# A comparative evaluation of machine learning methods for robot navigation
through human crowds

Anastasia Gaydashenko  Cisco, CTAO
<EMAIL_ADDRESS>Anastasia Gaydashenko conducted the research while
at internship at JetBrains Research, St. Petersburg, Russia. Daniel Kudenko
Department of Computer Science
University of York, UK
JetBrains Research
St. Petersburg, Russia
<EMAIL_ADDRESS>Aleksei Shpilman  JetBrains Research
St. Petersburg, Russia
<EMAIL_ADDRESS>
###### Abstract

Robot navigation through crowds poses a difficult challenge to AI systems,
since the methods should result in fast and efficient movement but at the same
time are not allowed to compromise safety. Most approaches to date were
focused on the combination of pathfinding algorithms with machine learning for
pedestrian walking prediction. More recently, reinforcement learning
techniques have been proposed in the research literature. In this paper, we
perform a comparative evaluation of pathfinding/prediction and reinforcement
learning approaches on a crowd movement dataset collected from surveillance
videos taken at Grand Central Station in New York. The results demonstrate the
strong superiority of state-of-the-art reinforcement learning approaches over
pathfinding with state-of-the-art behaviour prediction techniques.

###### Index Terms:

Robot Navigation, Human-Robot Interaction, Behavior Prediction, Reinforcement
Learning

##  I Introduction

Robots are predicted to increasingly participate in everyday life of humans
over the next decade, but so far most research applications deal with
interactions between a robot and a single human. However, if robots are to
perform general tasks such as pick-up and delivery in public spaces, they will
need to be able to cope and interact with large crowds. This poses a specific
challenge for navigation control, where robots are supposed to move fast and
efficient, while not endangering the safety of humans walking about in the
same location.

Most work so far on robot navigation in environments with moving obstacles has
been focused on a combination of pathfinding algorithms and human behaviour
prediction [1]. However, more recently, reinforcement learning has been
proposed for robot navigation through crowds [2], but there has been no
conclusive empirical comparison so far to pathfinding with prediction. In this
paper, we present an evaluation of various pedestrian prediction techniques
combined with the state-of-the-art pathfinding algorithm D* Lite [3], and a
recent reinforcement learning approach SA-CADRL [4].

We evaluate the different approaches on a dataset extracted from real crowd
movements recorded by CCTV at Grand Central Station in New York [5]. The
evaluation criteria are speed of movement from start to end point, and number
of collisions of the virtual robot with humans. Our empirical results
demonstrate that the reinforcement learning approach results in one of the
fastest robots, while significantly reducing the number of collisions with
humans, as measured in simulation based on real human movement data.

##  II Related work

Research in the area of socially aware robot motion follows three main
directions: behavior prediction, reinforcement learning, and behavior cloning.

Most of the work so far has focused on predicting humans’ trajectories and
plan the robot movement accordingly [6]. For the prediction of humans’
trajectories, clustering and typical behavior extraction was shown to be
effective (e.g. [7]). More recently, Deep Learning techniques have been
successfully applied to the prediction of individual and crowd motion [8],
which is currently the best performing technique. For robot route planning
with dynamic obstacles the approaches vary. The simplest approach is to use a
pre-defined set of rules to move the robot [9]. Another approach is to compute
attracting and repelling steering forces based on current and (where
available) predicted future positions of obstacles [10]. This steering
technique has been extended with Markov Decision Processes to deal with
uncertainty in the predictions [1]. Svenstrup et al. [11] iteratively improve
the current best path using a Rapidly-exploring Random Tree with obstacles
represented as potential fields.

In a recent reinforcement learning approach [2], the robot was trained in
simulation to move optimally through a crowd.

The third main approach to socially aware robot motion, behavior cloning,
relies on the fact that humans can navigate in human crowds quite successfully
and tries to imitate human behavior. This movement behaviour imitation problem
has recently been tackled with inverse reinforcement learning, computing a
reward function for crowd navigation from human crowd movement demonstrations,
that was subsequently used to train a reinforcement learning agent [12].

In this paper, we are focusing on the first two approaches, and perform a
comparative evaluation in a simulation based on real-world pedestrian crowd
recordings.

##  III Crowd movement dataset

In order to simulate realistic crowd movements, we used a dataset obtained
from surveillance video from Grand Central Station (New York, USA). The
dataset consists of 6001 frames of camera images taken from the top, and 12684
labeled pedestrian trajectories as obtained by Yi et al. [5]. An example frame
with a single pedestrian trajectory is shown in Figure 1. The frames were
sampled from a video of 4000 secs (i.e., a frequency is $1.5$ frames per
second), and each frame has a resolution of $1920\times 1080$. Trajectories
were filtered by minimum length, where only those with at least 10 frames were
retained. This left 12244 valid trajectories in the dataset with an average
length of $37$ frames.

Figure 1: Example frame with sample trajectory for one person.

In a data pre-processing step, we overlaid the frames with a $64\times 36$
grid, as shown in Figure 2. This allows discretization of the pedestrian
positions and reduces the size of the state space for the path planning. The
grid resolution was chosen based on average human speed from frame to frame
which are $35$ pixels for the horizontal coordinate and $30$ pixels for the
vertical one.

Figure 2: Overlaying a $64\times 36$ grid.

##  IV Path planning methods

We employed two general approaches to the path planning problem. In the first
approach pedestrians are considered as moving obstacles and the goal is to
predict their positions over the next five time steps. These positions are
then used as dynamic obstacles for the path planning algorithms such as D*
Lite [3], which will be described later. The second approach is to learn a
robot movement policy via reinforcement learning in simulation based on the
recorded pedestrian movements. We start with a discussion of prediction
methods for pedestrian positions.

###  IV-A Pedestrian position prediction approach

We compared several methods for position prediction in our study to
demonstrate their impact on the robot policy.

####  IV-A1 Baseline prediction

A very simple method to predict the future position of a pedestrian is to
assume that the pedestrian can move into any neighboring grid square in a time
step. All these squares are considered to be occupied by the pedestrian in the
next time step when performing the path planning.

####  IV-A2 Random forest regression prediction

A more sophisticated approach is to employ machine learning techniques for the
prediction task. We chose Random Forest Regressor [13] for this as a standard
technique which has demonstrated robust performance in the literature. In this
method, the prediction outputs are the five expected coordinates of the
pedestrian over the next five time steps, relative to their original position.

The state for which a prediction is made was represented using features based
on the five previous grid positions of the pedestrian as follows:

  * •

Absolute coordinates (10 features).

  * •

Coordinates relative to the previous point ($x$ and $y$ speed, 8 features).

  * •

Speed (distance between neighboring points, 4 features) and average speed (1
feature).

  * •

Acceleration (change in speed from one step to the next one, 3 features) and
average acceleration (1 feature).

  * •

Angle of movement for each step (4 features).

The features are illustrated on Fig. 3

Figure 3: Features for Random Forest Regressor. The top left shows the x and y
coordinates of the last five trajectory points. The top right picture shows
the distance between neighbouring trajectory points. The bottom left picture
shows the coordinates relative to the previous trajectory point. The bottom
right picture shows the changes in movement angle between two time steps.

####  IV-A3 Behavior-CNN prediction

Behavior-CNN [8] is a Neural Network architecture which was designed
especially for movement prediction. It takes as input the embedded positions
of the pedestrian over the last five time steps, and outputs the predicted
positions over the next five time steps.

These embedded positions are called a displacement volume and correspond to a
three-dimensional matrix of size $X\times Y\times 10$, where $X$ and $Y$
denote the area size (in our case $X=1920$ and $Y=1080$). The third dimension
corresponds to the distances between the pedestrian positions over the last 5
time steps, along the x and y dimension. This results in 10 values. The
distance values are first normalized to be between 0 and 1 according to $X$
and $Y$ respectively, and then 1 is added to is to avoid too small numbers.
More formally, the 10-dimensional vector is computed as given in equation 1,
where ($x_{t}$,$y_{t}$) is the $i$-th position of the pedestrian with $t=1$
being the time five steps ago, and $t=5$ the current time step:

$[1+\frac{x_{5}-x_{1}}{X},1+\frac{y_{5}-y_{1}}{Y},...,1+\frac{x_{5}-x_{5}}{X},1+\frac{y_{5}-y_{5}}{Y}]$
(1)

All other cells in the matrix are set to zero. The output the CNN has the same
shape as the input and represents the predicted embedded positions over the
next five time steps.

For other technical details we refer the reader to the original paper.

Figure 4: Behaviour-CNN framework [8].

####  IV-A4 Path planning with the D* Lite algorithm

D* Lite [3] is a heuristic path planning algorithm that takes as input a start
and goal position, and recalculates the current path at each step taking into
account the predicted coordinates of obstacles over a number of time steps.
The method uses the freespace assumption, i.e. it assumes that unknown space
is empty unless the predictions tell otherwise. We refer the reader to the
original paper for more details.

Realization in Python 3 was based on the java implementation GitHub
111https://github.com/daniel-beard/DStarLiteJava.

###  IV-B Reinforcement learning approach

####  IV-B1 SA-CADRL

A state-of-the-art RL approach to socially aware robot movement is SA-CADRL
(Socially Aware Collision Avoidance with Deep Reinforcement Learning) [4]
which aims to learn an optimal robot policy based on the observed state of the
robot and nearby agents (i.e. pedestrians). During training, the robot
receives a positive reward for reaching the destination and a negative reward
for delays and collisions. The policy is represented as a neural network whose
input is formed from state information on the robot and nearby agents
(pedestrians) and that outputs the robot velocity. This state information
includes the current position, size, and velocity of the agent/robot.

Formally speaking, by using SA-CADRL architecture authors are trying to
restore robot’s policy $\pi$ such as expected value to goal would be minimal:

$\arg\min_{\pi(s)}\mathbf{E}[t_{g}|\pi]$

With the following restrictions:

  * •

$||p_{t}-\hat{p}_{t}||_{2}\geq r+\hat{r}$ – for any frame distance between
robot and other agents should be greater than their radii sum.

  * •

$p_{t_{g}}=p_{g}$ – at the last frame robot should be in the goal coordinates.

  * •

$p_{t}=p_{t-1}+\Delta t\cdot\pi(s)$ – current robot’s position is defined by
previous position and chosen action (moving direction).

To evaluate this approach on the same trajectories and with the same metrics
as the other approaches, we used the pre-trained network provided by the Chen
et al. which can be found in the SA-CADRL project GitHub page
222https://github.com/mfe7/cadrl_ros.

Figure 5: Original SA-CADRL network architecture [4].

####  IV-B2 Grid-SA-CADRL

A limitation of the SA-CADRL approach is that the robot can’t move backwards.
We extended the SA-CADRL approach to provide the robot with these abilities.
Furthermore, we modified SA-CADRL to work on a grid-environment rather than a
coordinate space as the original approach. This means that rather than having
a continuous movement action, the actions are made discrete corresponding to a
move into one of the 8 neighboring grid squares or standing still.

The resulting network architecture is shown in Figure 6. It takes the robots
and three nearby pedestrians’ states as inputs into an LSTM layer (size 64),
followed by 3 fully connected layers with the last one outputting
probabilities of actions through a softmax transformation. The robot then
choose the action with the highest probability.

Figure 6: Grid-SA-CADRL network architecture.

