2D Disk Research Articles

Graphlet-based hyperbolic embeddings capture evolutionary dynamics in genetic networks.

Spatial Analysis of Functional Enrichment (SAFE) is a popular tool for biologists to investigate the functional organisation of biological networks via highly intuitive 2D functional maps. To create these maps, SAFE uses Spring embedding to project a given network into a 2D space in which nodes connected in the network are near each other in space. However, many biological networks are scale-free, containing highly connected hub nodes. Because Spring embedding fails to separate hub nodes, it provides uninformative embeddings that resemble a "hairball". In addition, Spring embedding only captures direct node connectivity in the network and does not consider higher-order node wiring patterns, which are best captured by graphlets, small, connected, non-isomorphic, induced subgraphs. The scale-free structure of biological networks is hypothesised to stem from an underlying low-dimensional hyperbolic geometry, which novel hyperbolic embedding methods try to uncover. These include coalescent embedding, which projects a network onto a 2D disk. To better capture the functional organisation of scale-free biological networks, whilst also going beyond simple direct connectivity patterns, we introduce Graphlet Coalescent (GraCoal) embedding, which embeds nodes nearby on a disk if they frequently co-occur on a given graphlet together. We use GraCoal to extend SAFE-based network analysis. Through SAFE-enabled enrichment analysis, we show that GraCoal outperforms graphlet-based Spring embedding in capturing the functional organisation of the genetic interaction networks of fruit fly, budding yeast, fission yeast and E. coli. We show that depending on the underlying graphlet, GraCoal embeddings capture different topology-function relationships. We show that triangle-based GraCoal embedding captures functional redundancies between paralogs. https://gitlab.bsc.es/swindels/gracoal_embedding. Supplementary data are available at Bioinformatics online.

High-resolution 3D spatial distribution of complex microbial colonies revealed by mass spectrometry imaging

IntroductionBacterial living states and the distribution of microbial colony signaling molecules are widely studied using mass spectrometry imaging (MSI). However, current approaches often treat 3D colonies as flat 2D disks, inadvertently omitting valuable details. The challenge of achieving 3D MSI in biofilms persists due to the unique properties of microbial samples. ObjectivesThe study aimed to develop a new biofilm sample preparation method that can realize high-resolution 3D MSI of bacterial colonies to reveal the spatial organization of bacterial colonies. MethodsThis article introduces the moisture-assisted cryo-section (MACS) method, enabling embedding-free sectioning parallel to the growth plane. The MACS method secures intact sections by controlling ambient humidity and slice thickness, preventing molecular delocalization. ResultsCombined with matrix-assisted laser desorption ionization mass spectrometry (MALDI)-MSI, the MACS method provides high-resolution insights into endogenic and exogenous molecule distributions in Pseudomonas aeruginosa (P. aeruginosa) biofilms, including isomeric pairs. Moreover, analyzed colonies are revived into 3D models, vividly depicting molecular distribution from inner to outer layers. Additionally, we investigated metabolite spatiotemporal dynamics in multiple colonies, observing changes over time and distinct patterns in single versus merged colonies. These findings shed light on the repel-merge process for multi-colony formation. Furthermore, our study monitored chemical responses inside biofilms after antibiotic treatment, showing increased antibiotic levels in the outer biofilm layer over time while maintaining low levels in the inner region. Moreover, the MACS method demonstrated its universality and applicability to other bacterial strains. ConclusionThese results unveil complex cell activities within biofilm colonies, offering insights into microbe communities. The MACS method is universally applicable to loosely packed microorganism colonies, overcoming the limitations of previously reported MSI methods. It has great potential for studying bacterial-infected cancer tissues and artificial organs, making it a valuable tool in microbiological research.

The Santa Barbara Binary−disk Code Comparison

We have performed numerical calculations of a binary interacting with a gas disk, using 11 different numerical methods and a standard binary−disk setup. The goal of this study is to determine whether all codes agree on a numerically converged solution and to determine the necessary resolution for convergence and the number of binary orbits that must be computed to reach an agreed-upon relaxed state of the binary−disk system. We find that all codes can agree on a converged solution (depending on the diagnostic being measured). The zone spacing required for most codes to reach a converged measurement of the torques applied to the binary by the disk is roughly 1% of the binary separation in the vicinity of the binary components. For our disk model to reach a relaxed state, codes must be run for at least 200 binary orbits, corresponding to about a viscous time for our parameters, 0.2(a 2Ω B /ν) binary orbits, where ν is the kinematic viscosity. The largest discrepancies between codes resulted from the dimensionality of the setup (3D vs. 2D disks). We find good agreement in the total torque on the binary between codes, although the partition of this torque between the gravitational torque, orbital accretion torque, and spin accretion torque depends sensitively on the sink prescriptions employed. In agreement with previous studies, we find a modest difference in torques and accretion variability between 2D and 3D disk models. We find cavity precession rates to be appreciably faster in 3D than in 2D.

