M: \underbrace{1:i:0}_{x}:
\underbrace{0:0:0}_{y}:0) , \]
\[ \beta = (0: \underbrace{0:0:0:0:0:0:0:0:0}_M: \underbrace{0:0:0}_{x}:
\underbrace{1:i:0}_{y}: 0). \]

\bigskip
We conclude the section about the boundary of $X$ by showing that we can
associate to each inversion, butterfly and similarity point a pair $(L,R)$ of
elements of $S^2$, namely oriented directions in $\R^3$. This piece of
information will play an important role in the main results of Section
\ref{bond_geometric}.

\smallskip
We recall that in the case of inversion and butterfly points the matrix $M$ is
of rank $1$, since it is non zero and has rank $\leq 1$, as implied by the
boundary condition $h = 0$. Hence $M$ is of the form $v w^t$ for two non zero
vectors whose coordinates satisfy
\[ v_1^2 + v_2^2 + v_3^2 \; = \; 0 \quad \mathrm{and} \quad w_1^2 + w_2^2 +
w_3^2 \; = \; 0 \]
Then we can think of $v$ and $w$ as points of the conic $C = \big\{ \alpha^2 +
\beta^2 + \gamma^2 = 0 \big\}$ in $\p^2_{\C}$. Notice that, although $v$ and $w$
are not unique, they always give the same pair of points on the conic. We would
like to think of $v$ and $w$ as directions in $\R^3$, and in order to do this we
provide an identification between $C$ and $S^2$. This identification is
accomplished in two steps, namely first we consider an isomorphism $C \cong
\p^1_{\C}$ and then we take the bijection between $\p^1_{\C}$ and $S^2$ given by
the stereographic projection\footnote{These identifications are very special
ones, since they become isomorphisms of real varieties when we consider,
respectively, componentwise complex conjugation on $C$, the map $(s,t) \mapsto
(-\overline{t}, \overline{s})$ on $\p^1_{\C}$ and the antipodal map on $S^2$
as real structures. This can be understood as a hint why these particular
choices work well, but we do not use this property in our investigations.}. \\
The isomorphism $C \cong \p^1_{\C}$ is given by the parametrization
\[ \p^1_{\C} \ni (s,t) \; \mapsto \; \big( (s^2 - t^2): i(s^2 + t^2): 2st
\big) \in C \]
and its inverse
\[
	\left\{
	\begin{array}{lcll}
		(\alpha: \beta : \gamma) & \mapsto & (\alpha - i \beta : \gamma) &
\mathrm{if\ } (i\alpha +
\beta, \gamma) \neq (0,0) \\
		(\alpha: \beta : \gamma) & \mapsto & (\gamma: - \alpha - i \beta) &
\mathrm{otherwise}
	\end{array}
	\right.
\]
The identification between $\p^1_{\C}$ and $S^2$ by stereographic projection
is provided by the following equations:
\[
\begin{array}{l}
	\left\{
	\begin{array}{lcll}
		(0,0,1) & \mapsto & (0:1) \in \p^1_{\C} \\ [2mm]
		(\lambda, \mu, \nu) & \mapsto & \left( 1: \frac{\lambda + i \mu}{1 - \nu}
\right) \in \p^1_{\C} & \mathrm{for\ all\ } (\lambda, \mu, \nu) \in S^2
\setminus \big\{ (0,0,1) \big\}
	\end{array}
	\right. \\ [8mm]
	\left\{
	\begin{array}{lcll}
		(0:1) & \mapsto & (0,0,1) \in S^2 \\ [2mm]
		(1: a + ib) & \mapsto & \left( \frac{2a}{a^2 + b^2 + 1}, \frac{2b}{a^2 +
b^2 + 1}, \frac{a^2 + b^2 - 1}{a^2 + b^2 + 1}  \right) & \mathrm{for\ all\ } a,b
\in \R
	\end{array}
	\right.
\end{array}
\]
For example if $v = (1:i:0) \in C$, then the corresponding element
of $S^2$ is the South pole $(0,0,-1)$.

\smallskip
In this way it is possible to assign to each inversion or butterfly point a
pair $(L,R)$ of elements in $S^2$. We would like to do the same for similarity
points. There is a glaring obstruction in doing this, namely the fact that for
similarity points both the $h$ and the $m_{ij}$--coordinates are zero. On the
other hand for all boundary points the two matrices $M$ and $xy^t$ are linear
dependent, and in the case of similarity points $x y^t$ is different from zero.
Moreover $x$ and $y$ satisfy $\langle x, x \rangle = \langle y, y \rangle = 0$.
So we can associate to a similarity point the pair of elements of $S^2$ coming
from the vectors $x$ and $y$.

\begin{definition}
\label{definition:left_right}
	Via these identifications we can associate to every inversion, butterfly or
similarity point $\beta$ in $B$ a pair $(L,R)$ of elements of $S^2$,
which are respectively called the \emph{left} and the \emph{right vector} of
$\beta$.
\end{definition}

\section{Geometric Interpretation of Bonds}
\label{bond_geometric}

This section represents an instance of a more general technique
called \emph{bond theory}: the goal is to extract information from boundary
points which arise as limits of self--motions of an $n$--pod. Boundary points do
not represent direct isometries of $\R^3$, but nevertheless we can give them
geometric meaning, since their presence as limits of self--motion determine
geometric conditions the base and platform points have to satisfy.

\medskip
Recall from Section \ref{compactification} that $\Phi(\SE)$ is an embedding of
$\SE$ in $\p^{16}_{\C}$, that we denoted by $\SEC$ its complexification and
that we defined $X$ as the Zariski closure of $\SEC$. Moreover, recall that we
think of an $n$--pod as a triple
		\[ \Pi = \Big( (p_1, \dots, p_n),(P_1, \dots, P_n),(d_1, \dots, d_n) \Big)
\]
Eventually, recall that in the new coordinates of $\p^{16}_{\C}$ the spherical
condition given by Equation \eqref{equation:spherical_condition} reads as
\begin{equation}
\label{equation:leg_condition}
	d_i^2 \tth h \; = \; \big( \langle p_i, p_i\rangle \tth - \langle P_i, P_i
\rangle \big) \tth h + r - 2 \langle p_i, x\rangle - 2 \langle y, P_i \rangle
-2 \langle M p_i, P_i \rangle,
\end{equation}
which gives a linear form $l_i$ on $\p^{16}_{\C}$.

\begin{remark}
	Let $\Pi$ be an $n$--pod, then the real points of the intersection
	\[ \SEC \cap \big\{ l_i \, = \, 0 \big\} \]
	are in bijective correspondence with the set of all $\sigma \in \SE$ such that
the distance between $\sigma(p_i)$ and $P_i$ is $d_i$, namely the set of direct
isometries satisfying the spherical condition for $p_i$ and $P_i$.
\end{remark}

\begin{definition}
	Let $\Pi$ be an $n$--pod, then the intersection of $X$ with the hyperplanes
defined by $\big\{ l_i = 0 \big\}$ for $i \in \{ 1, \ldots, n \}$ is called
the \emph{complex configuration set} of $\Pi$ and denoted by $K_\Pi$; the real
points of this intersection are called the \emph{real configuration set} of
$\Pi$.
The complex dimension of $K_\Pi \cap \SEC$ as a complex algebraic variety is
called the \emph{mobility} of $\Pi$. If the mobility of $\Pi$ is greater than or
equal to $1$, then $\Pi$ is said to be \emph{mobile}.
\end{definition}