We trained the Grid-SA-CADRL network for 1000 episodes, taking the pre-trained
SA-CADRL network as a starting point. In each training episode we randomly
picked a start and goal position for the robot, and a random starting time
from the recordings. The pedestrians in the simulation walk according to the
recording data, and an episode ends as soon as the robot has reached the goal
position, or the recording has reached its end.

##  V Evaluation

We first present the evaluation of the prediction method accuracy, followed by
the evaluation of the path planning approaches.

###  V-A Evaluation of prediction methods

We chose the Normalized Mean Square Error (NMSE, see equation below) to
evaluate the pedestrian position prediction methods presented in Section IV,
and optimize the hyper-parameters.

$NMSE=\frac{1}{n}\sum^{n}_{i=1}{\sqrt{(\frac{x_{i}-\hat{x_{i}}}{X})^{2}+(\frac{y_{i}-\hat{y_{i}}}{Y})^{2}}}$
(2)

Here $x_{i}$ and $y_{i}$ are ground truth values, $x_{i}$ and $y_{i}$ are
predicted. $X$ and $Y$ are upper limits for $x_{i}$ and $y_{i}$ respectively,
and in our case set to $X=1920$ and $Y=1080$.

We randomly split the data set of pedestrian trajectories into 80% training,
10% validation, and 10% test data. For Random Forest Regression, the NMSE was
0.27, and Behavioral CNN significantly lowered this to 0.02. To show how this
accuracy difference impacts on the performance of the robot, we used both
prediction methods in the D* lite path planning algorithm.

###  V-B Evaluation of path planning

All path planning algorithms were evaluated in terms of pedestrian safety and
time taken to move from the start to the goal position. These metrics were
measured as follows:

  * •

Number of collisions (i.e. states in which the robot ends up in the same cell
as a pedestrian) which were split into 3 groups as shown in Figure 8. These
represent a robot colliding a stationary pedestrian, a pedestrian colliding
with a stationary robot, and a moving robot and a moving pedestrian colliding
with each other.

  * •

Relative delay compared to the time of the optimal route with no pedestrians
present, as computed by $D=(\frac{\hat{t}}{t}-1)\times 100\%$, where $t$ is
the time taken to move along the optimal route from start to goal, and
$\hat{t}$ is the actual time taken.

In order to evaluate the path planning approaches, we generated 1000 episodes
with random start and goal positions for the robot, and random starting times
of the recording data.

While the SA-CADRL agent moves in a continuous coordinate space, this movement
needed to be transformed into the grid-space model of the environment. Figure
7 shows an example of this mapping. In the first time step the SA-CADRL agent
moves one step to the north, and then moves diagonally to the north east. In
the third step, however, the agent moves, but does not leave the current grid
cell. This means that the agent was stationary for one time step in the grid-
space representation. Note that the preferred speed of the SA-CADRL agent was
set to match the grid cell size.

Figure 7: Mapping of vector space movement of SA-CADRL agent to the grid-space
representation.

Note that the pedestrian movement was recorded without a robot being present,
which limits the realism of the simulation to some extent since it ignores the
effect that robot movements could influence pedestrian decisions. However, the
definition of the three different collision types has been specifically
designed to take this limitation into account. Specifically, a moving robot
colliding with a moving pedestrian is always an indicator for sub-optimal
robot behaviour, no matter whether the pedestrian movement was influenced by
the robot or not.

##  VI Results

The results of our evaluation are shown in table I. While the baseline
prediction approach may appear to be the most cautious one by assuming that a
pedestrian is going to occupy all neighboring squares in the next time step,
it caused the most collisions. There were due to some pedestrians moving
faster than one square per time step. Increasing the range of squares that are
assumed to be occupied to two does not improve the solution, since this causes
the robot to stop moving at all. This confirms that a simple over-cautious
solution is not feasible in a crowded space. In terms of movement speed, all
other techniques demonstrate comparable performance.

The D* Lite method does not cause any SR collisions because it never is
stationary. While the RL approaches do cause a relatively large number of
these collisions, these are not indicative of poor robot behaviour, since
pedestrians in a real-world situation are unlikely to collide with a
stationary robot, and if they do it would not be the robot’s fault. SP
collisions do not occur for any of the approaches.

Most importantly, MRP collisions are relatively frequent for the D* Lite
methods, corresponding to the accuracy of the chosen prediction method. The RL
methods clearly outperform D* Lite on this measure, and our adaptation of SA-
CADRL is more than halving the count of MRP collisions to a relatively small
number of 3 in 1000. This clearly shows the superiority of Grid-SA-CADRL in
this environment.

Figure 8: The three types of collisions between the robot (red square) and a pedestrian (blue square). TABLE I: Evaluation of different approaches to robots navigation in crowded environment (taken over 1000 paths) Approach |  delay | collisions
---|---|---
SR |  SP |  MRP
D* Lite based approach
Perfect prediction (0.0) |  $1.44\%$ |  0 |  0 |  0
Baseline |  $7.57\%$ |  0 |  0 |  143
RFR (0.27) |  $1.39\%$ |  0 |  0 |  67
Behavior-CNN (0.02) |  $1.42\%$ |  0 |  0 |  25
Deep Reinforcement Learning approach
SA-CADRL (checkpoint) |  $1.57\%$ |  183 |  0 |  8
Grid-SA-CADRL |  $1.46\%$ |  51 |  0 |  3

##  VII Conclusions

In this paper, we have presented a comparative evaluation of various path
planning methods for robot navigation through crowds. The evaluation was
carried out in a simulation based on real-world recorded pedestrian data in a
crowded space. The results show the clear superiority of reinforcement
learning in terms of number of collisions with moving pedestrians, while also
never colliding with stationary pedestrians.

## References

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# Symmedians as Hyperbolic Barycenters

Maxim Arnold and Carlos E. Arreche  Department of Mathematical Sciences, The
University of Texas at Dallas, Richardson, TX 75080
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The symmedian point of a triangle enjoys several geometric and optimality
properties, which also serve to define it. We develop a new dynamical
coordinatization of the symmedian, which naturally generalizes to other ideal
hyperbolic polygons beyond triangles. We prove that in general this point
still satisfies analogous geometric and optimality properties to those of the
symmedian, making it into a hyperbolic barycenter. We initiate a study of
moduli spaces of ideal polygons with fixed hyperbolic barycenter, and of some
additional optimality properties of this point for harmonic (and sufficiently
regular) ideal polygons.

##  1\. Introduction

The _symmedian point_ of a triangle (also called its _Lemoine point_ or its
_Grebe point_) has been aptly called “one of the crown jewels of modern
geometry” in [13, §7.1]. Let us recall one way to construct it (cf. [13,
§7.4(iii)]). Let $\mathbf{O}$ denote the circumcircle of $\Delta(ABC)$. Denote
by $A^{*}$ the intersection of the tangents to $\mathbf{O}$ at the vertices
$B$ and $C$, and define $B^{*}$ and $C^{*}$ analogously. Then the lines
$AA^{*}$, $BB^{*}$ and $CC^{*}$ are concurrent at the _symmedian point_ $S$ of
the triangle $ABC$.

Figure 1. Construction of the symmedian point $S$ of the triangle
$\Delta(ABC)$.

Among very many other properties enjoyed by the symmedian point $S$, there is
the following optimization property, which is included as Exercise 7.3 in
[13], and has been known (and reproved multiple times) since at least 1804
(cf. [15, p. 94]).

######  Theorem 1.1.

The symmedian point $S$ of a triangle minimizes the sum of the squares of the
distances from $S$ to the sides of the triangle.

This investigation originated from the humble observation that the above
construction of the symmedian point of a triangle (usually proved as a theorem
rather than used as a definition) makes it into the orthocenter of the ideal
hyperbolic triangle with “the same” vertices, and how strange it seems that
“the same” point should be meaningful from both the Euclidean and hyperbolic
vantage points simultaneously. We shall show in Theorem 7.5 that this
classical fact is only the first instance of a general property shared by all
harmonic polygons beyond triangles.

The main goal of this short note is to introduce a generalization of the
symmedian point $S$ of a triangle to inscribed polygons (or what is “the
same”, ideal hyperbolic polygons), which we call the _hyperbolic barycenter_ ,
and initiate a study of its properties. This point is uniquely defined in
three different ways: by explicit coordinates (3.2), by an optimality property
(see Theorem 3.2), and by a collection of geometric constructions (see §5),
all directly analogous to those identifying the symmedian point of a triangle.
We have the sense of having opened some long-forgotten Pandora’s box, and we
have now more questions than answers. We have striven to make this work as
accessible as possible so that other researchers (including students!) may
better address the questions that we leave unanswered. We hope the more
sophisticated reader will not begrudge us our choice to keep the exposition,
constructions, and proofs as elementary as we knew how.

Let us briefly describe the contents of this work in some more detail. In §2,
after recalling some basic facts from hyperbolic geometry and fixing some
conventions, we begin to refer to the _Kleinization_ of an inscribed Euclidean
polygon, which is the ideal hyperbolic polygon obtained by identifying the
open unit disc with the hyperbolic plane (in the Klein-Beltrami model), and to
the opposite process of _deKleinization_ of hyperpolic geometric objects into
Euclidean ones, and show in Theorem 2.1 the first of several hyperbolic
minimality properties of the Kleinization of the symmedian point $S$ of a
Euclidean triangle. In §3 we introduce a dynamical coordinatization of this
$S$, in terms of certain Hamiltonians introduced in [4] for the study of
cross-ratio dynamics on ideal polygons, and use it to prove another minimality
property in Theorem 3.2, which makes $S$ into a hyperbolic barycenter. In §4
we introduce the hyperbolic barycenters of general ideal hyperbolic polygons,
and prove several interesting and useful properties that, in particular, allow
for their several concurrent geometric (straightedge and compass)
constructions. In §5 we illustrate some of these geometric constructions for
the first few ideal hyperbolic $n$-gons beyond triangles, with $n=4,6,10,5$.
In §6 we initiate the study of the moduli spaces of ideal hyperbolic $n$-gons
with a fixed common hyperbolic barycenter, and describe it explicitly for the
smallest values of $n=3,4$ in terms of Poncelet conics. In §7 we show that the
classical Theorem 1.1 is the first instance of a more general Theorem 7.5 that
holds uniformly for all _harmonic_ ideal hyperbolic polygons. We prove a
partial converse in Theorem 7.6, that the deKleinization of the hyperbolic
barycenter of an ideal quadrilateral $\mathbf{P}$ minimizes the sum of squares
of distances to the sides if and only if either $\mathbf{P}$ is harmonic or
its deKleinization is a rectangle.

