# Learning Joint Surface Atlases

Theo Deprelle1∗
&Thibault Groueix2
Noam Aigerman2
&Vladimir G. Kim2
&Mathieu Aubry1
&1LIGM, Ecole des Ponts, Univ Gustave Eiffel, CNRS, 2Adobe Research

###### Abstract

This paper describes new techniques for learning atlas-like representations of
3D surfaces, i.e. homeomorphic transformations from a 2D domain to surfaces.
Compared to prior work, we propose two major contributions. First, instead of
mapping a fixed 2D domain, such as a set of square patches, to the surface, we
learn a continuous 2D domain with arbitrary topology by optimizing a point
sampling distribution represented as a mixture of Gaussians. Second, we learn
consistent mappings in both directions: charts, from the 3D surface to 2D
domain, and parametrizations, their inverse. We demonstrate that this improves
the quality of the learned surface representation, as well as its consistency
in a collection of related shapes. It thus leads to improvements for
applications such as correspondence estimation, texture transfer, and
consistent UV mapping. As an additional technical contribution, we outline
that, while incorporating normal consistency has clear benefits, it leads to
issues in the optimization, and that these issues can be mitigated using a
simple repulsive regularization. We demonstrate that our contributions provide
better surface representation than existing baselines.

Figure 1:  From a collection of shapes without annotations, we learn a 2D
domain which can be used to parameterize all shapes, parametrizations (blue)
and chart-mappings (red).

##  1 Introduction

This paper is concerned with 3D surfaces and their representation as atlases
or UV maps, i.e., their mappings to and from a domain of the 2D Euclidean
space. Surfaces and altases are closely related: surfaces are $2$-manifolds
embedded in $\mathbb{R}^{3}$ and are defined in topology by the existence of
_charts_ , homeomorphic mappings from the surface to a 2D Euclidian domain.
This relation is also central to many algorithms related to surfaces: on one
hand, the computation of UV maps of meshes is a highly-active research topic:
on the other hand, deep learning works have successfully used atlas-like
representation and learn local parametrizations to represent 3D surfaces [4,
10]. These last techniques typically learn to map a fixed sets of 2D squares
to 3D, which can approximate a 3D surface and makes it possible to use the 2D
domain to compute correspondences between _predicted_ surfaces [3] for which
parametrizations are learned jointly.

While these parametrization-based methods produce rather-accurate 3D surface
reconstructions, they do not lead to a well-defined homeomorphic map between a
2D domain and the predicted surface, which limits the scope of applications of
these techniques. For examples, when mapping a set of 2D squares to 3D,
AtlasNet [10] leads to many overlaps, and the chart-mappings from 3D to 2D are
thus not well defined. The predicted maps from 2D to 3D might also include
large amount of distortion, thus not yielding a good UV map of the generated
3D surface, and for example preventing using it to define mappings between
different input surfaces. Previous works have attempted to address these
limitations in various ways. Williams et al. [25] performs optimization for a
single shape on local neighbourhoods with Earth Mover Distance to obtain
homeomorphic parametrizations and consistent transitions maps, but this leads
to an heavy optimization, with many local parametrizations, and has no obvious
extension to multiple shapes. Rather than enforcing consistency between the
mappings of square patches AtlasNet v2 [7] attempts to learn the 2D domain,
e.g., by learning the positions of a fixed set of points, but loses the
continuous aspect of the mapping and can still lead to overlapping patches.
DSR [4] keeps the AtlasNet framework and its intrinsic limitations, but uses
several regularizations to encourage conformal mappings, to minimize the 3D
overlap between the images of the square patches and to prevent patch
collapse. However, it is still limited to square patches and we found it
difficult to use on complex shapes.

We argue that the two tasks of parameterizations and charts prediction are
complementary to one another, and that learning the 3D shape(s) reconstruction
should go together with learning a relevant 2D domain. We present an
architecture for such a joint optimization, where we learn the 2D domain by
learning a 2D probability distribution defined as a mixture of Gaussians and
from which we sample points for reconstructing the surface. These sampled
points are mapped to 3D and compared (via chamfer distance) to the target
point cloud; similarly, the point cloud is mapped to 2D and compared to the
sampled points. We optimize for cyclic consistency between the two mappings,
as well as for geometric losses such as isometric regularization. Through
experiments, we show that our method is able to better reconstruct surfaces
than existing baselines, in particular leading to more meaningful
parametrizations with fewer artefacts and yielding meaningful correspondences
between shapes in a collection. Our code is available on our project
webpage111https://imagine.enpc.fr/ deprellt/joint-surface/

##  2 Related work

Our work is related to prior work in optimizing chart-mappings for UV
parametrization and learning parameterization for surface reconstruction.

#### Optimizing charts-mappings for UV parametrization.

Identifying chart is a long-standing problem in geometry processing [22]. Most
prior techniques take a 2-manifold input represented as a mesh and map each
point to a 2D domain. These methods typically aim to produce a bijective
mapping [23], while also minimizing some distortion metric such as Dirichlet
[19], ARAP [15], LSCM [14], and symmetric Dirichlet [21]. Some techniques also
aim to predict consistent chart-mappings for a collection of related shapes,
so that semantically-similar points on different meshes map to the same point
in 2D. Doing so enables many applications such as correspondence estimation
[1], morphing [12], and texture transfer [20]. Unlike our method, these
techniques do not use deep learning and typically require manual input, such
as a sparse set of corresponding points. By using neural networks to represent
the UV map we can learn consistent charts without the user input or explicit
supervision. We can also co-parameterize point clouds without knowing the
underlying mesh. To the best of our knowledge, our work is the first method to
use neural networks to learn consistent chart-mappings.

#### Learning surface parameterization for reconstruction.

Previous learning-based techniques for surface parameterization mostly focused
on modeling 2D to 3D map for reconstruction. To learn such a mapping from a
collection of shape, AtlasNet [10] and FoldingNet [26] pioneered the idea of
using a Multi-Layered Perceptrons (MLP) as a class of continuous parametric
function that embed a 2-manifold into 3D. By conditionning the MLP on a latent
code, and using a large-scale 3D shape repository [5], AtlasNet [10]
demonstrated the possibility to predict the surface parameterization from an
input mesh, point-cloud or even a single image. Perhaps surprisingly, Williams
et al. [25] showed that parameterization did not necessarily need the
regularization brought by learning on a collection of shapes, and could also
be optimized on individual shapes.

Several approaches introduced novel losses to improve the parameterization.
Deng et al. [6] improve the global arrangement of the different parts of the
parameterization via a normal-aware reconstruction loss and a stiching loss.
In DSR, Bednarik et al. [4] regularize the smoothness of the reconstructed
surface by optimizing a conformal energy, based on the Jacobians of the
mappings. Remarkably, having access to analytical jacobians in a higher-order
differentiation procedure opens-up exciting applications. For instance,
Bednarik et al. [3] achieve temporally coherent parameterizations for each
frame in a video by regularizing the deformation to locally have a constant
metric tensor.

Particularly relevant to us is AtlasNet-v2 [7] which jointly optimize the
shape of the 2D manifold and learns the surface parameterization using two
proposed strategies. The first one consists in learning a deformation of a
fixed template into an elementary structure common to all shapes via an
additional AtlasNet-like MLP . This strategy leads to elementary structures
that can easily be meshed, but have the same topology as the initial template.
The second strategy consists in sampling a fixed set of points on the template
and adding them to the optimizer. After optimization, the sampled points can
form a complex elementary structure, without topology constraints, but meshing
it is not straight-forward. In this work, we propose a method to combine the
best features of both strategies, namely the ability to learn topologically
complex structures and the ability to mesh them. We achieve this via a novel
differentiable layer to sample points from 2D gaussians with a learnable mean
(see Sec. 3.2).

Our approach is different from AtlasNet [10] and its variants [2, 3, 4, 6, 7,
16, 25] in that: (i) we aim to jointly learn the surface parametrizations and
their inverse functions, the charts-mappings; (ii) we learn a 2D domain
relevant to a family of shapes by optimizing a probability density function in
the 2D Euclidean plane. We also differ in two novel losses that correctly
orient the normals of the reconstructed surface and fix point-collapse in 3D
which is a common artifact of AtlasNet-type of approaches.

Figure 2: Single shape architecture. Given a shape $\mathcal{S}$, we optimize
jointly a 2D probability distribution $\mathcal{P}$, a parametrization
network, $\varphi_{\mathcal{S}}$ and chart-mapping network
$\psi_{\mathcal{S}}$. The parameterisation network $\varphi_{\mathcal{S}}$
takes as input a point $\mathbf{x_{\mathcal{P}}}\in\mathbb{R}^{2}$ randomly
sampled according to the density probability function $\mathcal{P}$ to produce
the 6D vector (3D point and normal coordinates)
$\varphi_{\mathcal{S}}(\mathbf{x_{\mathcal{P}}})$. The chart-mapping network
$\psi_{\mathcal{S}}$ takes as input a point $\mathbf{x_{\mathcal{S}}}$ sampled
on the shape $\mathcal{S}$, its normal $\mathbf{n_{\mathcal{S}}}$ and outputs
a 2D points
$\psi_{\mathcal{S}}(\mathbf{x_{\mathcal{S}}},\mathbf{n_{\mathcal{S}}})$.