\begin{remark}\label{rem:cardinality}
	For a generic hexapod, the complex configuration set is finite of
cardinality~$40$. This has been shown by \cite{RongaVust}. It also follows from
the fact $\deg(X) = 40$ that was mentioned in Subsection
\ref{compactification:new}: in fact in the coordinates of $\p^{16}_{\C}$
the spherical equation \eqref{equation:sphere_condition} becomes linear, hence
every leg of an $n$--pod imposes a linear condition on $X$. For a
generic hexapod $\Pi$, its complex configuration set $K_{\Pi}$ is given by
the intersection of $6$ generic hyperplanes in $\p^{16}_{\C}$ with $X$. The
intersection of the hyperplanes gives a generic codimension $6$ linear space
$H_{\Pi}$. Now we use the following general fact from projective geometry: the
intersection of a complex projective variety of dimension $r$ and degree $d$
with a generic linear space of codimension $r$ consists of $d$ points. Since $X$
has dimension $6$, its intersection with $H_{\Pi}$, namely $K_{\Pi}$, is given
by a finite number of points whose cardinality equals the degree of $X$. Hence
in the generic case $K_{\Pi}$ is constituted by $40$ points.
\end{remark}

\begin{definition}
	Let $\Pi$ be an $n$--pod, we define its set of \emph{bonds} $B_{\Pi}$ as the
intersection of $K_{\Pi}$ and the hyperplane $\big\{ h = 0 \big\}$, namely
$B_{\Pi}$ is the intersection of $K_{\Pi}$ with the boundary $B$ of $X$, as
defined in Subsection \ref{compactification:boundary}.
\end{definition}

\begin{remark}
	For bonds, Equation \eqref{equation:leg_condition} reduces to
	\begin{equation}
	\label{equation:leg_condition_bonds}
		r - 2 \langle p_i, x\rangle - 2 \langle y, P_i \rangle
-2 \langle M p_i, P_i \rangle \; = \; 0.
	\end{equation}
\end{remark}

\begin{definition}
	We call the condition imposed by Equation
\eqref{equation:leg_condition_bonds} the \emph{pseudo spherical condition} for
the points $(p_i, P_i)$ at the bond $(0:M:x:y:r)$.
\end{definition}

\begin{remark}
\label{remark:vertex_no_bond}
	Recall that the vertex $v_0$ is the only real point of $B$ (see Subsection
\ref{compactification:boundary:vertex}). Since $v_0$ can never be a bond of an
$n$--pod (in fact, by instantiating $v_0$ in Equation
\eqref{equation:leg_condition_bonds} we would get the contradiction $1 = 0$),
then $B_{\Pi}$ has no real points.
\end{remark}

\begin{remark}
\label{remark:mobility_bonds}
	If an $n$--pod $\Pi$ is mobile, then by definition $\dim K_{\Pi} \cap \SEC
\geq 1$, so $\dim K_{\Pi} \geq 1$. Since $B_{\Pi}$ is an hyperplane section of
$K_{\Pi}$, it follows that the dimension decreases at most by $1$, so $B_{\Pi}$
is not empty. By the same argument we have that if the mobility is greater
than, or equal to $2$, then $\Pi$ admits infinitely many bonds.
\end{remark}

Before coming to the main results of this section, recall that at the end of
Subsection \ref{compactification:boundary} we associated to each inversion,
butterfly and similarity points a pair of directions in $S^2$, called the left
and right vector of the boundary point.

\begin{definition}
	Given a unit vector $\epsilon \in S^2$, we denote by $\pi_{\epsilon}: \R^3
\longrightarrow \R^2$ the \emph{orthogonal projection along $\epsilon$}, namely
for every $p = (a,b,c) \in \R^3$, the point $\pi_{\varepsilon}(p)$ is the
orthogonal projection of $p$ on the affine plane orthogonal to
$\varepsilon$, passing through the origin.
\end{definition}

\smallskip
We are ready to state and prove the main results of this section.
\begin{theorem}
\label{theorem:inversione-similarity}
	There is a one-to-one correspondence between inversion/simi-larity points
	$\beta$ with both left and right vectors $L$ and $R$ equal to the South pole
	$(0,0,-1) \in S^2$ and inversions/similarities $\kappa$ of the plane such that
	for any pair of points $(p,P)$ in $\R^3$ the pseudo spherical condition for
	$(p,P)$ at $\beta$ is equivalent to the fact that $\kappa(q) = Q$ where $q 	=
	\pi_L(p)$ and	$Q = \pi_R(P)$.
\end{theorem}
\begin{proof}
	We first treat the case of inversion points. Suppose that $\beta_{0} \in B$ is
an inversion point with $L = R = (0,0,-1)$. Suppose furthermore that $\beta_{0}$
is in the normal form (see Subsection
\ref{compactification:boundary:inversion}):
	\[ \beta_{0} \; = \; (0:\underbrace{1:i:0:i:-1:0:0:0:0}_M:
\underbrace{0:0:0}_{x}: \underbrace{0:0:0}_{y}: r), \]
	with $r \in \R_{>0}$. We get that $\pi_{L} = \pi_{R}$ are the projection on
the first two coordinates. Thus if $p = (a, b, c)$ and $P = (A, B, C)$, then
$q = (a, b)$ and $Q = (A, B)$. If we instantiate the pseudo spherical
condition for $(p, P)$ at $\beta_0$ given by Equation
(\ref{equation:leg_condition_bonds}) we get the relations:
	\begin{equation}
	\label{equation:inversion}
		\left\{
		\begin{array}{l}
			a A - b B = \nicefrac{r}{2} \\
			b A + a B = 0
	  \end{array}
		\right.
	\end{equation}
	which define an inversion $\kappa_0$ mapping $q$ to $Q$. Conversely, suppose
we are given an inversion $\kappa_0$ described by Equation
\eqref{equation:inversion}.
Then going backwards in the previous argument we can see that we obtain an
inversion point in normal form as in the thesis. \\
	Suppose now that the $\beta \in B$ is an inversion point with $L = R =
(0,0,-1)$, but not necessarily in normal form. Then, as we saw in
Subsection \ref{compactification:boundary:inversion}, we can find two isometries
$\sigma_1, \sigma_2 \in \SE$ which fix left and right vectors such that
$\sigma_1 \beta \sigma_2 = \beta_0$ is in normal form (here $\sigma_1 \beta
\sigma_2$ denotes the element of $X$ obtained by left action by $\sigma_1$ on
$\beta$ and then by right action of $\sigma_2$). Moreover $\sigma_1$ and
$\sigma_2$ induce isometries $\tau_1$ and $\tau_2$ of $\R^2$ such that the
following two diagrams commute:
\[ \xymatrix{\R^3 \ar[r]^{\sigma_1} \ar[d]_{\pi_L} & \R^3 \ar[d]^{\pi_L} \\ \R^2
\ar[r]^{\tau_1} & \R^2 } \qquad \qquad \xymatrix{\R^3 \ar[r]^{\sigma_2}
\ar[d]_{\pi_R} & \R^3 \ar[d]^{\pi_R} \\ \R^2 \ar[r]^{\tau_2} & \R^2 } \]

If $\kappa_0$ is the inversion associated to $\beta_0$, then we define $\kappa =
\tau_1 \kappa_0 \tau_2$, and one can check that the association $\beta
\leftrightarrow \kappa$ satisfies the requirements of the thesis.