Visualizing a bivariate discrete distribution and other distributions derived from it

AbstractA single discrete random variable is depicted by a stick diagram, a 2D picture. Naturally, to visualize a bivariate discrete distribution, one can use a bivariate stick diagram, a 3D picture. Unfortunately, many students have difficulty understanding and processing 3D pictures. Therefore, we construct an alternative 2D disc plot to depict the bivariate distribution of (, ), from which we obtain graphically the conditional distributions of given , and given ; the marginal distributions of , , , . Furthermore, we depict the mean and the standard deviation of each distribution using a single‐headed arrow. We hope these visualizations will help students better comprehend these concepts and avoid some misconceptions.

Mapping the Vertical Gas Structure of the Planet-hosting PDS 70 Disk

PDS 70 hosts two massive, still-accreting planets and the inclined orientation of its protoplanetary disk presents a unique opportunity to directly probe the vertical gas structure of a planet-hosting disk. Here, we use high-spatial-resolution (≈0.″1; 10 au) observations in a set of CO isotopologue lines and HCO+ J = 4−3 to map the full 2D (r, z) disk structure from the disk atmosphere, as traced by 12CO, to closer to the midplane, as probed by less abundant isotopologues and HCO+. In the PDS 70 disk, 12CO traces a height of z/r ≈ 0.3, 13CO is found at z/r ≈ 0.1, and C18O originates at, or near, the midplane. The HCO+ surface arises from z/r ≈ 0.2 and is one of the few non-CO emission surfaces constrained with high-fidelity in disks to date. In the 12CO J = 3−2 line, we resolve a vertical dip and steep rise in height at the cavity wall, making PDS 70 the first transition disk where this effect is directly seen in line-emitting heights. In the outer disk, the CO emission heights of PDS 70 appear typical for its stellar mass and disk size and are not substantially altered by the two inner embedded planets. By combining CO isotopologue and HCO+ lines, we derive the 2D gas temperature structure and estimate a midplane CO snowline of ≈ 56–85 au. This implies that both PDS 70b and 70c are located interior to the CO snowline and are likely accreting gas with a high C/O ratio of ≈ 1.0, which provides context for future planetary atmospheric measurements from, e.g., JWST, and for properly modeling their formation histories.

2D disc modelling of the JWST line spectrum of EX Lupi

We introduce a number of improvements to the thermo-chemical disc modelling code PRODIMO and new theoretical approaches that can be used to better predict and analyse the JWST line spectra of protoplanetary discs. We developed a new line escape probability method for disc geometries, and a new scheme for dust settling, and discuss how to apply UV molecular shielding factors to photo rates in 2D disc geometry. We show that these assumptions are crucial for the determination of gas heating and cooling rates and discuss how they affect the predicted molecular concentrations and line emissions. We apply our revised 2D models to the protoplanetary disc around the T Tauri star EX Lupi in quiescent state. We calculate infrared line emission spectra between 5 and 20 µm from CO, H2O, OH, CO2, HCN, C2H2, and H2, including lines of atoms and ions, using our full 2D predictions of molecular abundances, dust opacities, and gas and dust temperatures. We developed a disc model with a slowly increasing surface density structure around the inner rim that can simultaneously fit the spectral energy distribution, the overall shape of the JWST spectrum of EX Lupi, and the main observed molecular characteristics in terms of column densities, emitting areas, and molecular emission temperatures, which all result from one consistent disc model. The spatial structure of the line-emitting regions of the different molecules is discussed. High abundances of HCN and C2H2 are caused in the model by stellar X-ray irradiation of the gas around the inner rim.

Machine learning for interpreting coherent X-ray speckle patterns

Speckle patterns produced by coherent X-ray have a close relationship with the internal structure of materials but quantitative inversion of the relationship to determine structure from speckle patterns is challenging. Here, we investigate the link between coherent X-ray speckle patterns and sample structures using a model 2D disk system and explore the ability of machine learning to learn aspects of the relationship. Specifically, we train a deep neural network to classify the coherent X-ray speckle patterns according to the disk number density in the corresponding structure. It is demonstrated that the classification system is accurate for both non-disperse and disperse size distributions.