##  3 Method

#### Overview.

Given a collection of shapes, our goal is to learn for all shapes surface
parameterizations with their inverse chart-mappings and a join 2D domain on
which the parametrizations are defined. For simplicity, we first present our
approach in the case of a single shape $\mathcal{S}$: in Section 3.1 we
explain how to model surface parameterization and chart-mapping, and introduce
our main architectural blocks; in Section 3.2 we explain how we learn the 2D
UV domain as a probability distribution; in section 3.3 we discuss our losses.
Our pipeline for a single shape is illustrated in Figure 2. Finally, in
Section 3.4 we explain how to train our approach jointly on a family of
shapes.

#### Notations.

We use the following notations:

  * •

$\mathcal{S}$ : 3D shape of interest

  * •

$\mathcal{P}$ : learned probability distribution in 2D

  * •

$\phi_{\mathcal{S}}:\mathbb{R}^{2}\xrightarrow[]{}\mathbb{R}^{3}$ : surface
parameterization

  * •

$\varphi_{\mathcal{S}}:\mathbb{R}^{2}\xrightarrow[]{}\mathbb{R}^{6}$ : surface
parameterization with normals

  * •

$\psi_{\mathcal{S}}:\mathbb{R}^{6}\xrightarrow[]{}\mathbb{R}^{2}$ : chart-
mapping

We slightly abuse the brackets notation to indicate sampling a set of $M$
points, e.g.:

  * •

$\\{\mathbf{x_{\mathcal{S}}}\\}$ is a set of $M$ points sampled on
$\mathcal{S}$ using a uniform probability distribution

  * •

$\\{\mathbf{x_{\mathcal{P}}}\\}$ is a set of $M$ points sampled in
$\mathbb{R}^{2}$ according the probability distribution $\mathcal{P}$

###  3.1 Parametrization and chart-mapping

We explain the two main components of our architecture for a given shape
$\mathcal{S}$, a chart-mapping $\psi_{\mathcal{S}}$ and a surface
parameterization network $\varphi_{\mathcal{S}}$.

#### Chart-mapping.

We learn a single chart-mapping from the shape $\mathcal{S}$ to
$\mathbb{R}^{2}$, using a Multi-Layer Perceptron (MLP) with both point
coordinates and normals as input. We observe that naively learning an
$\mathbb{R}^{3}$ to $\mathbb{R}^{2}$ mapping leads to the collapse of thin
surfaces after their mapping to the 2D domain. Indeed, coordinate-based MLPs
are continuous functions and Tancik et. al. [24] showed that they have a prior
to learn smooth functions in the absence of positional encoding. Hence, two 3D
points with very close spatial coordinates but opposite normals tend to be
mapped closely in 2D. For chart-mappings, such smoothness is generally a
desirable feature that should be maintained, but distant normals should be a
strong cue to indicate that points are intrinsically far. To handle thin
surfaces, we propose to learn a mapping $\psi_{\mathcal{S}}$ from
$\mathbb{R}^{6}$ to $\mathbb{R}^{2}$ that takes as input a point
$\mathbf{x_{\mathcal{S}}}$ and its associated normal
$\mathbf{n_{\mathcal{S}}}$ scaled to have norm $\alpha$. The parameter
$\alpha$ is a hyperparameter of our approach which controls how much normals
contribute to distances compared to the 3D positions of the points.We set it
to $0.01$ in all our experiments.

#### Surface parameterization.

For a shape $\mathcal{S}$, we seek to learn the inverse function of the chart-
mapping,
$\psi_{\mathcal{S}}^{-1}:\mathbb{R}^{2}\xrightarrow[]{}\mathbb{R}^{6}$. We
first use an MLP
$\phi_{\mathcal{S}}:\mathbb{R}^{2}\xrightarrow[]{}\mathbb{R}^{3}$ to
parameterize the surface $\mathcal{S}$. The function $\phi_{\mathcal{S}}$ maps
a point $\mathbf{x}=(u,v)$ in $\mathbb{R}^{2}$ into a point
$\phi_{\mathcal{S}}(\mathbf{x})$ in $\mathbb{R}^{3}$. The Jacobian
$\mathbf{J}_{\mathbf{x}}$ of the mapping $\phi_{\mathcal{S}}$ is defined at
every point $\mathbf{x}$ by:

$\mathbf{J}_{\mathbf{x}}=\big{[}\mathbf{J}_{\mathbf{x},u},\mathbf{J}_{\mathbf{x},v}\big{]}=\big{[}\frac{\partial\phi_{\mathcal{S}}}{\partial\mathit{u}}(\mathbf{x}),\frac{\partial\phi_{\mathcal{S}}}{\partial\mathit{v}}(\mathbf{x})\big{]}~{}.$
(1)

The two partial derivatives are trivial to compute with Pytorch auto-
differentiation [18]. The normal $\mathbf{n}$ to the parametrizedsurface at
$\phi_{\mathcal{S}}(\mathbf{x})$ can be computed as the normalised cross
product of $\mathbf{J}_{\mathbf{x},u}$ and $\mathbf{J}_{\mathbf{x},v}$ and
scaled by the same hyperparameter $\alpha$ used for the chart-mapping
$\psi_{\mathcal{S}}$:

$\mathbf{n}=\alpha\frac{\mathbf{J_{\mathbf{x},\mathit{u}}}\times\mathbf{J_{\mathbf{x},\mathit{v}}}}{\|\mathbf{J_{\mathbf{x},\mathit{u}}}\times\mathbf{J_{\mathbf{x},\mathit{v}}}\|}.$
(2)

We define $\varphi_{\mathcal{S}}:\mathbb{R}^{2}\xrightarrow[]{}\mathbb{R}^{6}$
as the concatenation of an output point and its scaled normal:

$\varphi_{\mathcal{S}}(\mathbf{x})=[\phi(\mathbf{x}),\mathbf{n}]~{}.$ (3)

To summarize, the mapping $\varphi_{\mathcal{S}}$ is designed to represent the
inverse of the chart-mapping $\psi_{\mathcal{S}}$. However, it is not defined
for every point in $\mathbb{R}^{2}$, and in the next section we focus on
identifying the 2D domain for which it is defined.

###  3.2 Learning a 2D domain as a sampling probability distribution

To parametrize a shape $\mathcal{S}$, we want to define a domain in
$\mathbb{R}^{2}$ such that $\phi_{\mathcal{S}}$ defines a bijection from this
2D domain to $\mathcal{S}$. In practice, during training, we want to learn
this domain, sample points inside it, and map them using
$\varphi_{\mathcal{S}}$. Instead of handling explicitly the 2D domain
geometry, e.g., points or primitives similar to [7], we take a probabilistic
approach and learn the parameters of a probability distribution $\mathcal{P}$
from which to sample points. This enables us to easily deal with topological
changes. We now detail how we represent this probability distribution, learn
it, and use it to define a 2D domain.

#### Modelling 2D sampling probability as a mixture of Gaussians.

We sample 2D points according to a probability distribution $\mathcal{P}$
which we model as a mixture of $K$ 2D Gaussians with means
$\mathbf{\mu}_{i}\in\mathbb{R}^{2}$ for $i=1,...,K$ , a fixed standard
deviation $\mathbf{\sigma}\in\mathbb{R}$ and fixed mixing coefficients equal
to $1/K$:

$\mathcal{P}(\mathbf{x})=\frac{1}{K}\sum_{k=1}^{K}\mathcal{N}(\mathbf{x}|\mathbf{\mu}_{i},\mathbf{\sigma})\quad\text{with}\quad\sigma=\frac{1}{\sqrt{K}}~{}.$
(4)

During training, at every iteration, we sample $N$ 2D points
$\\{\mathbf{x_{\mathcal{P}}}\\}$ from $\mathcal{P}$, which are both the input
of the parameterisation network $\varphi_{\mathcal{S}}$ and the target of the
chart-mapping network $\psi_{\mathcal{S}}$.

#### Learning the Gaussian means.

To allow learning the 2D domain, we make the means $\mu_{i}$ learnable
parameters of the method. To do so, we need to see the sampled points as
differentiable with respect to the $\mu_{i}$. We achieve this with the
pathwise gradient estimator from [11], also called the reparameterization
trick. This consist in expressing a parameterized random variable via a
parameterized deterministic function of a parameter-free random variable. For
Gaussian Mixture Models (GMM), this simply amounts to sampling a GMM with zero
means and adding the means to the sampled points, i.e., defining each point
$\mathbf{x_{\mathcal{P}}}$ as the result of the following process: first
selecting the id $i$ of a mixture component using a uniform distribution; then
sampling a 2D point $\mathbf{x}_{\mathcal{N}}$ from a Gaussian distribution of
mean $0$ and standard deviation sigma
$\mathbf{x}_{\mathcal{N}}\sim\mathcal{N}(0,\mathbf{\sigma})$; finally,
defining the point $\mathbf{x_{\mathcal{P}}}$ as
$\mathbf{x_{\mathcal{P}}}=\mathbf{x}_{\mathcal{N}}+\mathbf{\mu}_{i}$, which is
trivially differentiable with respect to $\mathbf{\mu}_{i}$. Please see Figure
5 for examples of learned probability distributions.