	\smallskip
	We consider now the case of similarity points. Suppose that $\beta_0 \in B$
is a similarity point with $L = R = (0,0,-1)$. Suppose furthermore that
$\beta_{0}$ is in the normal form (see Subsection
\ref{compactification:boundary:similarity}):
	\[ \beta = (0: \underbrace{0:0:0:0:0:0:0:0:0}_M:
	\underbrace{\gamma:i\gamma:0}_{x}: \underbrace{1:i:0}_{y}: 0), \]
	with $\gamma \in \R_{>0}$. Again for this kind of points $\pi_L$ and $\pi_R$
are both the projection on the first two coordinates. Performing analogous
computations as before we get the relations:
	\begin{equation}
	\label{equation:similarity}
		\left\{
		\begin{array}{l}
			A = -\gamma \tth a  \\
			B = -\gamma \tth b
	  \end{array}
		\right.
	\end{equation}
	These define a similarity $\kappa_0$ mapping $q$ to $Q$. Conversely, and in
the general case of points not in normal form, we argue as for inversion points.
\end{proof}
\begin{remark}
	As pointed out in Subsection \ref{compactification:boundary:inversion}, the
complex dimension of the set of inversion points is $5$. Theorem
\ref{theorem:inversione-similarity} allows, as remarked in the Introduction, to
associate to it a real dimension, which can be computed as follows: $2$ degrees
of freedom for choosing the vector $L$ and the same for $R$, and $6$ degrees of
freedom for specifying an inversion from $\R^2$ to itself. So, in total, we get
$10$. We can argue analogously for similarity points: here the real dimension
is $8$.
\end{remark}
\begin{cor}
\label{cor:inversion-similarity}
	Assume that $\beta \in B_{\Pi}$ is an inversion/similarity bond of $\Pi$.
	Let $L,R \in S^2$ be the left and right vector of $\beta$.
	For $i = 1, \ldots, n$, set $q_i = \pi_L(p_i)$ and $Q_i = \pi_R(P_i)$.
	Then there is an inversion/similarity of $\R^2$ mapping $q_1, \ldots, q_n$ to $Q_1, \ldots, Q_n$.

	Conversely, let $L,R \in S^2$ be two unit vectors such that the images of
$(p_1, \dots, p_n)$ under $\pi_L$ and of $(P_1, \dots, P_n)$ under $\pi_R$
differ by an inversion/ similarity.
	Then $\Pi$ has an inversion/similarity bond with left vector $L$ and right
vector $R$.
	\end{cor}
\begin{proof}
	In both cases of inversion and similarity points we can apply suitable
isometries in order to put $\beta$ in normal form. Then it is enough to apply
Theorem \ref{theorem:inversione-similarity}.
\end{proof}

\begin{theorem}
\label{theorem:butterfly}
	There is a one-to-one correspondence between butterfly points $\beta$ and
	pairs $(g_L, g_R)$ of oriented lines in $\R^3$ such that for any pair of
	points $(p,P)$ in $\R^3$ the pseudo spherical condition for $(p,P)$ at $\beta$
	is equivalent to the fact that $p \in g_L$ or $P \in g_R$.
\end{theorem}
\begin{proof}
	Suppose that $\beta_0 \in B$ is a butterfly point in the normal form (see
Subsection \ref{compactification:boundary:butterfly}):
	\[ \beta_0 = (0: \underbrace{1:i:0:i:-1:0:0:0:0}_M: \underbrace{0:0:0}_{x}:
\underbrace{0:0:0}_{y}: 0). \]
	In this case we associate to $\beta_0$ the lines $g_L = g_R = \{
z\mathrm{-axis} \}$, both oriented to the South pole $(0,0,-1) \in S^2$. If we
instantiate the pseudo spherical condition for $(p, P)$ at $\beta$ given by
Equation (\ref{equation:leg_condition_bonds}) we get the relations:
	\begin{equation}
	\label{equation:butterfly}
		\left\{
		\begin{array}{l}
			a A - b B = 0 \\
			a B + b A = 0
	  \end{array}
		\right.
	\end{equation}
	Equation (\ref{equation:butterfly}) can be interpreted as: the vector $(a,
b)$ is parallel both to the vector $(A, -B)$ and to the vector $(B,
A)$. This is possible if and only if either $(a, b) = (0,0)$ or $(A,
B) = (0,0)$. Hence either $p$ is of the form $(0,0,c)$ (namely it lies on
$g_L$) or $P$ is of the form $(0,0,C)$ (namely it lies on $g_R$).

	If $\beta \in B$ is an arbitrary butterfly point, then from Subsection
\ref{compactification:boundary:butterfly} we know that there exist isometries
$\sigma_1, \sigma_2 \in \SE$ such that $\sigma_1 \beta \sigma_2 = \beta_0$ is
in normal form. Then we associate to $\beta$ the pair of lines
\[ (g_L, g_R) \; = \; \Big( \left(\sigma_1\right)^{-1}\big( \{ z\mathrm{-axis}
\} \big), \left(\sigma_2\right)^{-1}\big( \{ z\mathrm{-axis} \} \big) \Big) \]
	with orientation given by the left and right vectors of $\beta$. One can
check that the equivalence in the thesis holds. Conversely, given two oriented
lines $g_L$ and $g_R$ we can find isometries $\sigma_1, \sigma_2 \in \SE$ such
that $g_L = \sigma_1 \big( \{ z\mathrm{-axis} \} \big)$ and $g_R = \sigma_2
\big( \{ z\mathrm{-axis} \} \big)$, both oriented to the South pole $(0,0,-1)
\in S^2$. Then we associate to $(g_L, g_R)$ the butterfly point $\sigma_1
\beta \sigma_2$.
\end{proof}

\begin{cor}
\label{cor:butterfly}
	Assume that $\beta \in B_{\Pi}$ is a butterfly bond of $\Pi$.
	Let $L,R \in S^2$ be the left and right vector of $\beta$.
	Then, up to permutation of indices $1, \dots, n$, there exists $m \leq n$ such
	that $p_1, \ldots, p_m$ are collinear on a line parallel to $L$,
	and $P_{m+1}, \ldots, P_n$ are collinear on a line parallel to $R$.

	Conversely, let $L,R \in S^2$ be two unit vectors such that
	$p_1, \ldots, p_m$ are collinear on a line parallel to $L$, and
	$P_{m+1}, \ldots, P_n$ are collinear on a line parallel to $R$.
	Then $\Pi$ has a butterfly bond with left vector $L$ and right vector~$R$.
\end{cor}

\begin{notation}
	Recall from Subsection \ref{compactification:boundary:collinearity} that the
set of collinearity points is partitioned into two subsets: if the
$y$--coordinate of a collinearity point is zero we call it a \emph{left
collinearity point}, while if the $x$--coordinate is zero we call it a
\emph{right collinearity point}.
\end{notation}

\begin{theorem}
\label{theorem:collinear}
	There is a one-to-one correspondence between left (resp. right) collinearity
	points $\beta$	and oriented lines $g$ in $\R^3$ such that for any pair of
	points $(p, P)$ in	$\R^3$ the pseudo spherical condition for $(p, P)$ at
	$\beta$ is equivalent to	the fact that $p \in g$ (resp. $P \in g$).
\end{theorem}
\begin{proof}
	Suppose that $\beta_0 \in B$ is a left collinearity point and suppose that it
is in normal form (see Subsection
\ref{compactification:boundary:collinearity}):
	\[ \beta = (0: \underbrace{0:0:0:0:0:0:0:0:0}_M: \underbrace{1:i:0}_{x}:
	\underbrace{0:0:0}_{y}:0) , \]
	We associate to $\beta_0$ the line $g = \{ z\mathrm{-axis} \}$, directed to
the South pole $(0,0,-1) \in S^2$. If we instantiate the pseudo spherical
condition for $(p, P)$ at $\beta_0$ given by Equation
(\ref{equation:leg_condition_bonds}) we get the relations:
	\[ 0 \; = \; -2(a + i b) \quad \Leftrightarrow \quad a = b = 0 \]
	which is equivalent to $p \in g$.

	If $\beta \in B$ is an arbitrary left collinearity point we proceed as in the
proof of Theorem \ref{theorem:butterfly}. Analogous arguments prove the
statement about right collinearity points.
\end{proof}

\begin{cor}
\label{cor:collinear}
	Assume that $\beta \in B_{\Pi}$ is a collinearity bond of $\Pi$.
	Then either $p_1, \ldots, p_n$ are collinear or $P_1, \ldots, P_n$ are collinear (or both).