##  2\. Elements of hyperbolic geometry

In the _Klein-Beltrami model_ , the hyperbolic plane $\mathbb{H}$ is
identified with the interior of the unit disc in $\mathbb{R}^{2}$. The unit
circle boundary of this disc, called the _absolute_ , represents “ideal
points” at infinite hyperbolic distance from any given point in $\mathbb{H}$.
This absolute can be naturally identified with the real projective line
$\mathbb{RP}^{1}$ (see e.g. [8]), by associating with $p\in\mathbb{RP}^{1}$
the point

$P=\left(\frac{1-p^{2}}{1+p^{2}},\frac{2p}{1+p^{2}}\right)$ (2.1)

on the unit circle. Writing $p=\tan(\varphi/2)$ yields the usual
parametrization $P=(\cos\varphi,\sin\varphi)$. The hyperbolic distance between
two points $X,Y\in\mathbb{H}$ is defined by (see e.g. [6, §19.2.5])

$d_{\mathcal{K}}(X,Y)=\cosh^{-1}{\dfrac{1-X\cdot
Y}{\sqrt{(1-|X|^{2})(1-|Y|^{2}|)}}}$ (2.2)

where $\cosh(z):=(e^{z}+e^{-z})/2$ denotes the hyperbolic cosine, $X\cdot Y$
stands for the usual inner product of two vectors in $\mathbb{R}^{2}$, and
$|X|^{2}:=X\cdot X$. Geodesics are represented by straight lines, but the
hyperbolic angles between geodesics do not in general agree with the
corresponding Euclidean angles. Writing $\xi_{1}$ for the geodesic that
intersects the absolute at
$P_{\ell}=\left(\frac{1-p_{\ell}^{2}}{1+p_{\ell}^{2}},\frac{2p_{\ell}}{1+p_{\ell}^{2}}\right)$
for $\ell=1,2$ as in (2.1), the hyperbolic distance from
$X=(x,y)\in\mathbb{H}$ to $\xi_{1}$ is computed by

$d_{\mathcal{K}}(X,\xi_{1})=\cosh^{-1}\sqrt{\frac{\left(x(1-p_{1}p_{2})+y(p_{1}+p_{2})-(1+p_{1}p_{2})\right)^{2}}{(p_{1}-p_{2})^{2}(1-x^{2}-y^{2})}+1}.$
(2.3)

The construction of the symmedian point of a triangle, described in Figure 1,
is particularly suggestive to the hyperbolic geometer. As mentioned in the
introduction, if we interpret the circle $\mathbf{O}$ as the absolute in the
Klein-Beltrami model, we can interpret the Euclidean triangle $\Delta(ABC)$ as
an ideal hyperbolic triangle $\Delta_{\mathcal{K}}(ABC)$ in $\mathbb{H}$, with
“the same” vertices. We call $\Delta_{\mathcal{K}}(ABC)$ the Kleinization of
$\Delta(ABC)$, and $\Delta(ABC)$ the deKleinization of
$\Delta_{\mathcal{K}}(ABC)$. We emphasize that the idea of “Kleinizing”
Euclidean geometric objects is not new, see e.g. [2, 9].

The point $A^{*}$ is called the _polar_ to the line $BC$ with respect to
$\mathbf{O}$, and any geodesic through $A^{*}$ intersects $BC$ orthogonally in
$\mathbb{H}$. Therefore, the symmedian point of $\Delta(ABC)$ is the point of
intersection of the hyperbolic altitudes of $\Delta_{\mathcal{K}}(ABC)$, i.e.,
its hyperbolic orthocenter. This simple observation readily provides another
minimality property of the symmedian point $S$, as follows. For any Euclidean
triangle $\Delta(ABC)$ and a given point $X$ in the plane, consider its
_Cevian triangle_ $\Delta(A^{\prime}B^{\prime}C^{\prime})$, where $A^{\prime}$
is the intersection of the line $AX$ with the line $BC$, and the points
$B^{\prime}$ and $C^{\prime}$ are defined similarly.

######  Theorem 2.1.

The symmedian point of a Euclidean triangle minimizes the hyperbolic perimeter
of the Kleinization of its Cevian triangle.

Figure 2. Another minimality property of the Symmedian point, from both the
Klein-Beltrami (left) and the Poincaré (right) points of view.

###### Proof.

This follows from the classic reflection argument about Fargano’s orbit: under
sequential hyperbolic reflections of the triangle $ABC$ in its sides, the
sides of the triangle $A^{\prime}B^{\prime}C^{\prime}$ form a geodesic
interval (see Figure 2). ∎

##  3\. Coordinates and hyperbolic minimality property of the symmedian

For an ideal polygon $\mathbf{P}=(p_{1},\dots p_{n})$ the following three
quantities were introduced in [4, §6.2], and shown to be Hamiltonians for the
infinitesimal diagonal action of the Möbius group.

$I_{\mathbf{P}}=\sum_{\ell=1}^{n}\dfrac{1}{p_{\ell}-p_{\ell+1}},\qquad
J_{\mathbf{P}}=\frac{1}{2}\sum_{\ell=1}^{n}\dfrac{p_{\ell}+p_{\ell+1}}{p_{\ell}-p_{\ell+1}},\qquad
K_{\mathbf{P}}=\sum_{\ell=1}^{n}\dfrac{p_{\ell}p_{\ell+1}}{p_{\ell}-p_{\ell+1}}$
(3.1)

where indices are understood modulo $n$, that is, with $p_{n+1}:=p_{1}$.

######  Theorem 3.1.

The symmedian point $S_{\mathbf{P}}$ of a triangle
$\mathbf{P}=(p_{1},p_{2},p_{3})$ inscribed in the unit circle is

$S_{\mathbf{P}}=\left(\dfrac{I_{\mathbf{P}}-K_{\mathbf{P}}}{I_{\mathbf{P}}+K_{\mathbf{P}}},\dfrac{2J_{\mathbf{P}}}{I_{\mathbf{P}}+K_{\mathbf{P}}}\right).$
(3.2)

###### Proof.

This is an elementary computation. The polar
$P_{\ell}^{*}=(a_{\ell}^{*},b_{\ell}^{*})$ to the side $\xi_{\ell}$ is given
by

$P_{\ell}^{*}=(a_{\ell}^{*},b_{\ell}^{*})=\left(\dfrac{1-p_{\ell}p_{\ell+1}}{1+p_{\ell}p_{\ell+1}},\dfrac{p_{\ell}+p_{\ell+1}}{1+p_{\ell}p_{\ell+1}}\right).$
(3.3)

Writing $P_{\ell}=(a_{\ell},b_{\ell})$ as in (2.1) and
$S_{\mathbf{P}}=(\alpha,\beta)$ as in (3.2), we compute from (3.3) that

$\frac{\beta-b_{\ell-1}^{*}}{\alpha-
a_{\ell-1}^{*}}=\frac{b_{\ell-1}^{*}-b_{\ell}^{*}}{a_{\ell-1}^{*}-a_{\ell}^{*}}=\frac{b_{\ell}^{*}-\beta}{a_{\ell}^{*}-\alpha}.$

Hence $S_{\mathbf{P}}$ as given in (3.2) is the common intersection of the (at
least two) lines $P_{\ell-1}P_{\ell}^{*}$ such that
$p_{\ell}p_{\ell+1}\neq-1$, so it is the symmedian of $\mathbf{P}$.∎

We can now show a second minimality property of the Kleinization of
$S_{\mathbf{P}}$, which makes it into a kind of “hyperbolic barycenter”.

######  Theorem 3.2.

The point $S=S_{\mathbf{P}}$ in (3.2) minimizes the sum of the hyperbolic
sines of the hyperbolic distances from $S$ to the sides of the Kleinization of
$\,\mathbf{P}$.

###### Proof.

Using (2.3) together with the identity
$\cosh^{-1}(t)=\sinh^{-1}(\sqrt{t^{2}-1})$ yields, for the hyperbolic distance
$d_{\mathcal{K}}(X,\xi_{\ell})$ from a point $X=(x,y)$ to the side
$\xi_{\ell}=P_{\ell}P_{\ell+1}=\xi(p_{\ell},p_{\ell+1})$ (where we once again
consider indices modulo $n$ as in (3.1)), that

$q_{\ell}(X):=\sinh\bigl{(}d_{\mathcal{K}}(X,\xi_{\ell})\bigr{)}=\dfrac{((I_{\ell}-K_{\ell})x+2J_{\ell}y-(I_{\ell}+K_{\ell}))}{\sqrt{1-(x^{2}+y^{2})}}$
(3.4)

where $I_{\ell}:=\dfrac{1}{p_{\ell}-p_{\ell+1}}$;
$2J_{\ell}:=\dfrac{p_{\ell}+p_{\ell+1}}{p_{\ell}-p_{\ell+1}}$; and
$K_{\ell}:=\dfrac{p_{\ell}p_{\ell+1}}{p_{\ell}-p_{\ell+1}}$; so that

$\textstyle I:=\sum_{\ell}I_{\ell},\qquad J:=\sum_{\ell}J_{\ell},\qquad
K:=\sum_{\ell}K_{\ell}$ (3.5)

coincide with the expressions in (3.1). Writing
$\mathcal{Q}(X):=\sum_{\ell}q_{\ell}(X)$, we compute the gradient

$\nabla(\mathcal{Q}(X))=(1-|X|^{2})^{-\frac{3}{2}}\begin{pmatrix}\Bigl{(}(I-K)y-(2J)x\Bigr{)}y+(I+K)x-(I-K)\vphantom{\displaystyle\sum_{i}}\\\
\Bigl{(}(2J)x-(I-K)y\Bigr{)}x+(I+K)y-(2J)\end{pmatrix}.$ (3.6)

We claim that the only point $X$ with $|X|\neq 1$ that nullifies the
polynomial part of the gradient
$(1-|X|^{2})^{\frac{3}{2}}\nabla(\mathcal{Q}(X))$ is $X=S$. Denoting by
$W^{\perp}=(-z,w)$ the counterclockwise rotation of $W=(w,z)$ by $\pi/2$,
setting $\nabla(\mathcal{Q}(X))=0$ in (3.6) is equivalent to

$-(X\cdot S^{\perp})X^{\perp}+X-S=(0,0).$ (3.7)

For $X$ any solution to (3.7), taking the dot product of (3.7) with $X$ yields
$|X|^{2}=S\cdot X$, which implies that either $X=(0,0)$ (in which case
$S=(0,0)=X$, by (3.7), as desired), or else $S=X+sX^{\perp}$ for some
$s\in\mathbb{R}$, and therefore $S^{\perp}=X^{\perp}-sX$. Then (3.7) becomes
$s(|X|^{2}-1)X^{\perp}=(0,0).$ This further implies that either (i) $s=0$; or
(ii) $|X|^{2}=1$; or (iii) $X^{\perp}=(0,0)$. In case (i) we find that $X=S$;
case (ii) is forbidden; and in case (iii) we again find that $X=(0,0)=S$. Thus
$X=S$ is the only point not in $\mathbf{O}$ satisfying (3.7). To conclude,
observe that the Hessian determinant of $\mathcal{Q}(X)$ at $S$ is
$\frac{(I+K)^{2}}{(1-|S|^{2})^{2}}>0.$∎

##  4\. Hyperbolic barycenters of ideal polygons

The statement and the proof of Theorem 3.2 both generalize immediately to
ideal hyperbolic polygons $\mathbf{P}$ beyond triangles. Since
$S_{\mathbf{P}}$ in (3.2) minimizes the sum of the hyperbolic sines of the
hyperbolic distances to the sides of $\mathbf{P}$, we find it natural to call
it the _hyperbolic barycenter_ of $\mathbf{P}$.