#### From probability distribution to 2D continuous domain.

Once the distribution has been optimized, we simply threshold the probability
distribution function $\mathcal{P}$ to obtain a 2D domain. We can then compute
a 2D triangulation of the domain, and use the parameterization network $\phi$
to obtain a 3D mesh (as can be seen in Figure 3).

###  3.3 Training losses

We now explain the loss function we optimize. We write our loss for a single
shape:

$\mathcal{L}_{single}(\varphi_{\mathcal{S}},\psi_{\mathcal{S}},\mathbf{\mu})=\lambda_{\textrm{6D}}\mathcal{L}_{\textrm{6D}}+\lambda_{\textrm{2D}}\mathcal{L}_{\textrm{2D}}+\lambda_{\textrm{cycle}}\mathcal{L}_{\textrm{cycle}}+\lambda_{\textrm{iso}}\mathcal{L}_{\textrm{iso}}+\lambda_{\textrm{rep}}\mathcal{L}_{\textrm{rep}}~{},$
(5)

where $\mu=(\mu_{1},...,\mu_{K})$, the $\lambda$ are scalar hyper-parameters
and the different loss terms are detailed bellow. To compute distances between
two sets of points $\mathcal{X}$ and $\mathcal{Y}$, we base our losses on the
Chamfer distance defined as:

$\mathcal{L}_{\textrm{chamfer}}(\mathcal{X},\mathcal{Y})=\frac{1}{|\mathcal{X}|}\sum_{\mathbf{x}\in\mathcal{X}}\min_{\mathbf{y}\in\mathcal{Y}}\|\mathbf{x}-\mathbf{y}\|^{2}+\frac{1}{|\mathcal{Y}|}\sum_{\mathbf{y}\in\mathcal{Y}}\min_{\mathbf{x}\in\mathcal{X}}\|\mathbf{y}-\mathbf{x}\|^{2}~{},$
(6)

where $|\mathcal{X}|$ and $|\mathcal{Y}|$ are the number of points in
$\mathcal{X}$ and $\mathcal{Y}$ respectively. Our set of losses is designed to
enforce two objectives : (i) ensuring an accurate surface parameterization
$\varphi_{\mathcal{S}}$ with low distortion, (ii) ensuring that
$\varphi_{\mathcal{S}}$ and $\psi_{\mathcal{S}}$ are indeed inverse of each
other.

#### Surface parameterization reconstruction loss.

The surface parameterization $\varphi_{\mathcal{S}}$ takes as input a set of
2D points $\\{\mathbf{x_{\mathcal{P}}}\\}$ sampled from the probability
distribution $\mathcal{P}$ and outputs a set of 6D points (3D points with
scaled normals). We minimize the 6D Chamfer distance between the set of
generated points $\\{\varphi_{\mathcal{S}}(\mathbf{x_{\mathcal{P}}})\\}$ and a
set of points $\\{\mathbf{x_{\mathcal{S}}}\\}$ associated to normals
$\\{\mathbf{n_{\mathcal{S}}}\\}$ sampled on the target shape $\mathcal{S}$:

$\mathcal{L}_{\textrm{6D}}(\varphi,\mathbf{\mu})=\mathcal{L}_{\textrm{chamfer}}(\\{\varphi_{\mathcal{S}}(\mathbf{x_{\mathcal{P}}})\\},\\{\mathbf{x_{\mathcal{S}}},\mathbf{n_{\mathcal{S}}}\\})~{}.$
(7)

#### Chart-mapping reconstruction loss.

We encourage the overall 2D projection of the shape under $\psi_{\mathcal{S}}$
and the probability distribution $\mathcal{P}$ to be the same. Recall that
$\psi_{\mathcal{S}}$ takes as input
$\\{\mathbf{x_{\mathcal{S}}},\mathbf{n_{\mathcal{S}}}\\}$ (3D points
uniformely sanpled on $\mathcal{S}$ with scaled normals) and outputs a set of
2D points
$\\{\psi_{\mathcal{S}}(\mathbf{x_{\mathcal{S}}},\mathbf{n_{\mathcal{S}}})\\}$.
We simply minimize the Chamfer distance between the generated 2D points and a
set of points $\\{\mathbf{x_{\mathcal{P}}}\\}$ sampled from $\mathcal{P}$:

$\mathcal{L}_{\textrm{2D}}(\psi,\mathbf{\mu})=\mathcal{L}_{\textrm{chamfer}}(\\{\psi_{\mathcal{S}}(\mathbf{x_{\mathcal{S}}},\mathbf{n_{\mathcal{S}}})\\},\\{\mathbf{x_{\mathcal{P}}}\\})~{}.$
(8)

#### Cycle-consistency loss.

We want the two mappings $\varphi$ and $\psi$ to be inverse of one another. We
encourage this using a cycle-consistency loss on 2D points sampled from
$\mathcal{P}$ and on 3D points sampled uniformly on $\mathcal{S}$ with their
associated normals:

$\mathcal{L}_{\scriptsize\textrm{cycle}}(\varphi,\psi,\mathbf{\mu})=\frac{1}{M}\sum_{\mathbf{x}\in\\{\mathbf{x_{\mathcal{P}}}\\}}\|\mathbf{x}-\psi\circ\varphi(\mathbf{x})\|^{2}+\frac{1}{M}\sum_{(\mathbf{x},\mathbf{n})\sim\\{\mathbf{x_{\mathcal{S}}},\mathbf{n_{\mathcal{S}}}\\}}\|\mathbf{x}-\varphi\circ\psi(\mathbf{x},\mathbf{n})\|^{2}~{}.$
(9)

#### Distortion regularization loss.

We limit distortion in the parameterization with an isometric regularization.
Given the Jacobian $\mathbf{J}_{\mathbf{x}}$ of the transformation $\phi$ at
point $\mathbf{x}$ (Equation 1) and $\mathbf{I}$ the identity matrix, the
isometric loss can be written:

$\mathcal{L}_{\scriptsize\textrm{iso}}(\varphi_{\mathcal{S}},\mathbf{\mu})=\frac{1}{M}\sum_{\mathbf{x}\in\\{\mathbf{x_{\mathcal{P}}}\\}}\|\mathbf{J}_{\mathbf{x}}\mathbf{J}_{\mathbf{x}}^{T}-\mathbf{I}\|~{},$
(10)

where the sum is over points sampled according to $\mathcal{P}$. As already
observed in [3, 9], this type of regularization has the additional benefit of
making the parameterizations more consistent across shapes.

#### Probability distribution regularization loss.

Non-uniform density is a known failure mode of the Chamfer distance, as shown
in Figure 5 and also observed in [4]. In theory, a loss based on optimal
transport like the Earth Mover distance would be ideal to fix this problem.
However, in practice, we ran into optimisation, training time and parameter
tuning issues when using EMD. On the contrary the Chamfer loss is simple to
use and fast to compute. We thus use the Chamfer loss and introduce a
repulsive loss between the Gaussian means defining the probability
distribution $\mathcal{P}$ as a regularization:

$\mathcal{L}_{\textrm{rep}}(\mathbf{\mu})=\frac{1}{K^{2}}\sum_{i,j\in[0,K]}\exp({-\frac{\|\mu_{i}-\mu_{j}\|}{\sigma}})~{},$
(11)

where $\sigma$ is the Gaussian standard deviations in the definition of
$\mathcal{P}$. We found that this simple loss lead to much more uniform
distributions of points both in the 2D plane and in the reconstructed shapes.

###  3.4 Joint learning on a family of shapes.

We now explain how we learn jointly atlases on a collection of $N$ shapes
$\mathcal{S}_{1},...,\mathcal{S}_{N}$. Since we want to share the 2D domain
between the different shapes, we do not condition the probability distribution
$\mathcal{P}$ on the shape, and learn a single one for all shapes. On the
contrary, the parametrization and chart-mappings are expected to depend on the
shape. Rather than learning them completely independently for each shape, we
use the auto-decoder framework [17], where we optimize for each shape
$\mathcal{S}_{i}$ feature vectors $\mathbf{f}_{\psi,i}$ and
$\mathbf{f}_{\varphi,i}$ which we use to define respectively
$\varphi_{\mathcal{S}_{i}}$ and $\psi_{\mathcal{S}_{i}}$. More precisely, we
learn jointly for all shapes networks $\varphi$ and $\psi$, and define for
each shape $\mathcal{S}_{i}$ for all $\mathbf{x}\in\mathbb{R}^{2}$,
$\varphi_{\mathcal{S}_{i}}(\mathbf{x})=\varphi(\mathbf{x},\mathbf{f}_{\psi,i})$
and for all $\mathbf{x}\in\mathbb{R}^{6}$,
$\psi_{\mathcal{S}_{i}}(\mathbf{x})=\psi(\mathbf{x},\mathbf{f}_{\psi,i})$. We
then optimize $\psi$, $\varphi$ and $\mu$ by minimizing the loss:

$\mathcal{L}_{\text{full}}(\varphi,\psi,\mathbf{\mu},\mathbf{f_{\psi}},\mathbf{f_{\varphi}})=\sum_{i=1}^{N}\mathcal{L}_{\text{single}}(\varphi_{\mathcal{S}_{i}},\psi_{\mathcal{S}_{i}},\mathbf{\mu})~{},$
(12)

where $\mathbf{f_{\psi}}=(\mathbf{f}_{\psi,1},...,\mathbf{f}_{\psi,N})$ and
$\mathbf{f_{\varphi}}=(\mathbf{f}_{\varphi,1},...,\mathbf{f}_{\varphi,N})$.