	Conversely, if $p_1,\dots,p_n$ are collinear or $P_1, \ldots, P_n$ are
collinear (or both), then $\Pi$ has a collinearity bond.
\end{cor}

The following Corollary gives a necessary criterion for mobility of $n$--pods.
For the fist time (to the authors' knowledge) a necessary criterion
for the mobility of hexapods can be defined by the invariant
linkage parameters, irrespective of a specific configuration. (The well-known
criterion for infinitesimal mobility, see \cite{Merlet89}, refers to an explicit
relative pose of the platform with respect to the base.)

\begin{cor}
\label{corollary:mob1}
If an $n$--pod is mobile, then one of the following conditions holds:
	\begin{enumerate}[(i)]
		\item
			There exists at least one pair of orthogonal projections $\pi_L$ and
$\pi_R$ such that the projections of the platform points $p_1, \ldots, p_n$ by
$\pi_L$ and of the base points $P_1, \ldots, P_n$ by $\pi_R$ differ by an
inversion or a similarity.
		\item
			There exists $m \leq n$ such that $p_1, \ldots, p_m$ are collinear and
$P_{m+1}, \ldots, P_n$ are collinear, up to permutation of indices.
	\end{enumerate}
\end{cor}
\begin{proof}
	Since by hypothesis $K_{\Pi} \cap \SEC$ has dimension $\geq 1$, it follows
that $B_{\Pi}$ is not empty (see Remark \ref{remark:mobility_bonds}). Hence
there is at least one inversion/similarity/ collinearity/butterfly bond, and
then the result follows from Corollaries (\ref{cor:inversion-similarity}),
(\ref{cor:butterfly}) and (\ref{cor:collinear}).
\end{proof}

\begin{remark}
	As the bonds are determined by the invariant linkage parameters, they are
independent of the leg lengths. As a consequence a hexapod, which has $40$
solutions for the direct kinematics over $\mathbb C$ (see\ Remark
\ref{rem:cardinality}), is free of bonds and therefore also free of
self-motions.
	Due to the fact that condition (i) of Corollary \ref{corollary:mob1} is not
sufficient, the converse is not true; i.e.\ there exist hexapods with less
than $40$ solutions for the direct kinematics, which are free of self-motions
(e.g.\ hexapods where the platform and base are planar and projective --- but
not affine --- equivalent \cite{nawratilproj}).
\end{remark}

We conclude stating our last result, concerning constraints on base and
platform points of $n$--pods with higher mobility.

\begin{theorem}
\label{theorem:mobility_two}
	Let $\Pi$ be an $n$--pod with mobility $2$ or higher. Then one of the
following holds:
	\begin{itemize}
		\item[(a)]
			there are infinitely many pair $(L,R)$ of elements of $S^2$ such that the
points $\pi_L(p_1), \ldots, \pi_L(p_n)$ and $\pi_R(P_1), \ldots, \pi_R(P_n)$
differ by an inversion or a similarity;
		\item[(b)]
			there exists $m \leq n$ such that $p_1, \ldots, p_m$ are collinear and
$P_{m+1} = \ldots = P_n$, up to permutation of indices and interchange between
base and platform;
		\item[(c)]
			there exists $m \leq n$ with $1 < m < n-1$  such that $p_1, \ldots, p_m$
lie on a line $g \subseteq \R^3$ and $p_{m+1}, \ldots, p_n$ lie on a line $g'
\subseteq \R^3$ parallel to $g$, and $P_1, \ldots, P_m$ lie on a line $G
\subseteq \R^3$ and $P_{m+1}, \ldots, P_n$ lie on a line $G' \subseteq \R^3$
parallel to $G$, up to permutation of indices.
	\end{itemize}
\end{theorem}
\begin{proof}
	Since $\Pi$ has mobility at least $2$, it has infinitely many bonds (see
Remark \ref{remark:mobility_bonds}). Assume that $\Pi$ admits one collinearity
bond, then we have b) with $m = n$. Assume that it admits infinitely many
butterfly points, then in particular by Corollary \ref{cor:butterfly} there
exists $m \leq n$ such that $p_1, \ldots, p_m$ are collinear and $P_{m+1},
\ldots, P_n$ lie on infinitely many different lines, and therefore we have (b).
Hence we are left with the case when we have infinitely many inversion or
similarity bonds. If these bonds provide infinitely many different left and
right vectors, we are in case (a). Otherwise we have that there are infinitely
many inversion or similarity points with the same left and right vectors
$(L,R)$. We want to argue that in this case both sets $\mcal{U} = \big\{
\pi_L(p_1), \ldots, \pi_L(p_n) \big\}$ and $\mcal{V} = \big\{ \pi_R(P_1),
\ldots, \pi_R(P_n) \big\}$ consist of two points. In fact by Corollary
\ref{cor:inversion-similarity} the inversion/similarity associated to
these bonds maps $\pi_L(p_i)$ to $\pi_R(P_i)$, so $\mcal{U}$ and $\mcal{V}$
have the same cardinality; on the other hand any inversion or similarity is
completely specified if we prescribe the image of three points, so if the
cardinality of $\mcal{U}$ were greater than $2$ then we would have only one
inversion or similarity. Moreover we can exclude the case when both $\mcal{U}$
and $\mcal{V}$ are given by one point, since this falls in case (b). Hence $p_1,
\ldots, p_n$ are arranged on two parallel lines, and the same holds for $P_1,
\ldots, P_n$. From this and the fact that the inversions/similarities should
map $\pi_L(p_i)$ to $\pi_R(P_i)$ it follows that the only possible
configurations are the ones described in (c). As a side remark, since two
points fix a similarity it follows that in this case we have just one
similarity point and infinitely many inversion points.
\end{proof}

As already mentioned in the Introduction, we can also formulate some geometric
conditions on base and platform points in case (a) of Theorem
\ref{theorem:mobility_two}. This can be done by a new technique, called
\emph{M\"obius Photogrammetry}, which is developed by the authors in
\cite{GalletNawratilSchicho}. Moreover it should be noted that based on the
results obtained with this method a complete classification of pentapods with
mobility $2$ was achieved in \cite{NawratilSchicho1, NawratilSchicho2}.

\section*{Acknowledgments}

The first and third author's research is supported by the Austrian Science Fund (FWF): W1214-N15/DK9 and P26607 - ``Algebraic Methods in Kinematics: Motion Factorisation and Bond Theory''. The second author's research is funded by the Austrian Science Fund (FWF): P24927-N25 - ``Stewart Gough platforms with self-motions''.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

In a $\Lambda$CDM cosmology, primordial inhomogeneities in the density field of the Universe are expected to act as seeds for the later formation of structures. On small scales, gravity tends to make initially small inhomogeneities evolve into collapsed structures, most notably dark matter haloes, which later become the hosts of objects such as galaxies and galaxy clusters. In accordance with $\Lambda$CDM, such haloes often interact with each other through mergers; galaxy cluster mergers are the most extreme version of such interactions, and are the most energetic events in the universe since the Big Bang \citep{Sarazin2002}.

One prominent example of a system in interaction is the A901/2 multi-cluster, at $z \sim 0.165$. This is an unrelaxed system that contains four main cores -- A901a, A901b, A902 and the SW group -- and provides an ideal laboratory for probing galaxy evolution along different scales of environment and galaxy masses (\citealt{gray04}, \citealt{Gray2009}). All four subclusters are at similar redshifts \citep[see e.g.][]{Weinzirl2017}, and the two most massive cores (A901a and A901b) have overlapping virial radii, which indicates that the system is likely a multi-cluster merger in its early stages. This is reinforced by the fact that a system with the mass of A901/2 ($\sim$$3.5\times10^{14}$ M$_\odot$) is expected to collapse if its spacial extent is smaller than about 5 Mpc \citep[][]{busha03}, while most of the mass in A901/2 is within a spacial scale of a few Mpc.