######  Remark 4.1.

It is easy to see that $I_{\mathbf{P}}+K_{\mathbf{P}}\neq 0$, not just
generically, but always when $\mathbf{P}$ is convex. Under the
parameterization of §2, each
$I_{\ell}+K_{\ell}=\frac{1+p_{\ell}p_{\ell+1}}{p_{\ell}-p_{\ell+1}}=\cot\left(\frac{\varphi_{\ell}-\varphi_{\ell+1}}{2}\right)$.
At most one $I_{\ell}+K_{\ell}$ has different sign from the others (only in
case $|\varphi_{\ell}-\varphi_{\ell+1}|\geq\pi$). Even in this exceptional
case, since
$|\varphi_{\ell^{\prime}}-\varphi_{\ell^{\prime}+1}|<2\pi-|\varphi_{\ell}-\varphi_{\ell+1}|$
for any other $\ell^{\prime}\neq\ell$, we obtain immediately that
$|I_{\ell^{\prime}}+K_{\ell^{\prime}}|>|I_{\ell}+K_{\ell}|$.

For $\Xi=\\{\xi_{\ell_{1}},\dots,\xi_{\ell_{k}}\\}$ a non-empty set of (not
necessarily contiguous) sides of $\mathbf{P}$, define

$I_{\Xi}:=\sum_{\xi_{\ell}\in\Xi}I_{\ell};\qquad
J_{\Xi}:=\sum_{\xi_{\ell}\in\Xi}J_{\ell};\qquad
K_{\Xi}:=\sum_{\xi_{\ell}\in\Xi}K_{\ell}$

analogously as in (3.5). We similarly let, with
$q_{\ell}(X)=\sinh{\bigl{(}d_{\mathcal{K}}(X,\xi_{\ell})\bigr{)}}$ as in
(3.4),

$\mathcal{Q}_{\Xi}(X)=\sum_{\xi_{\ell}\in\Xi}q_{\ell}(X)\qquad\text{and}\qquad
S_{\Xi}=\left(\frac{I_{\Xi}-K_{\Xi}}{I_{\Xi}+K_{\Xi}},\frac{2J_{\Xi}}{I_{\Xi}+K_{\Xi}}\right).$
(4.1)

In case $\Xi$ is the set of all sides of $\mathbf{P}$, we still simply write
$S_{\mathbf{P}}$ (as in (3.2)) instead of $S_{\Xi}$,
$\mathcal{Q}_{\mathbf{P}}$ instead of $\mathcal{Q}_{\Xi}$, etc. The
computations in the proof of Theorem 3.2 immediately yield the following.

######  Corollary 4.2.

Let $\Xi=\\{\xi_{\ell_{1}},\dots,\xi_{\ell_{k}}\\}$ be a non-empty set of
sides. Then $S_{\Xi}$ nullifies the polynomial part of the gradient
$(1-|X|^{2})^{\frac{3}{2}}\nabla(\mathcal{Q}_{\Xi})$ as in (3.6).

######  Remark 4.3.

The point $S_{\Xi}$ need not be in $\mathbb{H}$. For example, $S_{\\{\xi\\}}$
is the polar to $\xi$, which is outside of the unit disk, and
$S_{\\{\xi_{1},\xi_{2}\\}}=P_{2}$, the common ideal vertex of the sides
$\xi_{1}$ and $\xi_{2}$, which lies on $\mathbf{O}$.

We set aside a more systematic discussion of the geometric meaning of
$S_{\Xi}$ for arbitrary $\Xi$, in favor of focusing on the most immediately
relevant configuration properties of the points $S_{\Xi}$ for different
interrelated choices of $\Xi$.

######  Lemma 4.4 (Interpolation Lemma).

Let $\bm{\Xi}=\\{\Xi_{1},\dots,\Xi_{\nu}\\}$, where the $\Xi_{j}$ are (not
necessarily disjoint) sets of sides of $\mathbf{P}$, such that
$\sum_{j=1}^{\nu}(I_{\Xi_{j}}+K_{\Xi_{j}})\neq 0$. Denote
$\mathcal{Q}_{\bm{\Xi}}(X):=\sum_{j=1}^{\nu}\mathcal{Q}_{\Xi_{j}}(X)$ and

$S_{\bm{\Xi}}:=\left(\sum_{j=1}^{\nu}(I_{\Xi_{j}}+K_{\Xi_{j}})\right)^{-1}\sum_{j=1}^{\nu}(I_{\Xi_{j}}+K_{\Xi_{j}})\cdot
S_{\Xi_{j}}.$ (4.2)

Then $S_{\bm{\Xi}}$ nullifies the polynomial part of the gradient
$(1-|X|^{2})^{\frac{3}{2}}\nabla(\mathcal{Q}_{\bm{\Xi}})$.

###### Proof.

As in the proof of Theorem 3.2, we see that $X$ nullifying the polynomial part
of the gradient $(1-|X|^{2})^{\frac{3}{2}}\nabla(\mathcal{Q}_{\bm{\Xi}})$ is
equivalent to $X$ satisfying (3.7) with $S=S_{\bm{\Xi}}$ as in (4.2). ∎

######  Corollary 4.5.

The points $S_{\\{\Xi_{1},\Xi_{2}\\}}$, $S_{\Xi_{1}}$, and $S_{\Xi_{2}}$ are
collinear. Moreover, if $\mathbf{P}$ is convex and each $|\Xi_{j}|\geq 2$ then
$S_{\bm{\Xi}}$ is in the convex hull of the $S_{\Xi_{j}}$.

###### Proof.

In (4.2), $S_{\bm{\Xi}}$ is a weighted sum of the $S_{\Xi_{j}}$ with weights
summing to $1$. Moreover, if $\mathbf{P}$ is convex and each $|\Xi_{j}|\geq 2$
then the $I_{\Xi_{j}}+K_{\Xi_{j}}$ all have the same sign (cf. Remark 4.1), so
these weights are positive. ∎

Corollary 4.5 is one of the main tools that allows for the geometric
construction of hyperbolic barycenters. It is educational to see it in action
in some trivial cases.

For any ideal $n$-gon $\mathbf{P}$, we can take
$\bm{\Xi}=\\{\Xi_{1},\dots,\Xi_{n}\\}$, where
$\Xi_{\ell}=\\{\xi_{\ell-1},\xi_{\ell}\\}$, the two sides that meet at the
ideal vertex $P_{\ell}$. Then
$\mathcal{Q}_{\bm{\Xi}}(X)=2\mathcal{Q}_{\mathbf{P}}(X)$ and each
$S_{\Xi_{\ell}}=P_{\ell}$ (cf. Remark 4.3). Thus if $\mathbf{P}$ is convex
then and $S_{\bm{\Xi}}=S_{\mathbf{P}}$ is not only in $\mathbb{H}$, but
actually in the interior of $\mathbf{P}$ by Corollary 4.5, which is neither
surprising nor immediately apparent from the algebraic expression (3.2).

In the tired case of triangles $\mathbf{P}=(p_{1},p_{2},p_{3})$, letting
$\Xi_{1}=\\{\xi_{\ell}\\}$ and $\Xi_{2}=\\{\xi_{\ell-1},\xi_{\ell+12}\\}$, we
have
$\mathcal{Q}_{\mathbf{P}}(X)=\mathcal{Q}_{\Xi_{1}}(X)+\mathcal{Q}_{\Xi_{2}}(X)$,
$S_{\Xi_{1}}$ is the polar $P_{\ell}^{*}$ to $\xi_{\ell}$, and
$S_{\Xi_{2}}=P_{\ell-1}$ (cf. Remark 4.3). Thus Corollary 4.5 says in this
case that $S_{\mathbf{P}}=S_{\\{\Xi_{1},\Xi_{2}\\}}$ lies on
$P_{\ell+2}P_{\ell}^{*}$, as expected. A first non-trivial consequence of
Corollary 4.5 is that this argument works all the same even if the first and
third sides do not meet to form a triangle, as formalized in the following
result.

######  Lemma 4.6.

Let $\Xi=\\{\xi_{\ell-1},\xi_{\ell},\xi_{\ell+1}\\}$ be any set of three
consecutive sides. Then $S_{\Xi}$ is the intersection of the two hyperbolic
altitudes, from $P_{\ell}$ to $\xi_{\ell+1}$, and from $P_{\ell+1}$ to
$\xi_{\ell-1}$.

###### Proof.

Partition $\Xi$ in two different ways: $\bm{\Xi}_{1}=\\{\Xi_{11},\Xi_{12}\\}$
and $\bm{\Xi}_{2}=\\{\Xi_{21},\Xi_{22}\\}$, where
$\Xi_{11}=\\{\xi_{\ell-1}\\}$ and $\Xi_{12}=\\{\xi_{\ell},\xi_{\ell+1}\\}$,
and $\Xi_{21}=\\{\xi_{\ell+1}\\}$ and
$\Xi_{22}=\\{\xi_{\ell-1},\xi_{\ell}\\}$. Then
$S_{\bm{\Xi}_{1}}=S_{\Xi}=S_{\bm{\Xi}_{2}}$ by Lemma 4.4, whence $S_{\Xi}$
lies on both geodesic segments $S_{\Xi_{11}}S_{\Xi_{12}}$ and
$S_{\Xi_{21}}S_{\Xi_{22}}$ by Corollary 4.5. To conclude, we observe as in
Remark 4.3 that $S_{\Xi_{11}}=P_{\ell-1}^{*}$ and $S_{\Xi_{12}}=P_{\ell+1}$,
and $S_{\Xi_{21}}=P_{\ell+1}^{*}$ and $S_{\Xi_{22}}=P_{\ell}$. ∎

######  Remark 4.7.

Lemma 4.6 is the third in a recursive series of constructions initiated in
Remark 4.3. For $\Xi=\\{\xi_{0},\dots,\xi_{\ell}\\}$ a set of $\ell+1$
consecutive sides, we find that $S_{\Xi}$ is the intersection of
$P_{0}^{*}S_{\Xi_{0}}$ and $S_{\Xi_{\ell}}P_{\ell}^{*}$, where
$\Xi_{m}:=\Xi-\\{\xi_{m}\\}$. More generally, the geodesic segments
$S_{\\{\xi_{0},\dots,\xi_{m-1}\\}}S_{\\{\xi_{m},\dots,\xi_{\ell}\\}}$ for
$m=1,\dots,\ell$ are all concurrent at $S_{\Xi}$.

As in [4, §2.1], we say that two ideal $n$-gons $\mathbf{P}$ and $\mathbf{Q}$
are $\alpha$_-related_ if the cross-ratio
$[p_{\ell},q_{\ell},p_{\ell+1},q_{\ell+1}]=\alpha$ for every $\ell$, with the
convention that the polygons have mutually orthogonal sides when $\alpha=-1$.
Note that $\mathbf{P}$ and $\mathbf{Q}$ are $\alpha$-related (for some
$\alpha$) if and only if the hyperbolic angles between the corresponding sides
of $\mathbf{P}$ and $\mathbf{Q}$ are all equal.

######  Theorem 4.8.

If two ideal polygons $\mathbf{P}$ and $\mathbf{Q}$ are $\alpha$-related for
some $\alpha\in\mathbb{R}$, then $S_{\mathbf{P}}=S_{\mathbf{Q}}$.