#### Implementation details.

We use latent codes $\mathbf{f}_{\varphi,i}$ and $\mathbf{f}_{\psi,i}$ of
dimension 256. The architectures we use for $\varphi$ and $\psi$ are MLP with
5 hidden layers of size 256 and ReLU activations. We do not use batch
normalization layers. $\mathcal{P}$ is defined using $K=10,000$ mixture
components. We sample $M=10^{4}$ points both on the shapes and according to
$\mathcal{P}$ at every training iteration. We train with
$\lambda_{\textrm{6D}}=1$, $\lambda_{\textrm{2D}}=10^{-2}$,
$\lambda_{\textrm{cycle}}=1/M$, $\lambda_{\textrm{iso}}=10^{-4}/M$ and
$\lambda_{\textrm{rep}}=1$.

##  4 Experiments

Figure 3: Top: single shape parametrization. On the shapes of [25], we
transfer a colored mesh of the learned 2D domain (left) to 3D, which enables
visualizing correspondence and cuts (right). Bottom: co-parameterization. For
’teddy’ and ’cup’ shapes of SHREC [8], we show the joint sampling
probabilities and 2D domain, and the reconstructions of 4 shapes with 3
different viewpoints and consistent coloring. Note the quality of the
parametrization compared to patch-based methods [4, 10]

#### Datasets.

We use individually the shapes of Williams et al. [25], which come from five
high-resolution scans with over a million points with associated normals. We
generate the manifold meshes from this data using screened Poisson Surface
Reconstruction. The resulting meshes have a variety of geometric details and
different topologies, providing interesting challenges for atlas-based
representations. The SHREC dataset [8] contains 400 manifold meshes with
sparse correspondence annotations, which enables us to quantitatively evaluate
the consistency of our joint atlases. For our experiments we selected
categories, ant, teddy, cups and armadillo, aiming for topological and
geometric diversity, and four shapes in each.

#### Reconstruction Metrics.

To evaluate how well our representation matches the input shape, we report two
commonly used metrics: 3D Chamfer Distance and the Earth Mover’s Distance
(EMD). Given two sets of points $\mathcal{X}$ and $\mathcal{Y}$ with
$M^{\prime}=2000$ randomly-sampled points each, we approximate the EMD as:

$\begin{split}\mathcal{L}_{\textrm{EMD}}(\mathcal{X},\mathcal{Y})=\frac{1}{M^{\prime}}\sum_{\mathbf{x}_{i}\in\mathcal{X}}\sum_{\mathbf{y}_{j}\in\mathcal{Y}}C_{i,j}\|\mathbf{x}_{i}-\mathbf{y}_{j}\|^{2}~{},\end{split}$
(13)

where the association matrix $C$ is such that $C_{i,j}=1$ if point
$\mathbf{x}_{i}$ is mapped to point $\mathbf{y}_{j}$ and $C_{i,j}=1$, and is
computed to minimize $\mathcal{L}_{\textrm{EMD}}$ via Hungarian Algorithm
[13]. To evaluate the reconstructions of the normals of the surfaces, we also
report the distances of the normals using the associations given either by the
chamfer distance or the EMD using the spatial coordinates.

#### Correspondence Metric.

Given a pair of shapes $\mathcal{S}_{1},\mathcal{S}_{2}$ and a set of
keypoints $\mathbf{p}_{i}\in\mathcal{S}_{1}$, we find the corresponding points
$\mathbf{q_{i}}\in\mathcal{S}_{2}$ by compositing the chart-mapping and
parameterization networks:
$\mathbf{q}_{i}=\varphi_{\mathcal{S}_{2}}\circ\psi_{\mathcal{S}_{1}}(\mathbf{p}_{i})$.
For the baselines that do not have a chart-mapping network, we obtain
correspondences for each point $\mathbf{p}_{i}$ using the following process:
we sample points according to $\mathcal{P}$, map them to $\mathcal{S}_{1}$,
select the one which image by $\varphi_{\mathcal{S}_{1}}$ is closest to
$\mathbf{p}_{i}$, and map it to $\mathcal{S}_{2}$ using
$\varphi_{\mathcal{S}_{2}}$. Given $M^{\prime\prime}$ ground truth
correspondences $\mathbf{q}_{i}^{\text{gt}}$ for each $\mathbf{p_{i}}$, we
measure the L2 loss on the spatial coordinates as
$\frac{1}{M^{\prime\prime}}\sum_{{i}}|\mathbf{q_{i}}-\mathbf{q_{i}^{\text{gt}}}|$.

#### Results and analysis.

Qualitative results for individual shapes of Williams et al. [25] are shown in
Figure 3 (top). Joint atlases for the SHREC dataset [8], can be seen for the
’ants’ are in the teaser Figure 1 and for ’cup’ and ’teddy’ in Figure 3
(bottom). Note how we manage to learn continuous 2D domains with complex
topology, which we can mesh to obtain a high quality 3D mesh for the shapes.
Also note that compared to the patch-based approaches we obtain few and
meaningful cuts in the parametrization, without any overlap. Finally, note the
consistency in the reconstruction of the different shapes, which can be
visualized by consistent colors transfered from the joint 2D domain to all
shapes.