Numerical simulations have often been employed to study mergers of galaxy clusters, both from a more general, theoretical point of view, and also in order to model specific objects. Binary cluster collisions are particularly well suited for this purpose, because the numerical resolution can be entirely focused on the objects of interest, as opposed to fully cosmological simulations of structure formation. For example, the Bullet Cluster has been studied in this way \citep{Springel2007, Mastropietro2008, Lage2014}, as have other so-called dissociative clusters \citep[e.g.][]{Donnert2014, MachadoMonteiro2015, Molnar2015}, in which gas and dark matter are offset as a result of the collision. Numerous other observed clusters have been modelled by dedicated simulations that aim to reconstruct their dynamical histories. Simulations have been used to study several phenomena related to collisions of galaxy clusters, such as radio relics \citep[e.g.][]{vanWeeren2011}, sloshing cold fronts \citep[e.g.][]{ZuHone2010, Machado2015, Walker2018}, turbulence \citep[e.g.][]{ZuHone2013a, Vazza2012}, thermal conduction \citep[e.g.][]{ZuHone2013b}, etc. Tailored simulations involving more than two initial objects are more uncommon. For example, a triple merger has been simulated by \cite{Bruggen2012} in order to model \mbox{1RXS~J0603.3+4214}.

Gas-rich galaxies which move within the environment of galaxy clusters are expected to have their evolution affected by the interaction with the intracluster medium (ICM). The ram pressure exerted by the ICM can lead to gas loss by ram pressure stripping \citep{1972ApJ...176....1G}, which, in more extreme cases, leads to the formation of ``jellyfish morphologies'', in which the galaxy is observed featuring a filamentary tail of stripped gas and stars. The phenomenon of ram pressure stripping of cluster galaxies has been extensively modelled through numerical simulations, which have explored e.g. the role of inclination angle in the rate of gas loss \citep{2006MNRAS.369..567R}, the changes in star formation rate which take place within the disks of affected galaxies \citep{2008A&A...481..337K,2012A&A...544A..54S,Ruggiero2017}, and the predicted emission features within their tails \citep{2009A&A...499...87K,2010ApJ...709.1203T}. Although most of the numerical work on ram pressure stripping has been based on idealised setups, cosmological simulations of galaxy formation have also been used to explore the phenomenon, as e.g. in \citet{2007ApJ...671.1434T} and more recently in \citet{2018arXiv181000005Y}.

Jellyfish galaxies have been found in large numbers in different cluster systems (see e.g. \citealt{ebeling14}, \citealt{Poggianti16}). However, the number of jellyfish galaxies found in single systems is usually small.
The numbers range from 21 in Coma (\citealt{smith10}, \citealt{yagi10}), 3 in Virgo (\citealt{abramson16}, \citealt{kenney14}, \citealt{kenney99}), 1 in A3627 (\citealt{sun06}) and 5 in A2744 (\citealt{rawle14}).
Nevertheless, the rich population of $\sim$70 jellyfish galaxy candidates found in the A901/2 system (\citealt{roman-oliveira18}) indicates that clusters in interaction may be an ideal environment to search for these galaxies.
Moreover, \cite{2016MNRAS.455.2994M} performs a large systematic search for such jellyfish morphologies and suggests that galaxy cluster mergers are more likely to be triggering extreme ram pressure stripping events. This scenario has also been suggested in \cite{owers12}, where four jellyfish galaxies were found near merger signatures of the gas. It is not surprising that such relation could exist: in galaxy cluster mergers, higher ICM velocities are found than in isolated clusters, making those environments favourable for the formation of jellyfish structures.

In this work, we attempt to probe the physical mechanism behind the formation of jellyfish galaxies in galaxy cluster mergers. For that, we model the diffuse gas in the A901/2 system with a galaxy cluster merger simulation, and then compare the gas conditions in this model to the location of a sample of jellyfish galaxies found in this system, allowing us to infer a scenario for the triggering of jellyfish morphologies both in the A901/2 system and in galaxy cluster mergers in general. Such theoretical treatment of ram pressure stripping in galaxy cluster mergers has not been given so far in the literature.

This paper is structured as follows. In Section \ref{sec:sample}, we describe the sample of jellyfish galaxy candidates we use and comment on how they were selected. Then we proceed to describe the setup and the results of our galaxy cluster merger simulation in Section \ref{sec:simulation}. The gas conditions in this simulation are correlated with the locations of the jellyfish galaxies in our sample in Section \ref{sec:results}, where we tentatively propose a physical mechanism for the generation of many of those jellyfishes. Finally, our results are discussed and summarised in Section \ref{sec:discussion}, where possible extensions of our work are also presented.

\section{Sample of galaxies} \label{sec:sample}

The galaxies used in this study come from an extensive search for galaxies with jellyfish morphological signatures in the A901/2 system \citep{roman-oliveira18}. This sample was selected through visual inspection of HST/ACS F606W images of galaxies in the parent sample of H$\alpha$ emitting galaxies in the OMEGA survey \citep{chies-santos15,rodriguezdelpino17,Weinzirl2017,wolf18}.
The visual inspection method applied follows the work of \citet{ebeling14} and \citet{Poggianti16}. A classification in JClasses was also employed, in which a number from 1 to 5 is assigned to a galaxy to evaluate its degree of asymmetry -- larger values are correlated with a greater likelihood of the galaxy being an actual jellyfish.

The final sample is restricted to the most reliable cases of jellyfish candidates, which we take as those classified as JClass 3 to 5. This sample contains the 73 galaxies that we use in this work. The image stamps are available at the OMEGA jellyfish candidates ATLAS\footnote{OMEGA jellyfish candidates ATLAS: \url{http://lief.if.ufrgs.br/~fernandavro/atlas.pdf}}.

\section{Simulations} \label{sec:simulation}

This work is based on a galaxy cluster merger simulation including the dark matter haloes and intracluster gas of the 4 subclusters in the A901/2 system, which was used to reproduce their positions on the plane of the sky, along with their observed X-ray properties. The simulations were run with the code \textsc{gadget-2} \citep{Springel2005}; in Appendix \ref{sec:codecomparison}, we also briefly compare the gas conditions in the main simulation with its results when it is run in \textsc{ramses} \citep{Teyssier2002}, in order to assess its robustness against a change in numerical methodology.

\subsection{Simulating the system as a whole}

Here we describe the simulation setup, in which the four subclusters were included with the goal of obtaining a suitable model of the system as a whole. Our aim here was chiefly to recover the relative distances between four subclusters having the known virial masses and also having plausible gas content. The main observational constraints are the virial masses of the subclusters, derived from gravitational weak lensing \citep{Heymans2008}.

The redshifts of the subclusters are close to each other \citep[e.g.][]{Weinzirl2017}, so we assume that they are on the same plane. We further assume, for simplicity, that the trajectories of the four subclusters are on the plane of the sky. Virial equilibrium would require velocity dispersions of roughly 1000\,km/s; we drew random velocities but choosing the signs of the Cartesian coordinates such that the subclusters are all incoming, i.e.\ falling towards the centre of mass. It should be noted that our explicit assumption here is that the subclusters are currently infalling towards their first approach, i.e.\ they have not previously collided.

As a preliminary step, we represent each cluster as a point mass having the $M_{200}$ from \cite{Heymans2008}. They are assigned velocities as described above, and position coordinates are known straightforwardly from observations. With this information we perform a simple gravitational $N$-body simulation (via direct summation) inverting the sign of time; i.e.\ we simply calculate the orbits backwards in time, for 5\,Gyr. This exercise provides a good approximation for the $t=0$ of the actual hydrodynamical simulation.