###### Proof.

By [4, Thm. 16], the values of $I,J,K$ in (3.1) coincide for $\alpha$-related
polygons. ∎

######  Remark 4.9.

Theorem 4.8 can also be seen in [4] from a complementary point of view. The
barycenter of an ideal hyperbolic polygon is _defined_ in [4, §6.2.1] to be
the limit as $\varepsilon\rightarrow 0$ of the fixed point of the composition
$L_{1+\varepsilon}(\mathbf{P})$ of infinitesimal rotations along the sides of
the polygon $\mathbf{P}$ – let us temporarily denote this point
$\tilde{S}_{\mathbf{P}}$. As explained in [4, Rmk. 6.9], the Möbius invariance
of $IK-J^{2}$ (cf. [4, Lem. 6.6]) implies
$\tilde{S}_{\mathbf{P}}=\tilde{S}_{\mathbf{Q}}$ for $\alpha$-related
$\mathbf{P}$ and $\mathbf{Q}$. Then in [4, Lem. 6.10] it is shown that
$\tilde{S}_{\mathbf{P}}$ minimizes $\mathcal{Q}_{\mathbf{P}}(X)$ given in
(4.1), thus _a posteriori_ $\tilde{S}_{\mathbf{P}}=S_{\mathbf{P}}$ as in
(3.2), by Theorem 3.2.

In the Euclidean setting, there is an analogous notion to the cross-ratio
dynamics considered in [4] called _bicycle correspondence_ , which also has a
fixed point called _circumcenter of mass_ (cf. [18, §1, Thm. 5, Rmk. 3.3]).
The next property of the hyperbolic barycenter $S_{\mathbf{P}}$ resembles the
similar Archimedian property of the Euclidean circumcenter of mass [19, Thm.
1].

######  Lemma 4.10 (Archimedean property).

If the ideal polygon $\mathbf{P}$ is partitioned along any of its diagonals
into ideal polygons $\mathbf{Q}_{1}$ and $\mathbf{Q}_{2}$ , then

$S_{\mathbf{P}}=\frac{I_{\mathbf{Q}_{1}}+K_{\mathbf{Q}_{1}}}{I_{\mathbf{P}}+K_{\mathbf{P}}}S_{\mathbf{Q}_{1}}+\frac{I_{\mathbf{Q}_{2}}+K_{\mathbf{Q}_{2}}}{I_{\mathbf{P}}+K_{\mathbf{P}}}S_{\mathbf{Q}_{2}}.$
(4.3)

###### Proof.

Indeed, let $\mathbf{Q}_{1}=(p_{1},\dots,p_{t})$ and
$\mathbf{Q}_{2}=(p_{1},p_{t},\dots,p_{n})$. One has

$I_{\mathbf{P}}=\left(\sum\limits_{\ell=1}^{t-1}\dfrac{1}{p_{\ell}-p_{\ell+1}}+\dfrac{1}{p_{t}-p_{1}}\right)+\left(\dfrac{1}{p_{1}-p_{t}}+\sum\limits_{\ell=t}^{n}\dfrac{1}{p_{\ell}-p_{\ell+1}}\right)=I_{\mathbf{Q}_{1}}+I_{\mathbf{Q}_{2}}.$

Similarly, $J_{\mathbf{P}}=J_{\mathbf{Q}_{1}}+J_{\mathbf{Q}_{2}}$ and
$K_{\mathbf{P}}=K_{\mathbf{Q}_{1}}+K_{\mathbf{Q}_{2}}$, and (4.3) follows from
(3.2). ∎

######  Corollary 4.11.

The points $S_{\mathbf{P}}$, $S_{\mathbf{Q}_{1}}$, and $S_{\mathbf{Q}_{2}}$
are collinear.

###### Proof.

In (4.3), $S_{\mathbf{P}}$ is a weighted sum of $S_{\mathbf{Q}_{1}}$ and
$S_{\mathbf{Q}_{2}}$ with weights summing to $1$. ∎

######  Remark 4.12.

Let us briefly compare the hyperbolic barycenter $S_{\mathbf{P}}$ with other
centers defined by integrable polygonal dynamics. As already mentioned, the
cross-ratio dynamics of [4] form a kind of ideal hyperbolic analogue of the
bicycle correspondence of [18], making $S_{\mathbf{P}}$ analogous to the
circumcenter of mass. Yet another infamous polygonal center is the fixed point
of the pentagram map, which was discovered in [16] and coordinatized in [12].
In [1, §1] it is suggested that this point may be the limit point as
$\varepsilon\rightarrow 0$ of an eigenvector for the infinitesimal monodromy
of a deformation into twisted polygons $\mathbf{P}_{1+\varepsilon}$ of a
closed polygon $\mathbf{P}_{1}$ (cf. Remark 4.9). And yet, it remains unknown
whether the circumcenter of mass or the fixed point of the pentagram map is
defined by an optimality property analogous to that of the hyperbolic
barycenter in Theorem 3.2. We refer to [14] for a systematic discussion of
polygonal centers defined by optimality conditions.

##  5\. Case studies: small-gons

The Interpolation Lemma and the Archimedian Property (and their corollaries)
established in §4 are especially useful for geometrically constructing the
hyperbolic barycenters of ideal hyperbolic $n$-gons recursively . As we have
already extensively addressed the particular case of triangles, let us now
illustrate these constructions in the next few particular cases of
$n=4,5,6,10$, where we begin to glimpse a cascade of concurrences culminating
at the hyperbolic barycenter.

###  5.1. Case study: Quadrilaterals

A first construction of the hyperbolic barycenter of an ideal quadrilateral
$\mathbf{P}$ (and in general, of an ideal polygon) is achieved via the
Archimedian Property recursively from the case of triangles (and in general,
from the case of smaller-gons). Let $S_{\ell}$ be the symmedian point of the
triangle $\Delta(P_{\ell-1}P_{\ell}P_{\ell+1})$. Then by Corollary 4.11
$S_{\mathbf{P}}$ lies on $S_{1}S_{3}$ and on $S_{2}S_{4}$ (see Figure 3).

Surprisingly, an alternative construction makes it easier to construct
hyperbolic barycenters of ideal quadrilaterals than those of ideal triangles,
as we see in the next result.

######  Theorem 5.1.

The hyperbolic barycenter of an ideal quadrilateral $\mathbf{P}$ lies on its
diagonals.

###### Proof.

Let $\Xi_{\ell}=\\{\xi_{\ell-1},\xi_{\ell}\\}$ be the set of sides meeting at
the ideal vertex $P_{\ell}$, where as usual we take indices modulo $n=4$. With
notation as in (4.1), we see that each $S_{\Xi_{\ell}}=P_{\ell}$ (cf. Remark
4.3), and

$\mathcal{Q}_{\Xi_{1}}(X)+\mathcal{Q}_{\Xi_{3}}(X)=\mathcal{Q}_{\mathbf{P}}(X)=\mathcal{Q}_{\Xi_{2}}(X)+\mathcal{Q}_{\Xi_{4}}(X).$

By Corollary 4.5,
$S_{\\{\Xi_{1},\Xi_{3}\\}}=S_{\mathbf{P}}=S_{\\{\Xi_{2},\Xi_{4}\\}}$ lies on
$P_{1}P_{3}$ and on $P_{2}P_{4}$.∎

Figure 3. The hyperbolic barycenter of an ideal quadrilateral is the
intersection point of its diagonals as well as the intersection of the
segments connecting symmedians of opposite triangles in each triangulation.

######  Corollary 5.2.

The hyperbolic barycenter of an ideal quadrilateral $\mathbf{P}$ is the
intersection of the diagonals $P_{1}^{*}P_{3}^{*}$ and $P_{2}^{*}P_{4}^{*}$,
of the quadrilateral formed by the polars to the sides.

###### Proof.

This is an immediate consequence of Theorem 5.1 and the well-known
(Euclidean!) fact [3, 11.1.7] that the diagonals of a circumscribed
quadrilateral intersect at the same point as the diagonals of the inscribed
quadrilateral whose vertices are the points of tangency. ∎

The trivial observation in Corollary 5.2 has the non-trivial consequence in
Lemma 5.3 below that the hyperbolic barycenter of an ideal quadrilateral
simultaneously and independently minimizes the sum of hyperbolic sines of
hyperbolic distances to each pair of opposite sides.

######  Lemma 5.3.

For $\mathbf{P}$ a convex ideal hyperbolic quadrilateral with sides
$\xi_{\ell}=P_{\ell}P_{\ell+1}$, let $\,\Xi_{o}=\\{\xi_{1},\xi_{3}\\}$ and
$\Xi_{e}=\\{\xi_{2},\xi_{4}\\}$. Then
$S_{\Xi_{o}}=S_{\mathbf{P}}=S_{\Xi_{e}}$.

###### Proof.

Writing $\bm{\Xi}_{\text{odd}}=\\{\\{\xi_{1}\\},\\{\xi_{3}\\}\\}$ and
$\bm{\Xi}_{\text{even}}=\\{\\{\xi_{2}\\},\\{\xi_{4}\\}\\}$, we conclude from
Lemma 4.4 that $S_{\bm{\Xi}_{\text{odd}}}=S_{\Xi_{o}}$ and
$S_{\bm{\Xi}_{\text{even}}}=S_{\Xi_{e}}$, whence by Corollary 4.5
$S_{\Xi_{o}}$ lies on $P_{1}^{*}P_{3}^{*}$ and $S_{\Xi_{e}}$ lies on
$P_{2}^{*}P_{4}^{*}$ (cf. Remark 4.3). Letting
$\bm{\Xi}_{\text{tot}}=\\{\Xi_{o},\Xi_{e}\\}$, then
$S_{\bm{\Xi}_{\text{tot}}}=S_{\mathbf{P}}$ again by Lemma 4.4, whence again by
Corollary 4.5 $S_{\mathbf{P}}$ lies on the geodesic segment
$S_{\Xi_{o}}S_{\Xi_{e}}$. Since $S_{\mathbf{P}}$ is the intersection of
$P_{1}^{*}P_{3}^{*}$ and $P_{2}^{*}P_{4}^{*}$ by Corollary 5.2, this forces
$S_{\Xi_{o}}$ to also lie on $P_{2}^{*}P_{4}^{*}$ and $S_{\Xi_{e}}$ to also
lie on $P_{1}^{*}P_{3}^{*}$. ∎

Besides the two constructions of the hyperbolic barycenter $S_{\mathbf{P}}$ of
an ideal quadrilateral $\mathbf{P}$, we witness in Figure 3 many additional
concurrences at $S_{\mathbf{P}}$, which we leave unaddressed as exercises for
the interested reader.

###  5.2. Case study: Pentagons

Our first construction of the hyperbolic barycenter of an ideal pentagon
$\mathbf{P}$ is obtained from the Archimedian Property. Let $S_{\ell}$ be the
symmedian point of the triangle $\Delta(P_{\ell-1}P_{\ell}P_{\ell+1})$, and
let $T_{\ell}$ be the intersection of the two diagonals not containing
$P_{\ell}$: $P_{\ell-1}P_{\ell+2}$ and $P_{\ell+1}P_{\ell-2}$. Then by Theorem
5.1, $T_{\ell}$ is the hyperbolic barycenter of the quadrilateral
$P_{\ell-2}P_{\ell-1}P_{\ell+1}P_{\ell+2}$. Thus by Corollary 4.11, the five
geodesic segments $S_{\ell}T_{\ell}$ are concurrent at $S_{\mathbf{P}}$.

On the other hand, we can use the Interpolation Lemma to provide an
alternative construction of the hyperbolic barycenter $S_{\mathbf{P}}$.