|  Williams et al. [25] |  SHREC [8]
---|---|---
| Spatial | Normal | Spatial | Normal | Corresp.
|  Ch. $\downarrow$ |  Emd $\downarrow$ |  Ch. $\downarrow$ |  Emd $\downarrow$ |  Ch. $\downarrow$ |  Emd $\downarrow$ |  Ch. $\downarrow$ |  Emd $\downarrow$ |  L2 $\downarrow$
Baselines |  |  |  |  |  |  |  |  |
$\scriptscriptstyle(1)$ |  ANv1-1 [10] | $2.3{\scriptscriptstyle\pm 0.0}$ | $\boldsymbol{5.0}{\scriptscriptstyle\pm 0.1}$ | $1.2{\scriptscriptstyle\pm 0.2}$ | $1.2{\scriptscriptstyle\pm 0.2}$ | $2.2{\scriptscriptstyle\pm 0.0}$ | $\boldsymbol{5.4}{\scriptscriptstyle\pm 0.5}$ | $1.1{\scriptscriptstyle\pm 0.2}$ | $1.1{\scriptscriptstyle\pm 0.2}$ | $2.6{\scriptscriptstyle\pm 0.8}$
$\scriptscriptstyle(2)$ |  ANv2-1 [7] | $\boldsymbol{2.2}{\scriptscriptstyle\pm 0.0}$ | $11.1{\scriptscriptstyle\pm 2.1}$ | $1.2{\scriptscriptstyle\pm 0.1}$ | $1.3{\scriptscriptstyle\pm 0.1}$ | $\boldsymbol{2.0}{\scriptscriptstyle\pm 0.0}$ | $7.1{\scriptscriptstyle\pm 1.4}$ | $1.3{\scriptscriptstyle\pm 0.2}$ | $1.3{\scriptscriptstyle\pm 0.2}$ | $1.3{\scriptscriptstyle\pm 0.1}$
$\scriptscriptstyle(3)$ |  ANv1-10 [10] | $2.4{\scriptscriptstyle\pm 0.0}$ | $11.4{\scriptscriptstyle\pm 0.9}$ | $1.1{\scriptscriptstyle\pm 0.0}$ | $1.1{\scriptscriptstyle\pm 0.0}$ | $2.3{\scriptscriptstyle\pm 0.1}$ | $11.1{\scriptscriptstyle\pm 0.4}$ | $1.1{\scriptscriptstyle\pm 0.1}$ | $1.1{\scriptscriptstyle\pm 0.1}$ | $3.5{\scriptscriptstyle\pm 1.4}$
$\scriptscriptstyle(4)$ |  ANv2-10 [7] | $2.2{\scriptscriptstyle\pm 0.0}$ | $11.1{\scriptscriptstyle\pm 2.1}$ | $1.2{\scriptscriptstyle\pm 0.1}$ | $1.3{\scriptscriptstyle\pm 0.1}$ | $2.1{\scriptscriptstyle\pm 0.0}$ | $13.4{\scriptscriptstyle\pm 0.5}$ | $1.0{\scriptscriptstyle\pm 0.1}$ | $1.1{\scriptscriptstyle\pm 0.1}$ | $2.8{\scriptscriptstyle\pm 1.6}$
$\scriptscriptstyle(5)$ |  DSR-10 [4] | $4.3{\scriptscriptstyle\pm 0.8}$ | $18.2{\scriptscriptstyle\pm 0.8}$ | $1.4{\scriptscriptstyle\pm 0.0}$ | $1.2{\scriptscriptstyle\pm 0.0}$ | $3.1{\scriptscriptstyle\pm 1.1}$ | $10.0{\scriptscriptstyle\pm 2.8}$ | $1.3{\scriptscriptstyle\pm 0.1}$ | $1.3{\scriptscriptstyle\pm 0.1}$ | $2.6{\scriptscriptstyle\pm 1.2}$
Ours |  |  |  |  |  |  |  |  |
$\scriptscriptstyle(6)$ | $\phi$ | $\boldsymbol{2.2}{\scriptscriptstyle\pm 0.0}$ | $15.2{\scriptscriptstyle\pm 5.9}$ | $1.0{\scriptscriptstyle\pm 0.2}$ | $1.1{\scriptscriptstyle\pm 0.2}$ | $\boldsymbol{2.0}{\scriptscriptstyle\pm 0.0}$ | $6.8{\scriptscriptstyle\pm 0.9}$ | $1.1{\scriptscriptstyle\pm 0.2}$ | $1.1{\scriptscriptstyle\pm 0.2}$ | $2.0{\scriptscriptstyle\pm 0.8}$
$\scriptscriptstyle(7)$ | $\phi,n$ | $2.4{\scriptscriptstyle\pm 0.1}$ | $20.3{\scriptscriptstyle\pm 3.3}$ | $0.3{\scriptscriptstyle\pm 0.0}$ | $0.7{\scriptscriptstyle\pm 0.0}$ | $2.1{\scriptscriptstyle\pm 0.3}$ | $13.1{\scriptscriptstyle\pm 8.0}$ | $\boldsymbol{0.2}{\scriptscriptstyle\pm 0.0}$ | $\pagecolor{orange}0.5{\scriptscriptstyle\pm 0.0}$ | $1.4{\scriptscriptstyle\pm 1.1}$
$\scriptscriptstyle(8)$ | $\phi,n,r$ | $\boldsymbol{2.2}{\scriptscriptstyle\pm 0.1}$ | $12.9{\scriptscriptstyle\pm 4.4}$ | $\boldsymbol{0.2}{\scriptscriptstyle\pm 0.1}$ | $0.6{\scriptscriptstyle\pm 0.1}$ | $\boldsymbol{2.0}{\scriptscriptstyle\pm 0.1}$ | $9.6{\scriptscriptstyle\pm 6.6}$ | $\boldsymbol{0.2}{\scriptscriptstyle\pm 0.0}$ | $\boldsymbol{0.4}{\scriptscriptstyle\pm 0.1}$ | $2.8{\scriptscriptstyle\pm 0.9}$
$\scriptscriptstyle(9)$ | $\phi,n,r,p$ | $2.3{\scriptscriptstyle\pm 0.2}$ | $13.2{\scriptscriptstyle\pm 4.0}$ | $0.3{\scriptscriptstyle\pm 0.1}$ | $0.6{\scriptscriptstyle\pm 0.1}$ | $\boldsymbol{2.0}{\scriptscriptstyle\pm 0.0}$ | $8.2{\scriptscriptstyle\pm 2.1}$ | $\boldsymbol{0.2}{\scriptscriptstyle\pm 0.0}$ | $\boldsymbol{0.4}{\scriptscriptstyle\pm 0.0}$ | $1.5{\scriptscriptstyle\pm 1.0}$
$\scriptscriptstyle(10)$ | $\phi,\psi,n,r,p$ | $2.6{\scriptscriptstyle\pm 0.0}$ | $\boldsymbol{5.1}{\scriptscriptstyle\pm 0.1}$ | $\boldsymbol{0.2}{\scriptscriptstyle\pm 0.0}$ | $\boldsymbol{0.5}{\scriptscriptstyle\pm 0.0}$ | $2.2{\scriptscriptstyle\pm 0.0}$ | $\boldsymbol{5.4}{\scriptscriptstyle\pm 0.2}$ | $\boldsymbol{0.2}{\scriptscriptstyle\pm 0.0}$ | $\boldsymbol{0.4}{\scriptscriptstyle\pm 0.0}$ | $1.0{\scriptscriptstyle\pm 1.0}$
$\scriptscriptstyle(11)$ | $\phi,\psi,n,r,p,i$ | $2.6{\scriptscriptstyle\pm 0.2}$ | $5.5{\scriptscriptstyle\pm 1.3}$ | $\boldsymbol{0.2}{\scriptscriptstyle\pm 0.3}$ | $\boldsymbol{0.5}{\scriptscriptstyle\pm 0.2}$ | $2.3{\scriptscriptstyle\pm 0.1}$ | $\pagecolor{orange}5.9{\scriptscriptstyle\pm 0.6}$ | $\boldsymbol{0.2}{\scriptscriptstyle\pm 0.0}$ | $\boldsymbol{0.4}{\scriptscriptstyle\pm 0.0}$ | $\boldsymbol{0.8}{\scriptscriptstyle\pm 1.0}$

Notations $n$: normal - $r$: repulsion loss - $p$: Probability distribution
function $\mathcal{P}$ \- $i$: Isometric regularisation of $\phi$

Table 1: Comparison and ablation study. We compare our method, successively
adding its different componenets, to AtlasNet [10], AtlasNetV2 [7] and DSR
[4]. We compute association between target and reconstructed 3D points using
Chamfer distance and EMD, then report for each association the average
distance between the points coordinates (’Spatial’) and their normals
(’Normals’). We also evaluate the quality of the correspondences when
available. All the values are scaled by a factor of $10^{-2}$. Please see text
for details.

We report quantitative results in Table 1. To account for randomness in
initialization and optimization, we re-run each method four times and report
mean and standard deviation. We report results for three baselines: vanilla
AtlasNet [10] (rows 1 and 3) which only learns the parameterization network,
AtlasNet v2-points [7] (rows 2 and 4) which learns elementary point structures
in the 2D domain, and DSR [4] (row 5) which uses several regularisation losses
including an isometry loss. We try the first two methods with 1 and 10 UV
patches. We found that methods using a single patch typically lead to similar
Chamfer distances but much lower EMD. We believe that this is because using a
single patch encourages points to be more uniformly distributed on the
surface. We also found that DSR [4] provided quantitativelty slightly worse
results, which is consistent with what was reported in this paper that mainly
aims at visual quality.

We evaluate our method by successively adding our different components. We
start (row 6) by only using our parametrization network, learning a fixed set
of input point positions with only 3D Chamfer distance as supervision, similar
to AtlasNet v2 with 1 patch and obtained similar results.
We then add normals to the Chamfer loss (row 7), which unsurprisingly gives a
very strong boost to the normal metrics. Qualitatively, the effect can be seen
in the left part of Figure 4 where the orientation of the normals is color-
coded for each point: without the normals in the loss, a significant part of
the normals are back-facing. Adding the normals in the loss however has a
second undesired effect and significantly increases the EMD. This can be
understood qualitatively by looking at the results, where we see greatly
varying density of points over the reconstructed shapes and in the 2D space.
We interpret this as the fact that adding normals in the loss complexifies the
loss landscape and adds bad local minima.
To avoid this effect, we add our repulsive regularization to the loss (row 8).
We can see this improves the EMD results without degrading the Chamfer results
and normal consistency. The effect on the points density is also striking
qualitatively, as visualized in the right part of Figure 4.
At this point, the parametrization is still only optimized on a set of points.
To recover an interpretation of the parametrization as a continuous mapping of
a 2D domain, we introduce our sampling probability distribution (row 9) which
has little quantitative effect but is crucial to define a continuous 2D
mapping.
We can then introduce our chart-mapping, together with the 2D reconstruction
and cycle losses (row 10). It can be seen that beside its theoretical interest
and benefits, it has a clear quantitative impact: both the EMD and the
correspondence metrics are clearly boosted.
Finally, adding the isometric regularization (row 11) provides a small
additional boost in term of correspondence quality, i.e., consistency between
different jointly learned atlases.

Figure 4: Impact of the normals and the repulsion loss. Note the variation of
the color-coded normal directions (left) and the variations in the point
density (right).

(a) Target

(b) ours

(c) w/o iso

(d) w/o normals

(e) w/o rep.

Figure 5: Qualitative ablation results. Given a target cube shape, we show the
probability distribution $\mathcal{P}$ learned by our method and three key
ablations: not using the isometric loss, not using the normals, and not using
the repulsive loss.

Note that all of our regularization losses remain necessary to the success of
our method, as can be visualized qualitatively in Figure 5, where we use our
architecture to learn the reconstruction of a simple cube and visualize the
learned 2D sampling distribution. Without the normal loss, we observe a
concentration of the density on edges, an equivalent of point collapse
described in [4]. Without the repulsion loss, the density is concentrated on a
single face. Without the isometry loss, the shape of the cube is highly
distorded. FWith our full method the net of the cube is clear and the point
density relatively uniform.

##  5 Conclusion

We propose a novel technique for representing a shape or a collection of
shapes, with two key differences with respect to prior work on atlas-like
representations. First, we joinlty learn two maps, a parameterization and a
chart-mapping. Second, we learn the 2D domain on which the parametrization is
defined. This makes our representation much closer than previous works to be a
proper atlas, i.e., one that defines an homeomorphism between 3D shape and a
2D domain. It also enables co-parameterization, where homeomorphisms are
learned between several shapes and a single 2D domain, which can have
applications for consistent texturing and correspondence estimation. We
further offer other technical contributions, such as learning the 2D domain by
optimizing a sampling probability distribution, analyzing the effect of
incorporating normals in the optimization, and introducing a repulsive loss to
have a more even point distribution. We believe our work is an important step
towards learning consistent atlases and it will inspire future work on further
improving quality of atlases, such as modeling transition maps, minimizing
seams, and distortion.