In the next step, we set up four actual subclusters including dark matter and gas. The method for generating initial conditions is similar to those used in \cite{Machado2015} or \cite{Ruggiero2017}, for example. The dark matter haloes follow a \cite{Hernquist1990} profile:
\begin{equation} \label{eq:hernquist}
\rho_{\rm h}(r) = \frac{M_{\rm h}}{2 \pi} ~ \frac{r_{\rm h}}{r~(r+r_{\rm h})^{3}} \, ,
\end{equation}
where $M_{\rm h}$ is the total dark matter mass, and $r_{\rm h}$ is a scale length. The gas is represented by a \cite{Dehnen1993} density profile (with gas mass $M_{\rm g}$ and scale lenght $r_{\rm g}$), adopting $\gamma=0$:
\begin{equation}
\rho_{\rm g}(r) = \frac{(3-\gamma)~M_{\rm g}}{4\pi} ~ \frac{r_{\rm g}}{r^{\gamma}(r+r_{\rm g})^{4-\gamma}} \, .
\end{equation}
The requirement of hydrostatic equilibrium determines the gas temperatures. Realisations of these initial conditions are created according to the procedures described in \cite{Machado2013}. The virial masses, virial radii and gas fractions of the initial conditions are given in Table~\ref{tb:ic}.

\begin{table}
\caption{Initial conditions of the simulated subclusters. The first column gives the names of the models and the objects they are meant to represent. The second and third columns give the virial mass and virial radius. The fourth column gives the overall gas fraction.}
\label{tb:ic}
\begin{center}
\begin{tabular}{l c c c c c}
\hline
         &   $M_{200}$     & $r_{200}$ &  $f_{\rm gas}$\\
         &  (${\rm M}_{\odot}$)  & (kpc) &     \\
\hline
subcluster A  (A901a)   & $1.3 \times 10^{14}$ & 1034 & 0.08 \\
subcluster B (A901b)    & $1.3 \times 10^{14}$ & 1036 & 0.15 \\
subcluster C (A902)     & $0.4 \times 10^{14}$ & 688  & 0.08 \\
subcluster D (SW Group) & $0.6 \times 10^{14}$ & 788  & 0.06 \\
\hline
\end{tabular}
\end{center}
\end{table}

Each of the four subclusters has $10^6$ gas particles and $10^5$ dark matter particles. Tests of the present simulations indicated convergence across three orders of magnitude in particle numbers, as far as the orbits are concerned. Moreover, since the subclusters are not interpenetrating, their gravitational potentials remain sufficiently spherical in the current stage of the approach. In this specific configuration, one could even attempt to model them by rigid analytic spherical potentials without much loss of detail. We opted to represent them as $N$-body particles. Here we employ the smoothed particle hydrodynamics (SPH) $N$-body code {\sc gadget-2} \citep{Springel2005}, and the evolution is followed for 5\,Gyr.

The four subclusters, created in the manner described above, are then placed at the locations that were reached by the end of the backwards point-mass integration. And then the subclusters are allowed to evolve forward in time for 5\,Gyr. They fall towards the centre of mass until the current observed separations are reached. However, they do not reach exactly the desired coordinates by the end, because the orbits of four point masses are not identical to the orbits of four extended objects. Some fine tuning of their initial positions and velocities was performed by trial and error until an acceptable agreement was reached. In the resulting preferred model, the instant when the coordinates best matched the observations was $t=4.3$\,Gyr.

\begin{figure}
\includegraphics[width=\columnwidth]{figs/density.pdf}
\caption[]{This is the snapshot that best reproduces the observed relative separations between the subclusters ($t=4.3$\,Gyr). Colours represent the projected gas density. Total projected mass is shown as contours.}
\label{fig:density}
\end{figure}

\subsection{X-ray mock image}

Observations indicate that A901b is the only one of the four with significant X-ray diffuse emission. A901a hosts a very bright AGN, so its extended emission is unclear. A902 is barely above the background noise, and the SW Group is essentially undetectable in X-rays \citep{Gilmour2007}. To ensure a higher X-ray emission, subcluster B in Table~\ref{tb:ic} has the highest gas content of the four. In the absence of detailed observational constraints, the other simulated subclusters were chosen to have a low gas fraction of approximately 8 per cent (or 6 per cent in the case of the SW group), towards the lower limit of what is expected for their masses \citep{Lagana2013}. The simulated gas densities of the best-matching instant are shown in Fig.~\ref{fig:density}. In this figure, one may also notice that the centroids of the projected total mass distributions of the four subhaloes reproduce the observed relative separations in a good approximation.

We performed a more quantitative test to ensure that the simulated gas densities were not excessive. Using the $t=4.3$\,Gyr snapshot of the simulation, we produced a mock X-ray image with the following procedure, assuming thermal emission from a hot plasma. We used \texttt{pyXSIM}\footnote{\url{http://hea-www.cfa.harvard.edu/~jzuhone/pyxsim/}}, a Python package for simulating X-ray observations from astrophysical sources. It is based on an algorithm of \cite{Biffi2012,Biffi2013}, but see also \cite{ZuHone2014}. In brief, it takes as input the simulated densities and temperatures of the gas, assumes a constant metallicity of 0.3 Z$_{\odot}$, and generates a photon sample assuming a spectral model ({\sc apec} from the AtomDB database\footnote{\url{http://www.atomdb.org/}}). The photon sample is then projected along the line of sight (the $z$ axis of the simulation). Given the coordinates of the cluster, a foreground Galactic absorption model is also applied, assuming a neutral hydrogen column of $N_{\rm H} = 4 \times 10^{20}~{\rm cm}^{-2}$. The photon list is exported to be used by the SIXTE\footnote{\url{http://www.sternwarte.uni-erlangen.de/research/sixte/}} (Simulation of X-ray Telescopes) package, to be convolved with the \textit{XMM} instrument response (the EPIC MOS camera, in this case). The effective exposure time was 67\,ks and the energy range was 0.2--7.0\,keV. Poissonian noise was added to the resulting 600$\times$600-pixel mock image, shown in Fig.~\ref{fig:A901mockXMM}. Note that the X-ray emission of the SW group is present in the first frame, albeit very faint. Once noise is added, it is lost in the background.

Our resulting model is approximate, and it cannot be expected to account for all details of the observed systems. Furthermore, there are no assurances that the solution we have found for the orbits is unique, as is always the case in such reconstructions. However, the gas properties in the model are physically well-motivated, and compatible with the observational expectation of X-ray detections -- two subclusters with significant emission are obtained, plus two near the threshold of detection (bearing in mind that the diffuse emission of A901a is somewhat inconclusive due to the very bright point source). Therefore, the resulting snapshot of the simulation should offer a sufficiently realistic environment in which to study ram pressure effects.

\begin{figure*}
\includegraphics[width=\textwidth]{figs/Mock_XMM_ESO_08_2018a.pdf}
\caption[]{Comparison between the the mock X-ray image and the observations. First frame: mock X-ray image produced from the simulation. Second frame: same as before, but with added noise. Third frame: \textit{XMM} observation. Fourth frame: Optical image from ESO/WFI.}
\label{fig:A901mockXMM}
\end{figure*}

\section{Local conditions of the jellyfish galaxies} \label{sec:results}

Now we turn to the analysis of the gas properties in the merger model presented in the previous section, with the goal of answering the question: what explains the presence of jellyfish galaxies at the locations where they are found in A901/2? Naturally, the two most important quantities to be analysed should be the diffuse gas density and the diffuse gas velocity across the system, since jellyfish morphologies are caused by ram pressure stripping events, while the ram pressure $P_{\mathrm{ram}}$ depends on those two quantities \citep{1972ApJ...176....1G}:
\begin{equation} \label{eq:rampressure}
P_{\mathrm{ram}} = \rho_{\mathrm{ICM}} v_{\mathrm{ICM}}^2,
\end{equation}
where $v_{\mathrm{ICM}}$ is the ICM velocity relative to a galaxy under consideration. In generating plots involving those quantities, we have used the Python package {\sc yt} \citep{Turk2011} to deposit the simulation particles into a space-filling grid with a ``cell-in-cloud'' approach.

In our analysis, we focus on the ram pressure calculated in the reference frame of each subcluster, as a first approximation for the ram pressure experienced by its member galaxies. The effect of peculiar velocities of member galaxies relative to their parent cluster will be discussed later. These reference frames are defined by the average speed of the dark matter particles within $r_{200}/3$ of the centre of a given subcluster, with the centre location defined as that of the density peak of this cluster's ICM. We have verified that the results that follow are not sensitive to the choice of the inner radius in the velocity calculation -- using the velocities within radii closer to $r_{200}$ yield similar results, but we find it more meaningful to restrict ourselves to the inner region of each subcluster since that region is in principle less disturbed by tidal effects.