Dependence of galactic bars on the tidal density field in the SDSS

ABSTRACT As a key driver of the secular evolution of disc galaxies, bar formation is potentially linked to the surrounding tidal field. We systematically investigate the dependence of bars on both the small (${\lt }2\, \mathrm{Mpc}\,h^{-1}$) and large (${\gt }5\, \mathrm{Mpc}\,h^{-1}$) scale tidal fields using galaxies observed in the range 0.01 < z < 0.11 by the Sloan Digital Sky Survey (SDSS). We characterize bar strength using the ellipticity of the isophote that corresponds to each bar, ebar, derived from its galaxy image after subtracting the 2D disc component. We demonstrate the efficacy of our bar detection method by performing an extensive comparison with the visual identifications from SDSS and the DESI Legacy Surveys. Using the Yang et al. SDSS group catalogue, we confirm the results from a recent study that the average ebar of galaxies within interacting clusters is higher than that within isolated ones at 0.01 < z < 0.06, but this small-scale tidal enhancement of bars disappears after we increase the cluster sample by a factor of 5 to z = 0.11. On large scales, we explore the dependence of ebar on α5, the tidal anisotropy of the density field defined over $5\, \mathrm{Mpc}\,h^{-1}$. We do not detect any such dependence for 98 per cent of the galaxies with α5 < 10. Intriguingly, among the 2 per cent with α5 ≥ 10, we detect some hint of a boost in bar strength in the underdense regions and a suppression in the overdense regions. Combining our results on both scales, we conclude that there is little evidence for the tidal dependence of bar formation in the local Universe, except for the extremely anisotropic environments.

A recipe for orbital eccentricity damping in the type-I regime for low-viscosity 2D discs

Context. It is well known that partial and deep gap opening depends on a disc’s viscosity; however, damping formulas for orbital eccentricities have only been derived at high viscosities, ignoring partial gap opening. Aims. In this work, we aim to obtain a simple formula to model eccentricity damping of the type-I regime in low-viscosity discs, where even small planets of a few to a few tens of Earth masses may start opening partial gaps in the gas surface density around their orbit. Methods. We performed high-resolution, 2D, locally isothermal hydrodynamical simulations of planets with varying masses on fixed orbits in discs with varying aspect ratios and viscosities. We determined the torque and power felt by the planet to ultimately derive migration and eccentricity damping timescales. Results. We first find a lower limit to the gap depths below which vortices appear; this happens roughly at the transition between type-I and classical type-II migration regimes. For the simulations that remain stable, we obtain a fit to the observed gap depth in the limit of vanishing eccentricities that is similar to the one currently used in the literature but accurate down to α = 3.16 × 10−5. We then record the eccentricity damping efficiency as a function of the observed gap depth and the initial eccentricity. When the planet has opened a deep enough gap such that the surface density is less than ~80% of the unperturbed disc surface density, a clear linear trend is observed independently of the planet’s eccentricity; at shallower gaps, this linear trend is preserved at low eccentricities, while it deviates to more efficient damping when e is comparable to the disc’s scale height. Both trends can be understood on theoretical grounds and are reproduced by a simple fitting formula. Conclusions. Our combined fits for the gap depth and eccentricity damping efficiency yield a simple recipe to implement type-I eccentricity damping in N-body codes in the case of partial gap opening planets that is consistent with high-resolution 2D hydrodynamical simulations. The typical error of the final fit is of the order of a few percent, and at most ~20%, which is the error of type-I torque formulas widely used in the literature. This will allow a more self-consistent treatment of planet-disc interactions of the type-I regime for population synthesis models at low viscosities.

A geometric probabilistic approach to random packing of hard disks in a plane.

In this paper, the random packing fraction of hard disks in a plane is analyzed, following a geometric probabilistic approach. First, the random close packing (RCP) of equally sized disks is modelled. Subsequently, following the same methodology, a simple, statistical geometric model is proposed for the random loose packing (RLP) of monodisperse disks. This very basic derivation of RLP leads to a packing value (≈0.66) that is in very good agreement with values that have been obtained previously for 2D disk packings. The present geometrical model also enables a closed-form expression for the contact (coordination) number as a function of the packing density at different states of compaction. These predictions are thoroughly compared with empirical and simulation results, among others the Rényi parking model, yielding good agreement.