Acknowledgments Thanks to F. Darmon, R. Loiseau, E. Vincent for their
feedbacks on the manuscript; E. Shechtman , D. Picard, Y. Siglidis and G.
Ponimatkin for inspiring discussions; and B. George for code suggestions; This
work was partly supported by ANR project EnHerit ANR-17-CE23-0008, Labex
Bézout, gifts from Adobe to École des Ponts and HPC resources from GENCI-IDRIS
(2021-AD011011937R1).

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11institutetext: Dipartimento di Fisica e Astronomia “Galileo Galilei”
dell’Università di Padova
Istituto Nazionale di Fisica Nucleare & INAF, Padova, Italy
Universidade de Lisboa & Instituto Superior Técnico/LIP, Lisboa, Portugal

# Galileo Galilei and a forgotten poem on the 1604 supernova

Alessandro De Angelis E-mail<EMAIL_ADDRESS>
###### Abstract

In October 1604, when SN1604, the last naked-eye visible supernova in our
Galaxy, exploded, Galileo Galilei was professor of mathematics at the
University of Padua, teaching the mechanics of planets. He was therefore the
figure of reference to whom all the doubts and questions that such an
apparition brought with it were addressed. The University asked Galilei to
outline the situation by exposing in three public conferences his point of
view, in order to answer the many questions that raged among the academic
community and the common people. Three conferences that Galilei held between
November and December, in the Aula Magna of the Bo, the central building of
the University. A month after Galilei’s lectures, a treatise on the supernova
appeared in Padua. The unknown Antonio Lorenzini, behind whose name it is easy
to see the inspiration of Cesare Cremonini, an Aristotelian professor of
natural philosophy in Padua, published a booklet entitled Discourse about the
new star which debunked the conclusions of Galilei. One month later was
published in Padua by the same editor the Dialogue by Cecco di Ronchitti from
Bruzene about the new star, a booklet in Paduan dialect replying to Lorenzini.
The Dialogue appears in the Italian national edition of Galilei’s works and is
attributed to Galilei himself with the help of the monk Girolamo Spinelli; it
does not appear instead a poem in octaves, by an unknown author (presumably
Galilei), published as an appendix to the first edition of 1605, and
immediately replaced in the second edition. We publish it here, with a
comment.

Nell’ottobre 1604, quando esplose la supernova SN1604, l’ultima visibile a
occhio nudo nella nostra Galassia, Galileo Galilei era professore di
matematica all’Università di Padova e insegnava la meccanica dei pianeti. Fu
quindi la figura di riferimento a cui vennero rivolti tutti i dubbi e le
domande che una tale apparizione portava con sé. L’Università chiese a Galilei
di fare il punto sulla situazione esponendo in tre conferenze pubbliche le sue
convinzioni, per rispondere alle tante domande che imperversavano tra la
comunità accademica e la gente comune. Tre conferenze che Galilei tenne tra
novembre e dicembre, nell’Aula Magna del Bo, l’edificio centrale
dell’Università. Un mese dopo le conferenze di Galilei, fu pubblicato a Padova
un trattato sulla supernova. Lo sconosciuto Antonio Lorenzini, dietro il cui
nome è facile scorgere l’ispirazione di Cesare Cremonini, professore
aristotelico di filosofia naturale a Padova, pubblicò un Discorso sulla nuova
stella, che contestava le conclusioni di Galilei. Un mese dopo fu pubblicato a
Padova presso lo stesso editore il Dialogo de Cecco di Ronchitti da Bruzene in
perpuosito de la stella nuova, un libretto in dialetto padovano che rispondeva
a quello di Lorenzini. Il Dialogo compare nell’edizione nazionale delle opere
di Galilei ed è attribuito a Galilei stesso con l’aiuto del monaco Girolamo
Spinelli; non vi compare invece un poema in ottave, di autore ignoto
(presumibilmente Galilei), pubblicato in appendice alla prima edizione del
1605, e subito sostituito nella seconda edizione. Lo pubblichiamo qui, con un
commento.

On the evening of October 9, 1604, curious and enthusiasts from all over the
world were observing a rare conjunction between Mars, Jupiter and Saturn, full
of astrological implications. Suddenly, near that place in the sky, which was
located in the Ophiucus constellation, near the foot of the Serpent bearer
(Fig. 1), a new light appeared, brighter than all the planets with the
exception of Venus. It remained there for a year and a half and then, as it
had appeared, it disappeared. That “new star”, as it was called, changed the
history of astronomy and cosmology: it was believed at the time that stars
were fixed, immutable and ungenerable, but obviously this was not true.
Scientists with different conceptions of the Universe competed and
collaborated to explain the nature and origin, including Galilei and Kepler,
and also Chinese, Korean and Arab astronomers reported observations [1]. Today
we know that that new star was a supernova, the last of the seven supernovae
observed with the naked eye in the Milky Way – the six previous ones of which
there is documentation were recorded in the years 185, 393, 1006, 1054, 1181,
1572. So not a new star, but a star that dies and explodes. What remains today
is a celestial object still full of mysteries: the cloud projected by the
explosion is expanding at very high speed, in some places ten thousand
kilometers per second, one thirtieth of the speed of light, and the shock
waves of this expansion accelerate cosmic particles up to very high energies.
We astrophysicists of today still study it carefully [1], always with an eye
to the first analysis of our illustrious predecessors of four hundred years
ago.

Figure 1: Illustration from Johannes Kepler’s book De stella nova in pede
Serpentarii (On the new star in Ophiuchus’s foot) indicating the location of
the 1604 supernova. The supernova is the star marked with a letter “N” on the
right foot of the Ophiuchus (Serpent Bearer) constellation. Source: Wikimedia
commons.

In October 1604, Galileo Galilei was professor of mathematics and astronomy at
the University of Padua, teaching the mechanics of planets. When the “stella
nova” appeared in the sky, he was therefore the figure of reference to whom
all the doubts and questions that such an apparition brought with it were
addressed.

That bright and pulsating sphere had generated wonder, but also terror and
curiosity. The University of Padua asked Galilei to outline the situation by
exposing in three public conferences his point of view, in order to answer to
the many questions that raged among the academic community and the common
people. Three conferences that Galilei held almost immediately, between
November and December, in the Aula Magna of the Bo, the central building of
the University – fragments of these lectures are in the Italian national
edition of the Opere di Galileo Galilei (Works of Galileo Galilei), edited by
Antonio Favaro [2], vol. II. Even if today it seems difficult to believe it,
Galilei reported that that a thousand people attended each lecture (vol. II of
[2], [3, 4]). Galilei demonstrated through the method of parallax that the new
star was beyond the Moon, and therefore in that Heaven that according to
Aristotle was immutable. Parallax is the apparent displacement of an object
due to a change in the observer’s point of view: looking at the tip of your
nose with the left eye closed, and then with the right eye closed, your nose
will appear to move; knowing the distance between eyes one can determine the
length of his/her nose. The same thing could be done from different points on
the Earth to measure the position of planets and stars, and Galilei did it in
collaboration with Neapolitan and Spanish correspondents. The new star was
shown to all observers to be at the same location with respect to the stars of
Sagittarius and Scorpio, and therefore it had to be farther away than the Moon
and the planets: precisely between the fixed stars. Today only a few fragments
of Galilei’s lecture notes remain (vol. II of [2]), and they are fascinating –
Galilei was a person of all-round culture, a man of letters as well as a
musician, painter and of course a scientist.

A month after Galilei’s lectures, a treatise on the supernova appeared in
Padua. The unknown Antonio Lorenzini, behind whose name it is easy to see the
inspiration of Cesare Cremonini, an Aristotelian professor of natural
philosophy in Padua and friend-rival of his colleague Galilei, published a
booklet entitled Discourse around the new star [5] which debunked the
conclusions of Galilei. In order to justify Aristotle of Stagira’s concept of
immutability of the sky, which seemed to be contradicted by the new star,
Lorenzini affirmed that looking at the stars to measure their distance was
useless: the principles of terrestrial physics did not apply to the sky.

The response to Lorenzini’s speech was rapid and original, but also very
detailed. On February 28, 1605 was published in Padua, by the same publisher
Tozzi who had published the book of Lorenzini, the Dialogo de Cecco di
Ronchitti da Bruzene in perpuosito de la stella nuova (Dialogue by Cecco di
Ronchitti from Bruzene about the new star) [6], a pseudonymous booklet in
Paduan dialect (a local form of Venetian dialect) written according to Antonio
Favaro and to many critics by Galilei together with his student and
Benedictine monk Girolamo Spinelli (who had a deeper knowledge of the Paduan
dialect). It is not known how many copies of the book (whose cover page is
shown in Fig. 2) were published; certainly a second edition was printed the
same year in Verona by the publisher Merlo [7], with minimal changes (not
substantial but with some improvements from the point of view of readability)
to the text and a substantial change in the appendix, which contained Alcune
ottave d’incerto, per la medesima stella, contra Aristotele (Some octaves by
unknown author, for the same star, against Aristotle), in Florentine
vernacular, which were substantially modified.

The Dialogo was an explicit mockery of what had been published by Lorenzini;
it defamed the entire Aristotelian dogma, which explains why the author(s)
preferred to hide under the pseudonym of Cecco di Ronchitti. Cecco di
Ronchitti from Brugine (a small village of peasants near Padua), an unknown
self-proclaimed farmer and land surveyor, gave in the book many indications of
being in reality an astronomer from Padua. The fact that the book was written
in Paduan dialect instead of Latin, which was the noble language used to deal
with relevant subjects, was a sign that the topic around which the booklet was
built was not worthy of consideration,

In addition to the fact that the dialect was perfect for light-hearted
teasing, the dialect of Padua was in vogue thanks to the writings of Angelo
Beolco, a sort of ante litteram “blogger” who had published in the first half
of the sixteenth century under the name of “Ruzante” or “Ruzzante”, which in
Paduan means “scratcher” or “mumbler”, about the events that took place in
Padua.