######  Lemma 5.4.

Let $\mathbf{P}$ be a convex ideal hyperbolic pentagon. Let $R_{\ell}$ be the
intersection of the two hyperbolic altitudes, from the vertex $P_{\ell+2}$ to
the side $P_{\ell-2}P_{\ell-1}$, and from the vertex $P_{\ell-2}$ to the side
$P_{\ell+1}P_{\ell+2}$, i.e., $R_{\ell}$ lies on $P_{\ell+2}P_{\ell+3}^{*}$
and on $P_{\ell+3}P_{\ell+1}^{*}$. Then the hyperbolic barycenter
$S_{\mathbf{P}}$ lies on the geodesic segment $P_{\ell}R_{\ell}$.

###### Proof.

By Corollary 4.5 applied to $\bm{\Xi}=\\{\Xi_{1},\Xi_{2}\\}$, with
$\Xi_{1}=\\{\xi_{\ell-1},\xi_{\ell}\\}$ and
$\Xi_{2}=\\{\xi_{\ell+1},\xi_{\ell+2},\xi_{\ell+3}\\}$,
$S_{\bm{\Xi}}=S_{\mathbf{P}}$ lies on $S_{\Xi_{1}}S_{\Xi_{2}}$. Now
$S_{\Xi_{1}}=P_{\ell}$ (cf. Remark 4.3), and $S_{\Xi_{2}}=R_{\ell}$ by Lemma
4.6. ∎

We witness in the case of ideal pentagons also, as in the case of ideal
quadrilaterals, the hyperbolic barycenter as the epicenter of several kinds of
concurrences.

Figure 4. Constructions of the hyperbolic barycenter of an ideal pentagon.
Left: via the Archimedian property. Right: via the Interpolation Lemma.

###  5.3. Case study: Hexagons

Surprisingly, it is relatively easier to construct the hyperbolic barycenter
of an ideal hexagon than that of an ideal pentagon. In fact, the construction
of the former arises from one of the variants of the Brianchon Theorem (see
[3, 11.1.9] and Figure 5).

######  Theorem 5.5.

Let $\mathbf{Q}$ be the hexagon in $\mathbb{H}$ whose sides lie on the the
short diagonals of the ideal hexagon $\mathbf{P}$. Then the long diagonals of
$\,\mathbf{Q}$ all intersect at $S_{\mathbf{P}}$.

###### Proof.

For each $\ell$, consider the ideal quadrilateral
$(p_{\ell-1},p_{\ell},p_{\ell+1},p_{\ell+2})$, and let $Q_{\ell}$ be its
hyperbolic barycenter. By Theorem 5.1, $Q_{\ell}$ is the intersection point of
$P_{\ell}P_{\ell+2}$ and $P_{\ell-1}P_{\ell+1}$, and by Corollary 4.10,
$S_{\mathbf{P}}$ lies on the geodesic segment $Q_{\ell}Q_{\ell+3}$.∎

Another construction of the hyperbolic barycenter of an ideal hexagon is
obtained from the Interpolation Lemma, as follows. Let $R_{\ell}$ be the
intersection of $P_{\ell}P_{\ell+1}^{*}$ and $P_{\ell+1}P_{\ell-1}^{*}$. Then
$R_{\ell}=S_{\Xi_{\ell}}$ for
$\Xi_{\ell}:=\\{\xi_{\ell-1},\xi_{\ell},\xi_{\ell+1}\\}$ by Lemma 4.6.
Applying Corollary 4.5 to $\bm{\Xi}=\\{\Xi_{\ell},\Xi_{\ell+3}\\},$ we find
that $S_{\mathbf{P}}$ lies on each geodesic segment $R_{\ell}R_{\ell+3}$.

Figure 5. Constructions of the hyperbolic barycenter of an ideal hexagon.
Left: via the Archimedian property. Right: via the Interpolation Lemma.

###  5.4. Case study: $(2^{m}+2)$-gons

The (initially surprising) relative simplicity in the construction of
hyperbolic barycenters for ideal hexagons versus ideal pentagons is part of a
general phenomenon, which leads to a layered family of interesting
concurrences arising from the Archimedian Property. We can subdivide an ideal
$(2^{m}+2)$-gon $\mathbf{P}$ along any of its $2^{m-1}+1$ long diagonals into
two $(2^{m-1}+2)$-gons $\mathbf{Q}_{1}$ and $\mathbf{Q}_{2}$, and conclude by
Corollary 4.11 that $S_{\mathbf{P}}$ lies on
$S_{\mathbf{Q}_{1}}S_{\mathbf{Q}_{2}}$. This gives a concurrence of
$2^{m-1}+1$ lines, each of whose endpoints is itself a concurrence of
$2^{m-2}+1$ lines, each of whose endpoints is itself a concurrence …, etc.
This recursively reduces the construction of $S_{\mathbf{P}}$ to any one of
the many equivalent constructions of the hyperbolic barycenter of an ideal
quadrilateral in §5.1 (see Figure 3). In Figure 6 we depict the next case
$m=3$ of this construction, to obtain the hyperbolic barycenter of an ideal
decagon as the point of concurrence of the five line segments connecting
opposite hyperbolic barycenters of ideal hexagons constructed according to
Theorem 5.5.

Figure 6. Recursive construction of the hyperbolic barycenter of an ideal
decagon.

##  6\. Moduli of ideal polygons with fixed hyperbolic barycenter

Let us denote by $\mathcal{M}_{n}(S)=\mathcal{M}_{n}(\alpha,\beta)$ the moduli
space of ideal $n$-gons having a fixed $S=(\alpha,\beta)\in\mathbb{H}$ as
their hyperbolic barycenter. We immediately see from the coordinatization of
$S_{\mathbf{P}}$ in (3.2) that

$\begin{cases}I_{\mathbf{P}}-K_{\mathbf{P}}=\alpha\cdot(I_{\mathbf{P}}+K_{\mathbf{P}});\\\
2J_{\mathbf{P}}=\beta\cdot(I_{\mathbf{P}}+K_{\mathbf{P}})\end{cases}$ (6.1)

defines a subspace of codimension at most $2$ in the moduli space
$\mathcal{P}_{n}$ of ideal hyperbolic $n$-gons. We would like to have a more
geometrically meaningful description of $\mathcal{M}_{n}(S)$ than merely the
algebraic conditions in (6.1). Here we accomplish this only for the smallest
values of $n=3,4$.

###  6.1. Moduli of ideal triangles with fixed hyperbolic barycenter

The space $\mathcal{P}_{3}$ of all ideal hyperbolic triangles is three-
dimensional, hence the subspaces $\mathcal{M}_{3}(S)$ are expected to be one-
dimensional. It is clear from the minimality property of $S_{\mathbf{P}}$ in
Theorem 3.2 that all the triangles obtained from $\mathbf{P}$ by a hyperbolic
rotation about $S_{\mathbf{P}}$ share the same hyperbolic barycenter, yielding
a one-dimensional family, as expected. It is easy to see that this is indeed
all of $\mathcal{M}_{3}(S_{\mathbf{P}})$, as we show next.

######  Lemma 6.1.

Two ideal triangles $\mathbf{P}_{1}$ and $\mathbf{P}_{2}$ share the same
hyperbolic barycenter $S_{\mathbf{P}_{1}}=S=S_{\mathbf{P}_{2}}$ if and only if
they are obtained from one another by a hyperbolic rotation about $S$.

###### Proof.

It suffices to show that if an ideal triangle $\mathbf{P}=(p_{1},p_{2},p_{3})$
has $S_{\mathbf{P}}=(0,0)$ then it is regular. This follows from a simple
algebraic computation. If we rotate such a triangle so that one of its
vertices is at $0\in\mathbb{R}\mathbb{P}^{1}$, say $p_{3}=0$, then
$2J_{\mathbf{P}}=0$ if and only if $p_{1}+p_{2}=0$, and
$I_{\mathbf{P}}-K_{\mathbf{P}}=0$ if and only if
$p_{1}^{2}p_{2}^{2}-p_{1}p_{2}-(p_{1}-p_{2})^{2}=0$, which together imply
$\\{p_{1},p_{2}\\}=\\{\pm\sqrt{3}\\}$, as required. ∎

An alternative construction of $\mathcal{M}_{3}(S)$ is as follows. Given an
ideal hyperbolic triangle $\mathbf{P}$, consider the ideal hyperbolic triangle
$\hat{\mathbf{P}}$, whose ideal vertex $\hat{P}_{\ell}$ is the hyperbolic
reflection of $P_{\ell}$ with respect to its opposite side
$\xi_{\ell+1}=P_{\ell+1}P_{\ell+2}$. Then the points $P_{\ell}$,
$\hat{P}_{\ell}$, and $P_{\ell+1}^{*}$ are collinear by construction. Since
$S_{\mathbf{P}}$ is the hyperbolic orthocenter of $\mathbf{P}$, it also lies
on $P_{\ell}P_{\ell+1}^{*}$, and by symmetry (since
$\hat{\hat{\mathbf{P}}}=\mathbf{P}$) we see that $\hat{P}_{\ell+1}^{*}$ is
also on this line. Hence $S_{\mathbf{P}}$ is the hyperbolic orthocenter of
$\hat{\mathbf{P}}$.

######  Lemma 6.2.

The six polars
$P_{1}^{*},P_{2}^{*},P_{3}^{*},\hat{P}_{1}^{*},\hat{P}_{2}^{*},\hat{P}_{3}^{*}$
all lie on the same conic $\Gamma_{\mathbf{P}}$.

###### Proof.

Consider the unique conic $\Gamma$ passing through the first five of these
points, except possibly $\hat{P}_{3}^{*}$. Then we see that
$P_{1}^{*},P_{2}^{*},P_{3}^{*}$ form a $3$-periodic Poncelet trajectory on
$\Gamma$ circumscribed about $\mathbf{O}$. By the Poncelet Theorem [10, Thm.
2.14], the Poncelet trajectory on $\Gamma$ containing $\hat{P}_{1}^{*}$ and
$\hat{P}_{2}^{*}$ is also $3$-periodic, whence $\hat{P}_{3}^{*}$ also lies on
$\Gamma$. ∎

######  Theorem 6.3.

The moduli space $\mathcal{M}_{3}(S_{\mathbf{P}})$ consists of the ideal
triangles whose ideal vertices are the tangency points of polar triangles
circuminscribed in the $3$-periodic Poncelet pair
$(\Gamma_{\mathbf{P}},\mathbf{O})$, where $\Gamma_{\mathbf{P}}$ is the conic
associated in Lemma 6.2 with $\mathbf{P}$. The conic
$\Gamma_{\mathbf{P}}=\Gamma(S_{\mathbf{P}})$ depends only on $S_{\mathbf{P}}$,
and not on $\mathbf{P}$.