Adjoint optimisation for wind farm flow control with a free-vortex wake model

Wind farm flow control aims to improve wind turbine performance by reducing aerodynamic wake interaction between turbines. Dynamic, physics-based models of wind farm flows have been essential for exploring control strategies such as wake redirection and dynamic induction control. Free-vortex methods can provide a computationally efficient way to model wind turbine wake dynamics for control optimisation. We present a control-oriented free-vortex wake model of a 2D and 3D actuator disc to represent wind turbine wakes. The novel derivation of the discrete adjoint equations allows efficient gradient evaluation for gradient-based optimisation in an economic model-predictive control algorithm. Initial results are presented for mean power maximisation in a two-turbine case study. An induction control signal is found using the 2D model that is roughly periodic and supports previous results on dynamic induction control to stimulate wake mixing. The 3D model formulation effectively models a curled wake under yaw misalignment. Under time-varying wind direction, the optimisation finds solutions demonstrating both wake steering and a smooth transition to greedy control. The free-vortex wake model with gradient information shows potential for efficient optimisation and provides a promising way to further explore dynamic wind farm flow control.

Forced eccentricity in circumbinary discs

ABSTRACT We analyse the eccentric response of a low-mass coplanar circumbinary disc to secular tidal forcing by a Keplerian eccentric orbit central binary. The disc acquires a forced eccentricity whose magnitude depends on the properties of the binary and disc. The largest eccentricities occur when there is a global apsidal resonance in the disc. The driving frequency by the binary is its apsidal frequency, which is equal to zero. A global resonance occurs when the disc properties permit the existence of a zero apsidal frequency free eccentric mode. Resonances occur for different free eccentric modes, which differ in the number of radial nodes. For a disc not at resonance, the eccentricity distribution has somewhat similar form to the eccentricity distributions in discs at resonance that have the closest matching disc aspect ratios. For higher disc aspect ratios, the forced eccentricity distribution in a 2D disc is similar to that of the fundamental free mode. The forced eccentricity distribution in a 3D disc is similar to that of higher order free modes, not the fundamental mode, unless the disc is very cool. For parameters close to resonance, large phase shifts occur between the disc and binary eccentricities that are locked in phase. Forced eccentricity may play an important role in the evolution of circumbinary discs and their central binaries.

Photoelastic stress response of complex 3D-printed particle shapes

Stress visualization within 3-dimensional particles could provide significant insights for understanding of the behaviors of complex particles. However traditional photoelastic methods are only able to produce simple particle shapes. Recently, 3D-printing has created new possibilities for enhancing the scope of stress analysis of physically representative particles. We report the results of X-ray computed tomography and 3D-printing, combined with traditional photoelastic analysis, to visualize strain for particles ranging from simple 2D discs to complex 3D-printed coffee beans, including internal voids. The relative orientation of the print layers and the loading force can affect the optical response of the discs without affecting the mechanical properties. Furthermore, a semi-quantitative measurement of the generated stresses within 3D-printed complex particle is presented. Potential limitations and areas of future interest for stress visualization of 3-dimensional particles are also outlined. • 3D printing photoelastic UV-curable resin enables stress analysis of complex particles. • Ray tracing confirms shadow at multiple faces of complex particles is from refraction. • Print orientation affects optical properties of 3D printed discs with applied load. • Semi-quantitative stress measurement can be obtained from complex 3D printed particles.

Epitaxial Orientation Angle Tuned Disk-on-Rod Nanoheterostructures for Boosting Charge Transfer.

Controlling the compositions of Se(VI) and Te(VI) ions in a 2D disk on 1D structures of Sb(V) chalcogenides, disk-on-rod heterostructures having three different epitaxial angles with different surface facets are reported. Te injection temperature determined the composition, ensuring heterostructure formation with trigonal Sb2SexTe3-x disks on orthorhombic Sb2Se3 rods having orientation angles 180°, 135°, and 90°. The growth kinetics of disks connected at one/two heads of parent rods is manipulated using an Se precursor as a limiting reagent. Theoretical calculations established the energy minimization of different orientations, their possible formation, and suitability in energy transfer applications. Electrochemical measurements were also in agreement with theoretical calculations. Hence, this is a case study of advanced modular synthesis of disk-on-rod nanostructures, leading a step further in nanocrystal engineering for more desirable complex structures and their charge transfer property.