With those four reference frames defined, we are then able to calculate the ram pressure of the system as a whole in each of them. This is shown in Fig.~\ref{fig:rampressures}, where the ram pressure is shown in slices along the plane of the four subclusters in our model, overlaid with streamlines of diffuse gas velocity. At the centre of each subcluster the ram pressure is low, since in that region the diffuse gas is on average moving along with the cluster halo. On the other hand, the ram pressure is intense far from the cluster centre, since the diffuse gas from other clusters is moving in the opposite direction at high speed. It turns out that a reasonably narrow ($\sim$100 kpc) boundary exists between those two regions, where a significant increase in the ram pressure takes place. The dashed contours in Fig.~\ref{fig:rampressures} are the approximate locations of those boundaries, which were obtained using ram pressure isocontours, and in each subplot the positions of the galaxies in our sample closest (in projected space) to the subcluster considered in that plot than to any of the other three are shown.

Figure \ref{fig:id} shows a gas density slice of the simulation, in which all the ram pressure boundaries are shown simultaneously, along with the locations of all our jellyfish candidates and some examples of HST images for those galaxies. This plot shows that the density in the system does not feature any pronounced structure at the locations of the boundaries, which implies that they emerge exclusively due to the velocity structure of the diffuse gas around their locations. It is not surprising that this should be the case, since the clusters are approaching each other (and thus their diffuse gas is moving at opposing directions), while the ram pressure depends very strongly on the diffuse gas speed, more so than on its density (see Eq.~\ref{eq:rampressure}). In this way, the ram pressure boundaries can be identified as regions where gas moving along each subcluster and gas from the remainder of the system meet.

\begin{figure*}
\includegraphics[width=\textwidth]{figs/rampressures.png}
\caption[]{Ram pressure intensities in the reference frames of each of the four subclusters. Each subplot is a midplane slice. The dashed lines show the approximate locations of the ram pressure boundaries in each subcluster, and triangles, squares and pentagons represent the locations of jellyfish galaxies classified as JClass 3, 4 and 5, respectively. Ram pressure profiles along the four black lines are shown in Fig.~\ref{fig:discontinuity}.}
\label{fig:rampressures}
\end{figure*}

It can be visually noted in Fig.~\ref{fig:rampressures} that many galaxies are located in the vicinities of the ram pressure boundaries. This is not always the case -- for instance, many galaxies in the A902 subcluster are found in a region without an apparent connection to the boundary we find for that subcluster. Still, a correlation seems to exist. We quantify this effect in the following manner. First, we measure the projected distance from each jellyfish galaxy to its respective nearest boundary. Then, we generate a random cloud of points occupying the same area as those galaxies and also measure their distances to the nearest boundaries. The comparison between these two distributions of distances is shown in Fig.~\ref{fig:randomdistances}, which makes it evident that the jellyfishes are systematically closer to a boundary than what would be expected from a random distribution. The Kolmogorov-Smirnov test applied to cumulative, normalised histograms for both samples indicates that the chance of the two distributions being equivalent is very small, of 1 in $\sim$85 million (p-value of $10^{-6}$ per cent). We also make the same comparison using the STAGES sample \citep{Gray2009} of galaxies in the A901/2 cluster instead of a random sample, filtered for member galaxies with stellar mass between $10^{8}$\,M$_\odot$ and $10^{12}$\,M$_\odot$. The upper quartile, lower quartile and median for the jellyfish distribution are all lower than for the STAGES distribution, with a chance of 1 in 757 (p-value of 0.13 per cent) of the two distributions being equivalent, further reinforcing our thesis that the jellyfishes are systematically closer to the ram pressure boundaries we report.

\begin{figure}
\includegraphics[width=\columnwidth]{figs/randomdistances.png}
\caption[]{Violin plot showing the distributions of distances to the nearest ram pressure boundary for a random set of points, the STAGES sample of galaxies in the A901/2 cluster, and our sample of jellyfishes. The jellyfishes are systematically closer to the boundaries than it would be expected if their locations were random, and they are also systematically closer than the non-jellyfish galaxies in the system.}
\label{fig:randomdistances}
\end{figure}

The distribution of distances to the nearest boundary can also be analysed as a function of JClass. This is shown in Fig.~\ref{fig:distances}. The three distributions are overall quite similar, but an interesting feature is that the median distance to the nearest boundary decreases systematically with JClass. This could be an indication that jellyfish morphologies are more pronounced when a galaxy has just encountered a boundary, and then on a short timespan after that, they become less intense. The median distance for JClass 5 galaxies is 53\,kpc smaller than for JClass 3 galaxies; assuming that the galaxies move at 1000\,km/s, this would imply that the transition from JClass 5 to 3 happens on a timescale of 53\,Myr in this scenario. Despite this being a tantalising hypothesis, Kolmogorov-Smirnov tests applied to the distributions for JClass 3 and 4, 3 and 5 and 4 and 5 galaxies yield p-values of 6.75, 13.5 and 7.06 per cent respectively, meaning we have a low confidence that a difference actually exists between the three distributions.

\begin{figure}
\includegraphics[width=\columnwidth]{figs/distances3.png}
\caption[]{Box plot showing the distances between the jellyfish galaxies and the nearest ram pressure boundaries as a function of JClass. The median distances decrease with JClass: they are 279 kpc, 245 kpc and 226 kpc for JClass 3, 4 and 5 galaxies, but a Kolmogorov-Smirnov test applied to the pairs of distributions lead us to conclude that this is not a statistically significant result.}
\label{fig:distances}
\end{figure}

As a more quantitative illustration of the ram pressure variations along the ram pressure boundaries we report, we show in Fig.~\ref{fig:discontinuity} ram pressure profiles along the four black lines in Fig.~\ref{fig:rampressures}, which were chosen arbitrarily for the sake of illustration. Before the boundaries are crossed, the ram pressure profiles feature some noise, but overall they remain somewhat constant. After the boundary is crossed, an increase of a factor of 10 -- 1000 (depending on the cluster) takes place in the ram pressure within a few hundred kpc. The two largest subclusters (A and B) feature ram pressure increments larger than that of the remaining two (C and D), mainly due to their proximity to each other.

\begin{figure}
\includegraphics[width=\columnwidth]{figs/discontinuity.png}
\caption[]{Ram pressure variation along the four black lines shown in Fig.~\ref{fig:rampressures}, which are all perpendicular to their respective ram pressure boundaries. This illustrates the ram pressure increase which takes place in those boundaries, which can be of a factor of up to 1000.}
\label{fig:discontinuity}
\end{figure}

An initial hypothesis of the analysis so far was to calculate ram pressure in the reference frames of each subcluster, as an approximation for the velocities of its member galaxies. But the galaxies in reality feature peculiar velocities relative to their parent clusters. In Fig.~\ref{fig:differentspeeds} we show the same map as in Fig.~\ref{fig:rampressures} for subcluster C for reference frames with different velocities added in the same direction as that of the cluster average. We have chosen this subcluster for the sake of illustration because it is the one with the simplest ram pressure boundary. We find that the ram pressure boundary is only pronounced for velocities within roughly 100 km/s of the average cluster velocity; beyond that, the boundary fades away and the ram pressure intensity becomes correlated with the gas density at each location. This adds up to our picture so far in the following manner: our scenario should involve galaxies moving at relatively low speed relative to their parent subcluster, perhaps close to their apocentric passage. Those galaxies are still moving at high speed in the reference frame of the system as a whole, allowing them to cross the ram pressure boundary within a short timescale, of $\sim$100 Myr, and then become jellyfishes after that.