###### Proof.

After applying an isometry sending $S$ to $(0,0)$, it suffices to prove this
for the special case $S=(0,0)$, which is an immediate consequence of Lemma
6.1. ∎

In order to find $\Gamma(S)$ explicitly it suffices to find $\Gamma(r,0)$ for
$0\leq r<1$, and then rotate by $\theta$ to reach any desired
$S=(r\cos\theta,r\sin\theta)\in\mathbb{H}$. There are two special triangles
$\mathbf{P}$ and $\hat{\mathbf{P}}$ such that
$S_{\mathbf{P}}=S_{\hat{\mathbf{P}}}=(r,0)$ with $p_{3}=0$ and
$\hat{p}_{3}=\infty$, respectively. We find
$p_{1}+p_{2}=0=\hat{p}_{1}+\hat{p}_{2}$ (from $J=0$), and then that
$p_{1}=\pm\sqrt{3\frac{1-r}{1+r}}$ and $\hat{p}_{1}=\frac{-p_{1}}{3}$ (from
$I-J=r(I+J)$). We then compute from (3.3) that

$P_{1}^{*}=\left(\tfrac{r-2}{1-2r},0\right);\ \ P_{2}^{*}=(1,-p_{1});\ \
P_{3}^{*}=(1,p_{1});\ \ \hat{P}_{1}^{*}=\left(\tfrac{r+2}{2r+1},0\right);\ \
\hat{P}_{2}^{*}=(-1,3p_{1}^{-1});\ \ \hat{P}_{3}^{*}=(-1,-3p_{1}^{-1}).$

It is trivial to verify that

$\Gamma(r,0)\ :\
\bigl{(}1-4r^{2}\bigr{)}x^{2}+6rx+\bigl{(}1-r^{2})y^{2}+r^{2}-4=0$

is the unique (overdetermined!) conic passing through all six polars as in
Lemma 6.2. The following amusing observation is then immediate.

######  Corollary 6.4.

The Poncelet conic $\Gamma(S)$ is an ellipse (resp., a parabola; resp., a
hyperbola) if and only if $|S|<1/2$ (resp., $|S|=1/2$; resp., $|S|>1/2$),
where $|S|$ denotes the Euclidean norm of $S$.

Figure 7.  Left: Poncelet conic of all ideal triangles with a common
hyperbolic barycenter. Right: one representative of the pencil of Poncelet
conics of ideal quadrilaterals with a common hyperbolic barycenter.

###  6.2. Moduli of ideal quadrilaterals with fixed hyperbolic barycenter

The space $\mathcal{P}_{4}$ of all ideal hyperbolic quadrilaterals is four-
dimensional, hence the subspaces $\mathcal{M}_{4}(S)$ are expected to be two-
dimensional. Theorem 5.1 immediately yields a compact and satisfying
description of $\mathcal{M}_{4}(S)$: choose any two distinct hyperbolic lines
$\delta_{1}$ and $\delta_{2}$ intersecting at $S$ to be the diagonals of the
arbitrary ideal quadrilateral $\mathbf{P}$ with $S_{\mathbf{P}}$. This
construction exhausts all of $\mathcal{M}_{4}(S)$, and is roughly analogous to
our first description of $\mathcal{M}_{3}(S)$ in terms of rotations about $S$
in Lemma 6.1. Inspired by the description of $\mathcal{M}_{3}(S)$ in terms of
Poncelet conics afforded by Theorem 6.3, we seek a similar description of
$\mathcal{M}_{4}(S)$.

Given an ideal hyperbolic quadrilateral $\mathbf{P}$, its four polars
$P_{\ell}^{*}$ determine a unique pencil of conics $\bm{\Gamma}_{\mathbf{P}}$
passing through these four points. For a generic fifth point $Q$, let
$\Gamma_{\mathbf{P}}(Q)$ be the unique conic on this pencil that also passes
through $Q$. Then $(\Gamma_{\mathbf{P}}(Q),\mathbf{O})$ forms a $4$-periodic
Poncelet pair, whence by the Poncelet Theorem [10, Theorem 2.14] $Q=Q_{1}^{*}$
is polar to the side of a unique ideal quadrilateral $\mathbf{Q}$ all of whose
polars $Q_{\ell}^{*}$ are also on $\Gamma_{\mathbf{P}}(Q)$. It is not obvious
(and for us, not even expected) that we should always have
$S_{\mathbf{Q}}=S_{\mathbf{P}}$ as apparently depicted in Figure 7. In the
next result we show that this is indeed the case, and that this construction
exhausts all of $\mathcal{M}_{4}(S_{\mathbf{P}})$.

######  Theorem 6.5.

The moduli space $\mathcal{M}_{4}(S_{\mathbf{P}})$ consists of the ideal
quadrilaterals whose ideal vertices are the tangency points of polar
quadrilaterals circuminscribed in a $4$-periodic Poncelet pair
$(\Gamma,\mathbf{O})$, for some conic $\Gamma$ in the pencil
$\bm{\Gamma}_{\mathbf{P}}$.

###### Proof.

After applying an isometry sending $S$ to $(0,0)$, it suffices to prove this
for the special case $S=(0,0)$. By Corollary 5.2, it suffices to prove the
claim that for any $\mathbf{P}\in\mathcal{M}_{4}(0,0)$, any $\Gamma$ in
$\bm{\Gamma}_{\mathbf{P}}$, and any $\mathbf{Q}^{*}$ circuminscribed in
$(\Gamma,\mathbf{O})$, the diagonals of $\mathbf{Q}^{*}$ intersect at the
origin.

To prove the claim, first note that $S_{\mathbf{P}}=(0,0)$ if and only if
$\mathbf{P}$ is a rectangle, which after a harmless rotation about the origin
may be assumed to have its sides parallel to the coordinate axes. Then the
polar quadrilateral $\mathbf{P}^{*}=P_{1}^{*}P_{2}^{*}P_{3}^{*}P_{4}^{*}$ is a
rhombus whose vertices lie on the coordinate axes. By comparing the general
form of a conic evaluated at each pair $P_{\ell}^{*}$ and $P_{\ell+2}^{*}$, we
discover immediately that every (non-degenerate) $\Gamma$ in the pencil
$\bm{\Gamma}_{\mathbf{P}}$ must be a central ellipse. Our claim follows
immediately from the following general Lemma 6.6.∎

We have severed the following configuration result from the rest of the proof
of Theorem 6.5 due to its independent interest. We provide a short conceptual
proof because we have not found a reference for it in the literature, even
though it is surely well-known to the experts.

######  Lemma 6.6.

If $\Gamma$ is a central conic such that $(\Gamma,\mathbf{O})$ forms a
$4$-periodic Poncelet pair then every circuminscribed polygon in
$(\Gamma,\mathbf{O})$ is a rhombus centered at the origin.

###### Proof.

Let us first show this in the special case where our circuminscribed
$\mathbf{Q}$ has one of its vertices, call it $Q_{1}$, on a vertex of
$\Gamma$. Begin constructing the Poncelet orbit in both directions starting
from $Q_{1}$, to obtain $Q_{2}$ and $Q_{4}$. By symmetry, these are both the
next vertices of the ellipse (as we claim), or both in the hemisphere
containing $Q_{1}$, or both in the hemisphere containing $-Q_{1}$. We see
again by symmetry that the latter two impossibilities would contradict the
$4$-periodicity of the Poncelet pair $(\Gamma,\mathbf{O})$.

We can assume without loss of generality that the axes of $\Gamma$ coincide
with the coordinate axes. We ignore from now on the special case already
established above. Let us label the vertices of any other $\mathbf{Q}$
circuminscribed in $(\Gamma,\mathbf{O})$ such that $Q_{\ell}$ lies on the
$\ell$-th quadrant. For any other such $\hat{\mathbf{Q}}\neq\mathbf{Q}$
circuminscribed in $(\Gamma,\mathbf{O})$, the vertex $\hat{Q}_{\ell}$ in the
$\ell$-th quadrant lies before (resp., after) $Q_{\ell}$, relative to the
standard counterclockwise orientation of $\Gamma$, for all $\ell$
simultaneously. Suppose that $Q_{3}$ lies before (resp., after) $-Q_{1}$ for
some $\mathbf{Q}$, contrary to our contention. Then every vertex of
$\mathbf{Q}$ lies before (resp., after) every vertex of $-\mathbf{Q}$. But
this would imply, by symmetry, that every vertex of $-\mathbf{Q}$ must lie
before (resp., after) every vertex of $-(-\mathbf{Q})=\mathbf{Q}$, which is
absurd. ∎

######  Remark 6.7.

We see from Theorem 6.5 that for any two ideal quadrilaterals $\
\mathbf{P}\neq\mathbf{Q}$ such that $S_{\mathbf{P}}=S_{\mathbf{Q}}$, the eight
distinct polars of $\mathbf{P}$ and $\mathbf{Q}$ belong to a conic, which is
in fact the unique common conic to both pencils $\bm{\Gamma}_{\mathbf{P}}$ and
$\bm{\Gamma}_{\mathbf{Q}}$.

In the case of triangles, $\mathcal{M}_{3}(S_{\mathbf{P}})$ is triply covered
by the conic $\Gamma_{\mathbf{P}}$ traced out by the polars of the triangles
obtained by rotating $\mathbf{P}$ about $S_{\mathbf{P}}$. This exceptional
phenomenon does not occur in general for ideal quadrilaterals $\mathbf{P}$,
with an important exception. After an isometry sending $S_{\mathbf{P}}$ to the
origin, it we see that the polars of the rotations of $\mathbf{P}$ about
$S_{\mathbf{P}}$ in general trace out two disjoint conics corresponding to
pairs of opposite sides. These two conics coincide precisely when $\mathbf{P}$
is _harmonic_ , which case we begin to study systematically in the next
section.

######  Remark 6.8.

The five polars of an ideal pentagon $\mathbf{P}$ define a unique conic
$\Gamma$, which again by the Poncelet porism yields a one-parameter family of
pentagons circuminscribed in $(\Gamma,\mathbf{O})$. Though it might seem
natural to hope for all these pentagons to share the same hyperbolic
barycenter $S_{\mathbf{P}}$, we have verified experimentally that they do not.
We leave open the tantalizing question of how to describe, in a similarly
geometric way as for ideal triangles and quadrilaterals, the corresponding
moduli spaces for ideal hyperbolic $n$-gons for $n\geq 5$, beyond the obvious
algebraic conditions (6.1).

##  7\. Least-squares points and harmonic polygons

Thus far we have been mainly concerned with entities and phenomena occurring
within hyperbolic geometry. In Theorem 3.2, the hyperbolic barycenter
$S_{\mathbf{P}}$ of an ideal triangle $\mathbf{P}$ is uniquely defined by the
property that it minimizes the sum of hyperbolic sines of hyperbolic distances
to the sides. Now the classical Theorem 1.1 states that the deKleinization of
this same $S_{\mathbf{P}}$ happens to coincide with the _least-squares point_
$L_{\mathbf{P}}$, uniquely defined by the property that it minimizes the sum
of squares of Euclidean distances to the sides. This strange and unexpected
coincidence begs the natural question, for which other ideal polygons it might
happen that $S_{\mathbf{P}}=L_{\mathbf{P}}$.

In this section we keep considering “the same” $\mathbf{P}$ intermittently as
an ideal hyperbolic polygon and as an inscribed Euclidean polygon, the first
(resp., latter) being the (de)Kleinization of the latter (resp., former). To
avoid cumbersome notation, we keep using the same symbol $\mathbf{P}$, making
it clear from context or explicitly each time which geometry we are working
with.

An inscribed Euclidean (resp., ideal hyperbolic) polygon $\mathbf{P}$ is
_harmonic_ if it is obtained from a regular polygon by a projective
transformation (resp., hyperbolic isometry). These Euclidean and hyperbolic
notions agree under the (de)Kleinization correspondence. Harmonic polygons
have been extensively studied since the late 1800s – see for example [7, 17,
20], and the recent [11].