The Dialogue takes place between two peasants, Matthio (Matteo) and Nale
(Natale), who confront each other after Natale read Lorenzini’s book and
summarizes to Matteo its contents. The booklet analyzes point by point the
contradictions and distortions expressed by Lorenzini in his Discourse. It is
like Galilei did not want to react personally to what Lorenzini had written,
but did so through the dialectal, intelligent and frank voices of two
peasants.

It is easy to recognize in the content of the Dialogo many characteristics of
Galilei, so that already at the time of its publication the treatise was
attributed to him. In addition to this, no family named Ronchitti is recorded
in the birth registers of the parish of Brugine or in the neighboring towns;
moreover, Cecco is also a Tuscan nickname. Galilei had good knowledge of the
Paduan vernacular: Niccolò Gherardini, in his Vita del signor Galileo Galilei
(Life of Galileo Galilei) [8], writes: “he was still very familiar with a book
entitled Il Ruzzante, written in the rustic Paduan language, taking pleasure
in those crude tales and ridiculous incidents” (Gherardini confuses the author
with the title). Among the Miscellanea galileiana inedita (Unpublished
Galilei’s works) published by Favaro in Venice in 1887 [9], one can read a
thought of Galilei that contains words of the rustic Paduan dialect. Finally,
in his letter to Paolo Gualdo of June 16, 1612, published in volume XI of the
Opere [2], Galilei writes whole sentences using the same dialect, and
Galilei’s Venetian correspondents often wrote to him in various flavors of the
Venetian dialect. In a letter to Paolo Gualdo dated August 16, 1614, Girolamo
Spinelli was called by Galilei “one of my students, monk of Santa Giustina,
companion of Cecco de’ Ronchetti”. A focus on the form of the text caused
Galilei’s contribution to be greatly diminished [10] in editions following
Favaro’s (especially from Paduan critics), in favor of Girolamo Spinelli. This
gradual deauthorization has certainly also depended on an excessive attention
to form, admittedly so particular, which has unfortunately come at the expense
of attention to specific content, which is equally (if not even more)
important [11].

Figure 2: The cover page of the first edition of the Dialogo de Cecco di
Ronchitti da Bruzene in perpuosito de la stella nuova.

In the mouth of one of the two interlocutors, Matteo, there are Galilean
concepts; we can say that Galilei’s ideas about the new star are all in
Matteo’s words. The Copernican idea of the author of the Dialogue is in the
explanation of the concept of parallax made in the booklet, which also
contemplates the seasonal displacement of the Earth linked to its revolution
around the Sun. Two drawings reproduced from Galilean manuscripts kept at the
Biblioteca Nazionale Centrale di Firenze, presumably contemporary to the
publication of the Dialogue, and included in a section entitled by Galilei De
stella anni 1604 (About the star of 1604), clarify even better the meaning of
parallax applied to the Earth’s revolution.111Manoscritti galileiani 70, c.
16v e 47, c. 10v. Images reproduced by permission of the Italian Ministero
della Cultura, Biblioteca Nazionale Centrale di Firenze.

Note the Copernican essence of the drawings, and the outer circle of the
Zodiac with the signs. In the first drawing the Earth, third planet from the
Sun which is the black point in the center, is indicated by a sign badly
written, as if Galilei was afraid that his notes would fall into the wrong
hands, but it is still between Mars and Venus and is the center of the
observations – a drawing similar will be reproduced in the Dialogue concerning
the two chief world systems in 1632. In the second you can see the seasonal
variation of parallax. The drawings are not in the original first edition of
the Opere [2], but were added in an appendix to the second volume in later
editions.

In addition to being more compelling than a treatise, as Plato had already
shown and as prescribed in numerous courses on rhetoric of Galilei’s times, a
dialogue allows for circumventing formalism in certain conclusions that
Galilei was probably not able to develop rigorously: a dialogue allows its
participants to forego certain rigorous demonstrations and replace them with
assumptions of sufficient plausibility, made stronger by the use of common
sense and of humour. In this sense the technique of persuasion in this
dialogue anticipates by three decades the two great Galilei’s dialogues: the
Dialogue concerning the two chief world systems (1632) and the Discourses and
mathematical demonstrations relating to two new sciences (1638).

The present Dialogue appears as Galilei’s work in the second volume of the
Italian national edition of the Opere [2]. In this edition it does not appear
instead a poem in octaves (i.e., groups or “stanzas” of eight verses), by an
unknown author, published as an appendix to the first edition of 1605, and
immediately replaced in the second edition. An English translation of the
Dialogue appears in a book by Drake [12]; a commented translation in modern
Italian is published in [4]. Some (see [13], also for notes on the litterary
production by Galilei) possibly attribute it to Antonio Querengo222The Paduan
Antonio Querengo (or Querenghi, or Quarenghi, or Antuogno Squerengo in the
dialectal mispronunciation) was a very learned diplomat and poet in Latin, a
friend of Torquato Tasso. After the death in 1601 of Giovanni Vincenzo
Pinelli, a good friend and sponsor of Galilei since his arrival in Padua,
Querengo’s house replaced Pinelli’s as a mecca for all important visitors and
a meeting place for local scholars and men of letters. Among these were Paolo
Gualdo and Lorenzo Pignoria, who were also good friends of Galilei. Querengo
continued to follow with interest the career of Galilei after he left Padua,
and it is from his pen that we have the most amusing letters concerning the
unfortunate campaign of Galilei in Rome in 1615-1616 to support the Copernican
point of view., to whom the Dialogue is dedicated.

The following elements strengthen the attribution of the poem to Galilei:

  * •

Galilei had likely a strong participation to the Dialogo de Cecco di
Ronchitti; being a poet (the full volume IX of the national edition of his
Opere [2] is dedicated to his litterary activity), it appears strange that the
poem in the appendix of the Dialogo itself was written by somebody else, or at
least that Galilei did not heavily rework it;

  * •

the style, including the verbal violence, is coherent with Galilei, and there
is quite a strong attack to the Aristotelians;

  * •

as Milani observes in her 1992 translation of the Dialogo [14] (the
translation of the text is done from the second edition [7]), some rhymes are
common with those of Galilei, and one stanza begins with the imperative
“Dunque”, as in Galilei’s reworking of Andrea Salvadori’s Canzone per le
Stelle Medicee (vol. IX of Galilei’s Opere [2]);

  * •

a description of the method of parallax is appropriate for an astronomer, and
a Copernican statement is present in the fourth stanza of the poem.

In any case, attribution always contains some degree of arbitrariness.

Here follows the poem published in the appendix of the Dialogue, in the first
edition of 1605.

STANZE D’INCERTO CONTRA ARISTOTELE

PER LA STELLA NUOVAMENTE APPARSA

> Che più vaneggi, o Stagirita stolto:
> e puro il Cielo e ingenerabil credi?
> Stella nuova, in lui fissa, il chiaro volto
> discopre scintillando, e non la vedi?
> O più che mai ne’ primi errori involto
> il senso neghi ed altre prove chiedi?
> Il senso neghi, onde i principii certi
> dicesti avere de le scienze aperti?
>
>
>
>
>
> Ma che nel Cielo sia la nata stella
> stimerai forse non verace assunto;
> e chiaro è pur che a questa gente e a quella
> si mostra fissa in un medesmo punto.
> Vede ciascun che da la punta bella
> di Sagittario ha il suo splendor disgiunto
> e quasi fugga di Scorpion la coda
> s’erge tremante e sovra lei s’inchioda.
>
>
>
>
>
> Or s’ella è sotto il ciel, sotto la Luna,
> sì lungi a quelle del sublime tetto,
> come di un altro clima in parte alcuna
> non si nasconde o cangia almen l’aspetto,
> e al variar de’ siti or sembra in una
> or in altra figura aver ricetto,
> anzi pur, come a noi torna mostrarse
> nel mattutino ove la sera apparse?
>
>
>
>
>
> Né cessan qui le vere mie parole,
> che a ragion ferme e nove ecco ritorno.
> Dimmi, se volge la terrestre mole
> ventidue mila miglia intorno intorno
> e questa nova luce a par del Sole
> tutta la gira in una notte e un giorno,
> come tre ore e più sull’orizzonte
> nostro nel tramontar mostrò la fronte?
>
>
>
>
>
> Esser non può che, s’ella umile è tanto,
> così il passo allentar da noi sia vista,
> ché in breve spazio di mirarla il vanto
> la retta cederebbe umana vista;
> né patria sempre assimigliar (se, quanto
> da noi s’allunga, mole non acquista
> e quella perde poi mentre s’appressa)
> a l’occhio nostro la grandezza stessa.
>
>
>
>
>
> Ma poi ch’ella è fissa nel ciel sublime,
> provan l’altezza sua sì ferme prove,
> che come posta infra le stelle prime
> a ragion seco si mostra e move;
> né d’esser può che in altre parti o in ime
> varia di mole o sito, ella si trove,
> ché nulla face il mutar loco in terra
> all’ampiezza del Cielo, in cui si serra.
>
>
>
>
>
> Dunque di cecità squarciando il panno
> omai la lingua sciogli in vere note
> e se credesti con tuo scorno e danno
> dianzi immutabil le soperne rote,
> or che fiamma novella apre l’inganno
> conosci il Ciel che generar si puote,
> e grazie rendi a la Natura madre,
> ond’hai luci del ver tanto leggiadre.
>