\begin{figure}
\includegraphics[width=\columnwidth]{figs/different_speeds.png}
\caption[]{Effect of peculiar velocities relative to the parent galaxy cluster. Each panel is a midplane ram pressure slice in a reference frame defined by the average velocity of the A902 subcluster plus the indicated velocity, added in the same direction. A ram pressure increase in the boundary we report is pronounced for velocities within 100 km/s of the average cluster velocity.}
\label{fig:differentspeeds}
\end{figure}

One could also wonder whether our results would be different for off-plane ram pressure slices, i.e. planes parallel to the plane of the centres of the 4 subclusters, but at a certain height -- so far we have limited ourselves to a mid-plane slice. We have verified that the locations of the ram pressure boundaries are very close to that in the mid-plane slice for heights of up to $\sim$500 kpc; the main difference is that the off-plane densities are lower, making the off-plane ram pressure increments also smaller.

\begin{figure*}
\includegraphics[width=\textwidth]{figs/id.png}
\caption[]{Identification of our sample of galaxies over a midplane density slice of our simulation. The dashed lines are the locations of the ram pressure boundaries shown in Fig.~\ref{fig:rampressures}. Some HST thumbs in arbitrary scale are shown for the sake of illustration, and, also as in Fig.~\ref{fig:rampressures}, triangles, squares and pentagons are used for JClass 3, 4 and 5 galaxies respectively.}
\label{fig:id}
\end{figure*}

\section{Discussion and summary} \label{sec:discussion}

We have employed in our analyses a tailored simulation of the A901/2 system. Even though it consists of four subclusters, we were able to reach a satisfactory model. Dedicated simulations of galaxy cluster mergers involving three or more objects have not often been attempted -- an example is \citet{Bruggen2012}. In a general situation, the large number of degrees of freedom in the initial conditions would render the exploration of the parameter space nearly impracticable. However, in the particular case of A901/2, this approach was feasible due to some simplifying assumptions that were adopted --  apart from the usual setup of such idealised simulations (initially spherical objects, hydrostatic equilibrium, absence of small-scale substructures, etc). Regarding the dynamics of the system, our assumption was that the four objects are incoming for their first approach. This seems to be justifiable, given that they currently display no noticeable large-scale disturbances in their morphologies. Furthermore, we assumed for simplicity that the orbits are all on the plane of the sky. In our best model, the relative separations between the four subclusters are recovered. More importantly, the diffuse X-ray emissions are also quite well reproduced. This suggests that, in spite of the simplifying assumptions adopted, the gas properties in the simulation must be realistic within a good approximation.

From comparing the gas conditions in this model to the locations of the jellyfish galaxies in our adopted sample, we have inferred a possible mechanism for the triggering of jellyfish morphologies in this interaction environment, which is that the galaxies become jellyfishes when they cross a boundary within its parent subcluster where a significant increase in the ram pressure takes place, due to gas originally from the subcluster and gas from the remainder of the system meeting. A ram pressure increment of a factor of up to 1000 takes place on a scale of $\sim$100 kpc, while the galaxies are expected to be moving at speeds greater than 1000 km/s, implying a timescale of $\sim$100 Myr to cross the boundaries. We believe that this combination of a large increment in the ram pressure and a relatively short timescale to cross the boundaries makes it reasonable to conjecture that those boundaries could markedly affect the evolution of the gas content within the galaxies and turn them into jellyfishes. One caveat which should be pointed out is that this mechanism does not necessarily apply to all of the jellyfishes in the system. Indeed, in each subcluster a subset of galaxies far away from the respective boundary exists, which could have had their jellyfish morphologies triggered by a mechanism unassociated to the merger altogether. Still, we find significant differences in the distributions of distance to nearest boundary between the jellyfish sample and both a random sample and the STAGES sample of member galaxies in the system. This suggests that it is reasonable to assume that a significant fraction of the jellyfishes are indeed being generated by the aforementioned mechanism.

Another caveat is that the exact locations of the ram pressure boundaries in the real system could be different from the ones we report due to a variety of factors -- for instance, the exact positioning of clusters in the line of sight, deviations from our adopted initial density profiles and deviations from hydrostatic equilibrium in the clusters prior to the merger are all factors which could lead to different ram pressure distributions. Indeed, regarding deviations from hydrostatic equilibrium, cosmological simulations have pointed out that random bulk motions are relevant in the ram pressure stripping of galaxies in isolated galaxy clusters \citep{2008ApJ...684L...9T}. One illustrative implication of this is the following: if, for some reason, the boundary for subcluster C shown in Fig.~\ref{fig:rampressures} is in reality located some 100 kpc lower than what we find, then it would be located right on top of a concentration of about 8 jellyfish galaxies, which are all behind the boundary in our model.

Although this entire work is devoted to the particular case of the A901/2 system, we expect the ram pressure boundaries we report inside each subcluster to be present in all galaxy cluster mergers which are still in the early stages of the collision, before their centres have crossed each other. In this way, a similar analysis to the one presented here could be carried out for other observed galaxy clusters, in order to probe the universality of the mechanism. Perhaps the biggest difficulty in this is finding jellyfish galaxies in clusters in the first place -- they are rare and their identification is very dependent on visual inspection.
Previous observational work on jellyfish galaxies, such as \citet{2016MNRAS.455.2994M} and \citet{owers12}, have hinted that such galaxies could actually be preferentially found in galaxy cluster merger systems; our findings are consistent with interacting galaxy clusters being a favourable environment for searching for such galaxies.

The summary of this paper is as follows. We have developed a hydrodynamic model for the A901/2 system using a multi-cluster merger simulation, consistent with their positions relative to each other, their masses and their X-ray emissions. This model was used to correlate the gas conditions in the system with the locations of the jellyfish galaxy candidates in it identified by \mbox{\citet{roman-oliveira18}}. We have found that at each subcluster, a boundary exists where gas moving along the cluster and gas from the remainder of the system meet; in those boundaries, an increment of a factor of 10 -- 1000 in the ram pressure takes place within a few hundred kpc, due to a large increment in the diffuse gas velocity. More importantly, we have found that jellyfish galaxies in the system seem to be preferentially located near those boundaries, which could mean that the crossing of those boundaries is the mechanism behind the formation of jellyfishes at those locations. We propose that this mechanism could be common in galaxy cluster mergers which are at the beginning stages of their encounter, possibly making those environments particularly favorable for searching for jellyfish galaxies. This is the first theoretical treatment of ram pressure stripping in the environment of galaxy cluster mergers which has been presented in the literature.

\section*{Acknowledgements}

RR thanks the S\~ao Paulo Research Foundation, FAPESP, for the financial support (grant 15/13141-7). This work has made use of the computing facilities of the Laboratory of Astroinformatics (IAG/USP, NAT/Unicsul), whose purchase was made possible by the Brazilian agency FAPESP (grant 2009/54006-4) and the INCT-A. Simulations were carried out in part at the Centro de Computa\c c\~ao Cient\'ifica e Tecnol\'ogica (UTFPR). This study was financed in part by the \textit{Coordena\c{c}\~{a}o de Aperfei\c{c}oamento de Pessoal de N\'ivel Superior - Brasil} (CAPES) - Finance Code 001.
ACS and FRO acknowledge funding from the Brazilian agencies \textit {Conselho Nacional de Desenvolvimento Cient\'ifico e Tecnol\'ogico} (CNPq) and the Rio Grande do Sul Research Foundation (FAPERGS) through grants PIBIC-CNPq, CNPq-403580/2016-1, CNPq-310845/2015-7, PqG/FAPERGS-17/2551-0001, PROBIC/FAPERGS and L'Or\'eal UNESCO ABC \textit{Para Mulheres na Ci\^encia}.
GBLN thank financial support from CNPq and FAPESP (2018/17543-0). BRP acknowledges financial support from the Spanish Ministry of Economy and Competitiveness through grants ESP2015-68964 and ESP2017-83197. We thank the anonymous reviewer for the feedback, which helped make the presentation of our results more clear.

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