As mentioned in the introduction, one of our initial motivations for this work
was to find an explanation for the classical but strange Theorem 1.1. Our main
goal in this section is to provide a simple proof of Theorem 7.5: for every
harmonic polygon $\mathbf{P}$, the hyperbolic barycenter $S_{\mathbf{P}}$ and
the least-squares point $L_{\mathbf{P}}$ coincide. From this we propose the
provocative explanation: $S_{\mathbf{P}}=L_{\mathbf{P}}$ for all triangles
$\mathbf{P}$ only because all triangles are harmonic.

######  Remark 7.1.

The coordinates of the symmedian $S$ of a triangle have been obtained
explicitly in [5], in terms of the equations of the lines formed by the sides
of the triangle, using the optimality property in Theorem 1.1 as a defining
property of $S$. In principle, the coordinatization of [5] and ours in (3.2)
could be deduced directly from one another by a series of unenlightening
algebraic manipulations. Theorem 7.5, which contains Theorem 1.1 as its first
special case, can be understood as a more conceptual explanation for the
agreement of the two coordinatizations of the same point.

Denoting by $d_{\ell}(X)$ the Euclidean distance from the point $X$ to the
line $\xi_{\ell}$, we compute that

$d_{\ell}(X)=\dfrac{1}{|P_{\ell}^{*}|}\left(1\mp P_{\ell}^{*}\cdot X\right),$
(7.1)

where the sign depends on whether $X$ and $P_{\ell}^{*}$ are on opposite sides
or on the same side of $\xi_{\ell}$, respectively. For $X$ in the interior of
$\mathbf{P}$, we then compute the gradient

$\nabla\left(\sum_{\ell}d_{\ell}^{2}(X)\right)=2\sum\limits_{\ell}\left(\dfrac{P^{*}_{\ell}\cdot
X-1}{|P_{\ell}^{*}|^{2}}\right)P^{*}_{\ell}.$ (7.2)

From now on we denote by $\sigma_{\ell}$ the Euclidean length of $\xi_{\ell}$.

For the following basic result we include an (unnecessary, but short) argument
in the spirit of some of our other computations, to ensure non-circularity.

######  Lemma 7.2.

The locus of points $X$ satisfying the Euclidean proportionality condition

$[d_{\ell-1}(X):\sigma_{\ell-1}]=[d_{\ell}(X):\sigma_{\ell}]$ (7.3)

is the hyperbolic altitude from $P_{\ell}$ to its short diagonal
$P_{\ell-1}P_{\ell+1}$.

###### Proof.

This is an elementary fact about any triangle $P_{1}P_{2}P_{3}$. Under the
parametrization of §2, each
$|P^{*}_{t}|=\sec\left(\frac{\varphi_{t}-\varphi_{t+1}}{2}\right)$ and
$\sigma_{t}=2\sin\left(\frac{\varphi_{t}-\varphi_{t+1}}{2}\right)$. Obviously,
$P_{2}$ is on the line $d_{1}(X)\sigma_{2}=d_{2}(X)\sigma_{3}$. That
$P_{3}^{*}$ is also on this line is reduced by (7.1) to the short computation
that

$(I_{1}+K_{1})(I_{3}+K_{3})-(I_{1}-K_{1})(I_{3}-K_{3})-4J_{1}J_{3}=1=(I_{2}+K_{2})(I_{3}+K_{3})-(I_{2}-K_{2})(I_{3}-K_{3})-4J_{2}J_{3}.\qed$

######  Theorem 7.3.

A polygon $\mathbf{P}$ is harmonic if and only the hyperbolic altitudes from
the vertices to their short diagonals are all concurrent. In this case, the
point of concurrence is $S_{\mathbf{P}}$.

###### Proof.

One direction is obvious: if $\mathbf{P}$ is harmonic then all the hyperbolic
altitudes to short diagonals are concurrent at $S_{\mathbf{P}}$, since these
entities are all preserved by hyperbolic isometry.

To prove the opposite implication, suppose the hyperbolic altitudes to the
short diagonals of the $n$-gon $\mathbf{P}$ are concurrent at some $S$. After
a hyperbolic isometry we can assume without loss of generality that $S=(0,0)$
and
$P_{\ell}=\left(\cos\left(\frac{2\pi\ell}{n}\right),\sin\left(\frac{2\pi\ell}{n}\right)\right)$
for $\ell=1,2$. The remaining vertices are now recursively determined: having
already fixed $P_{\ell-1}$ and $P_{\ell}$, the short diagonal
$P_{\ell-1}P_{\ell+1}$ is uniquely determined by its polar, which lies on the
tangent line to $\mathbf{O}$ at $P_{\ell-1}$ and on $SP_{\ell}$. The vertices
of the regular $n$-gon satisfy the same recurrence and initial conditions, so
$\mathbf{P}$ is harmonic. ∎

######  Remark 7.4.

Our Euclidean definition of harmonicity, taken from [20, §4], is not the same
as the one given in [7, VI] and [17, §1], which asks for the existence of a
point whose distances to the sides are proportional to the side lengths,
called the symmedian of $\,\mathbf{P}$ in [7, VI]. Mediated by Lemma 7.2,
Theorem 7.3 essentially states, from a hyperbolic barycentric perspective,
that the two definitions of harmonicity agree. This is not a new result. The
first implication in the above proof is essentially already contained in [20,
Thm. 4]. A different proof of the opposite implication is given in [17, §27]
and attributed to Neuberg. Of course, these earlier results simply state that
one of the definitions of harmonicity implies the other, and do not refer to
hyperbolic altitudes at all. Theorem 7.3 allows us to eponymously interpret
all symmedians as hyperbolic barycenters. The former are defined only for
harmonic polygons, whereas the latter are defined in (3.2) for all ideal
polygons.

######  Theorem 7.5.

If $\,\mathbf{P}$ is harmonic then $S_{\mathbf{P}}=L_{\mathbf{P}}$.

###### Proof.

Let $\bm{\sigma}:=(\sigma_{1},\ldots,\sigma_{n})$ denote the vector of
Euclidean side-lengths of $\mathbf{P}$. For any $X$, let
$\mathbf{d}(X):=(d_{1}(X),\ldots,d_{n}(X))$ denote the vector of Euclidean
distances from $X$ to the sides of $\mathbf{P}$. Then
$\mathbf{d}(X)\cdot\bm{\sigma}=2A$, twice the Euclidean area of $\mathbf{P}$,
provided that $X$ is in the interior of $\mathbf{P}$. Now let
$\bm{\sigma}^{\perp}(X):=\mathbf{d}(X)-\frac{2A}{|\bm{\sigma}|^{2}}\bm{\sigma}$
denote the orthogonal projection of $\mathbf{d}(X)$ onto the orthogonal
complement of $\mathbb{R}\bm{\sigma}$. Then
$|\mathbf{d}(X)|^{2}=4A^{2}|\bm{\sigma}|^{-2}+|\bm{\sigma}^{\perp}(X)|^{2}$.
Therefore $L_{\mathbf{P}}$, which is by definition the minimizer of
$|\mathbf{d}(X)|^{2}$, is also the minimizer of
$|\bm{\sigma}^{\perp}(X)|^{2}$.

In case $\mathbf{P}$ is harmonic, Theorem 7.3 and Lemma 7.2 together imply
that
$\frac{d_{\ell-1}(S_{\mathbf{P}})}{\sigma_{\ell-1}}=\frac{d_{\ell}(S_{\mathbf{P}})}{\sigma_{\ell}}$
for each $\ell$. Therefore $\mathbf{d}(S_{\mathbf{P}})$ is a scalar multiple
of $\bm{\sigma}$, whence $|\bm{\sigma}^{\perp}(S_{\mathbf{P}})|^{2}=0$ and
$S_{\mathbf{P}}=L_{\mathbf{P}}$. ∎

We see from the proof of Theorem 7.5 that $L_{\mathbf{P}}$ is equivalently
characterized as the minimizer of $|\bm{\sigma}^{\perp}(X)|$, which achieves
the minimum zero precisely when $\mathbf{P}$ is harmonic, by Theorem 7.3. In
principle, one can express $L_{\mathbf{P}}$ algebraically in terms of
$p_{1},\dots,p_{n}$ by setting (7.2) to zero. The condition
$S_{\mathbf{P}}=L_{\mathbf{P}}$ defines a subspace of codimension at most $2$
in the space $\mathcal{P}_{n}$ of all ideal $n$-gons (cf. §6). On the other
hand, the subspace of $\mathcal{P}_{n}$ consisting of harmonic $n$-gons is
three-dimensional, and therefore the space of $n$-gons for which
$S_{\mathbf{P}}=L_{\mathbf{P}}$ must strictly contain the space of harmonic
$n$-gons for $n>5$. This can maybe be understood as some kind of “regularity
condition” on $\mathbf{P}$, which is in general strictly weaker than
harmonicity. And yet, according to the following result, harmonicity is
equivalent to having $S_{\mathbf{P}}=L_{\mathbf{P}}$ in the next case $n=4$.

######  Theorem 7.6.

If $\,\mathbf{P}$ is a quadrilateral, then $S_{\mathbf{P}}=L_{\mathbf{P}}$ if
and only if $\,\mathbf{P}$ is harmonic or $S_{\mathbf{P}}=(0,0)$.

###### Proof.

($\Leftarrow$). It follows from Theorem 5.1 that $S_{\mathbf{P}}=(0,0)$ if and
only if the Euclidean polygon $\mathbf{P}$ is a rectangle, in which case it is
clear that $L_{\mathbf{P}}=(0,0)$. If $\mathbf{P}$ is harmonic then
$S_{\mathbf{P}}=L_{\mathbf{P}}$ by Theorem 7.5.

($\Rightarrow$). Setting to zero the evaluation of (7.2) at
$X=S_{\mathbf{P}}$, after some algebraic manipulation, we conclude that either
$I_{\mathbf{P}}-K_{\mathbf{P}}=0=J_{\mathbf{P}}$, or else the cross-ratio
$[p_{1},p_{2},p_{3},p_{4}]=1$. These conditions are equivalent to having
$S_{\mathbf{P}}=(0,0)$ and to $\mathbf{P}$ being harmonic, respectively.∎

######  Remark 7.7.

The conclusion of Theorem 7.6 may seem strange, because the origin is only
special from the Euclidean point of view, but hyperbolically unremarkable. But
the origin remains meaningful when comparing the two geometries, since the
endofunctions of the unit disc which are simultaneously hyperbolic and
Euclidean isometries is precisely the set of rotations about the origin.

We expect that a similar algebraic analysis should yield the following case
$n=5$. One subtlety in this case is that the regular pentagram $\mathbf{P}$
still satisfies the proportionality condition
$\bm{\sigma}^{\perp}(S_{\mathbf{P}})=\mathbf{0}$, and therefore so do all of
its images under hyperbolic isometry, even though this $\mathbf{P}$ is not
technically “harmonic” in our sense, and yet the subspace of $\mathcal{P}_{5}$
defined by $S_{\mathbf{P}}=L_{\mathbf{P}}$ does not care about the convexity
of $\mathbf{P}$. We still optimistically hope for the best possible result.

######  Conjecture 7.8.

If $\,\mathbf{P}=(p_{1},\dots,p_{5})$ is a pentagon, then
$S_{\mathbf{P}}=L_{\mathbf{P}}$ if and only if either the five cross-ratios
$[p_{\ell},p_{\ell+1},p_{\ell+2},p_{\ell+3}]$ are all the same or
$S_{\mathbf{P}}=(0,0)$.

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