This is my translation:

STANZAS BY AN UNKNOWN AUTHOR TO THE NEWLY APPEARED STAR

AGAINST ARISTOTLE

> What raving, o foolish Stagirite,
> your belief that creations are forbidden in Heaven!
> A newborn star, in it fixed, its clear face
> shows glittering, and you don’t see it?
> Or you deny sense more than ever
> getting stuck in your first mistakes, and ask for other proofs?
> But how to deny this simple evidence
> when all your science was based on sense?
>
>
>
>
>
> That in Heaven the star was born
> maybe you think not to be true;
> though every nation sees it
> fixed in the same point up on the sky.
> Everybody sees it in the beautiful location
> where Sagittarius has its splendor separated
> and as if from Scorpion’s tail escaping
> rises trembling, and it is nailed over it.
>
>
>
>
>
> Now if it be under the Heavens, under the Moon,
> so far from the stars of the sublime roof,
> as of another climate, how it does not partly hide,
> or at least changes its aspect,
> and in a variety of sites seems now
> in the same constellation to have shelter?
> Indeed, how does it return to us in the morning
> where it appeared in the evening?
>
>
>
>
>
> Nor here cease my true words,
> which firm and new now return.
> Tell me, if the Earth
> turns twenty-two thousand miles around
> and this new light, like for the Sun
> turns it all round in a night and a day,
> how could it for three hours and more
> on our horizon show its face at sunset?
>
>
>
>
>
> It cannot be that, if it is so humble,
> it sets its pace as to be seen,
> since in a short time
> it would become invisible;
> nor it could look always similar, unless, as it lengthens from us,
> it does acquire mass
> and then loses it while it draws near
> to look the same in bulk at every place.
>
>
>
>
>
> But since it is fixed in the sublime sky,
> Its height proves so firmly, that
> as placed between the first stars
> naturally shows itself there and moves accordingly;
> nor it could be found in lower parts
> varying in size or site,
> since its place on Earth does not change
> given the enormous circumference of the Heavens.
>
>
>
>
>
> Therefore, ripping the cloth blinding you,
> now let your tongue melt into true notes:
> if you believed, at your scorn and harm
> the upper wheels to be immutable,
> now that this new flame shows your mistake,
> you know that Heaven can be generated,
> and thank Mother Nature,
> who sent you such graceful light of truth.
>

A second edition of the Dialogo de Cecco di Ronchitti was published later the
same year in Verona [7]. The poem was largely amended, with some uncertainties
in the poetic language. In addition, the dangerous Copernican allusion to
Earth rotation in the fourth stanza was removed, and some criticisms to
Aristotle were softened and became “politically correct” (for example, in the
first verse the sentence “stolto Stagirita”, where “stolto” literally means
“fool”, “foolish”, or even “stupid” [15], became “per altro Stagirista
saggio”, i.e., “otherwise sage Stagirite”). This new version, which I leave
here untranslated, when compared to other poems by Galilei, lacks the usual
verbal aggressivity.

> Dunque, o per altro Stagirita saggio,
> vaneggi ancora, e l’alte rote credi
> immutabili e pure? Un novo raggio
> tra quei zaffiri è nato, e non lo vedi?
> O pur facendo a te medesmo oltraggio
> il senso neghi, ed altre prove chiedi?
> Il senso neghi, onde i principii certi
> dicesti aver de le scienze aperti?
>
>
>
>
>
> Ma che in Ciel sia la generata stella
> stimerai forse tu pensier fallace;
> e chiaro è pur che questa gente e quella
> un solo osserva in lei sito verace;
> vede tra’ segni luminosi ch’ella
> di Sagittario al dardo aggiunge face,
> e quasi fugga di Scorpion la coda,
> scintilla e s’alza e sovra lei s’inchioda.
>
>
>
>
>
> Or quando, al tuo parer, sotto la Luna
> resti l’obliquo suo camin ristretto,
> come non mai s’asconde, e non s’imbruna
> a’ varii Climi opposta, o cangia aspetto?
> E non rassembra a noi dal Cielo in una,
> ad altri in altra parte haver ricetto?
> Anzi come il mattin torna mostrarse
> fissa nel punto ove la sera apparse?
>
>
>
>
>
> Aggiungo: e come all’or, che parte o riede
> e si mostra or d’appresso, et or distante
> al Zenit nostro, non mai muta o cede
> la figura, e la luce ogn’or tremante?
> Come in lungo passaggio non la vede,
> e meno o più distinta, occhio costante?
> E non cresce e non scema, al guardo esposta,
> qual’or, girando, s’avvicina e scosta?
>
>
>
>
>
> Meglio dirò: s’ella il suo corso stende
> pel vicino sentier de gli Elementi,
> come sopra di noi tant’ore splende
> quante del sommo Giro i lumi ardenti?
> Ed essi regolato il moto apprende,
> né vien che più s’affretti, o più s’allenti,
> come l’infimo raggio col supremo
> ne va del par da l’uno a l’altro estremo?
>
>
>
>
>
> Esser non può che in cerchio angusto e basso
> de’ corpi eccelsi al gran viaggio assista,
> e mentre tien su l’Orizonte il passo
> di mille aspetti al variar resista,
> ché d’umil Cerchio una gran parte il crasso
> globo ben toglie a la terrena vista;
> ma la spera stellata ei non ingombra,
> in cui riguardo egli d’un punto è l’ombra.
>
>
>
>
>
> Or questi effetti indubitati e veri
> son del nostro saper fondate prove,
> poiché la stella unita a i segni alteri
> tra quei scintilla, e si dimostra, e move;
> né da gl’Indi scorrendo a i lidi Iberi,
> fia chi di mole o forma altra la trove,
> ché simil varietà non cape il senso,
> né la distanza di quell’Orbe immenso.
>
>
>
>
>
> Dunque, o Maestro di color che sanno,
> sgombra le nubi omai da l’intelletto,
> e se credesti, con tuo scorno e danno,
> dianzi immutabil quel soperno oggetto,
> or che fiamma novella apre l’inganno
> fatto il confessa, e a corruttion soggetto,
> e rendi gratie a la Natura Madre,
> ond’hai luci del ver tanto leggiadre.
>

## Acknowlegedments

I thank Selenia Broccio and Marianna Giannicolo for their help, and Sandro
Bettini, Michele Camerota, Luisa Cifarelli, Ivano Paccagnella and Bill Shea
for their comments and suggestions (Bill also for many pleasant discussions).

This article is dedicated to the 800th anniversary of the University of
Padova, and has been written in the framework of the celebrations. Universa
universis patavina libertas.

## References

  * [1] Alessandro De Angelis and Selenia Broccio First Observations of SN 1604 (Kepler’s Supernova), Universe74302021
  * [2] Antonio Favaro ed. Edizione Nazionale delle Opere di Galileo Galilei, vol. I-XX (Barbera, Firenze 1890–1909)
  * [3] Alessandro De Angelis I diciotto anni migliori della mia vita (Castelvecchi, Rome 2021)
  * [4] Alessandro De Angelis Galileo e la supernova del 1604, con una traduzione del Dialogo de Cecco di Ronchitti da Bruzene (Castelvecchi, Rome 2022)
  * [5] Antonio Lorenzini Discorso intorno alla nuova stella (Tozzi, Padua 1605)
  * [6] Dialogo de Cecco di Ronchitti da Bruzene in perpuosito de la stella nuova (Tozzi, Padua 1605)
  * [7] Dialogo de Cecco di Ronchitti da Bruzene in perpuosito de la stella nuova (Merlo, Verona 1605)
  * [8] Niccolò Gherardini Vita del Signor Galileo Galilei (Firenze 1653-1564), in G. Targioni Tozzetti, Notizie degli aggrandimenti delle scienze fisiche accaduti in Toscana nel corso di anni LX del secolo XVII, II (Firenze 1780)
  * [9] Antonio Favaro ed. Miscellanea galileiana inedita (Antonelli, Venice 1887)
  * [10] Lorenzo Tomasin “Galileo e il pavano: un consuntivo”, Lingua Nostra XXXIX 22 2008
  * [11] Matteo Cosci “Disputatio accademica e poesia”, in I generi dell’aristotelismo volgare nel Rinascimento, M. Sgarbi ed. (La filosofia e il suo passato 66, CLEUP, Padova 2018, and other works by Cosci
  * [12] Stillman Drake Galileo Against the Philosophers (Zeitlin & VerBrugge, Los Angeles 1976)
  * [13] Antonio Daniele Intorno a Galileo (CLEUP, Padova 2022)
  * [14] Marisa Milani (editor and translator) Dialogo de Cecco di Ronchitti da Bruzene in perpuosito de la stella nuova (Editoriale Programma, Padova 1992)
  * [15] Cambridge Italian-English Dictionary, https://dictionary.cambridge.org