Tuning Crystal Plane Orientation in Multijunction and Hexagonal Single Crystalline CsPbBr3 Perovskite Disc Nanocrystals

Two-dimensional-shaped CsPbBr3 platelet nanocrystals are widely studied for their bright high energy emission and self-assembly. These nanostructures are in orthorhombic phase, have a square shape, and have the vertical axis [001] perpendicular to the basal plane. Moreover, these are mostly single-crystalline structures with a continuous lattice and appear like slices of cube nanocrystals. In contrast, herein, multijunction and hexagonal single crystalline 2D discs of CsPbBr3 are reported to have all their vertical axes [100]. These are obtained by using the perovskite derivative of tetragonal Cs3MnBr5 as the parent material and subsequent B-site Pb(II) introduction in the presence of phenacyl bromide at different reaction temperatures. At low temperature, multijunction discs having random orientations of two horizontal axes [010] and [001] from one to another segment are observed. Orientations of planes remained random as both coherent and incoherent twin planes were observed at their boundaries. However, the number of junctions/segments was reduced at higher temperature, and finally hexagonal single crystalline discs remained as the ultimate product. Analysis suggested that the crystal nature of parent Cs3MnBr5 and temperature-dependent variation in the rate of Pb(II) insertions determined the nature of discs having randomly oriented or static planes in the entire nanostructure. Not only in 2D discs but also, 3D nanocrystals having similar segments with different orientations are formed upon Pb(II) exchange with Mn(II) alloyed cubic CsBr. Hexagonal single crystalline and segmented multijunction CsPbBr3 discs remain unique among 2D perovskites nanostructures, and their formation mechanism indeed introduced new fundamentals of the crystallization process of these emerging energy materials.

Investigating Cancerous Exosomes' Effects on CD8+ T-Cell IL-2 Production in a 3D Unidirectional Flow Bioreactor Using 3D Printed, RGD-Functionalized PLLA Scaffolds.

Exosomes from cancer cells are implicated in cancer progression and metastasis, carrying immunosuppressive factors that limit the antitumor abilities of immune cells. The development of a real-time, 3D cell/scaffold construct flow perfusion system has been explored as a novel tool in the study of T-cells and exosomes from cancer cells. Exosomes from human lung cancer (H1299 and A549) cells were co-cultured in a unidirectional flow bioreactor with CD8+ T-cells immobilized onto 3D-printed RGD-functionalized poly(L-lactic) acid (PLLA) scaffolds and assessed for IL-2 production. The IL-2 production was investigated for a wide range of T-cell to exosome ratios. With the successful incorporation of the RGD binding motif onto the PLLA surface at controllable densities, CD8+ T-cells were successfully attached onto 2D disks and 3D printed porous PLLA scaffolds. T-cell attachment increased with increasing RGD surface density. The diameter of the attached T-cells was 7.2 ± 0.2 µm for RGD densities below 0.5 nmoles/mm2 but dropped to 5.1 ± 0.3 µm when the RGD density was 2 nmoles/mm2 due to overcrowding. The higher the number of cancer exosomes, the less the IL-2 production by the surface-attached T-cells. In 2D disks, the IL-2 production was silenced for T-cell to exosome ratios higher than 1:10 in static conditions. IL-2 production silencing in static 3D porous scaffolds required ratios higher than 1:20. The incorporation of flow resulted in moderate to significant T-cell detachment. The portions of T-cells retained on the 3D scaffolds after exposure for 4 h to 0.15 or 1.5 mL/min of perfusion flow were 89 ± 11% and 30 ± 8%, respectively. On 3D scaffolds and in the presence of flow at 0.15 ml/min, both H1299 and A549 cancerous exosomes significantly suppressed IL-2 production for T-cell to exosome ratios of 1:1000. The much higher level of exosomes needed to silence the IL-2 production from T-cells cultured under unidirectional flow, compared to static conditions, denotes the importance of the culturing conditions and the hydrodynamic environment, on the interactions between CD8+ T-cells and cancer exosomes.

Survival of planet-induced vortices in 2D disks

Context. Several observations of protoplanetary disks reveal non-axisymmetric features, which are often interpreted as vortices. Numerical modeling has repeatedly shown that gap-opening planets are capable of producing large and long-lasting vortices at their outer gap edge, making massive planets popular candidates as the source of such features. Aims. We explore the lifetime of vortices generated by Jupiter-sized planets as a function of the thermal relaxation timescale, the level of turbulence, and the effect of disk self-gravity. Methods. We conduct 2D numerical simulations using the hydrodynamics codes PLUTO and FARGO, scanning through several physical and numerical parameters. Vortex properties are automatically extracted from thousands of simulation snapshots. Results. We find that vortices that spawn at the outer gap edge can survive for about 100–3000 planetary orbits, with the shortest lifetimes occurring for moderately efficient dissipation and cooling. However, we also observe a different regime of long-lasting vortices with lifetimes of at least 15 000 orbits for very low viscosity and very short thermal relaxation timescales. Disk self-gravity significantly shortens the lifetime of regular vortices but still allows long-lived ones to survive. Conclusions. Our results suggest that the cooling timescale plays an important role in vortex formation and lifetime and that planet-generated vortices should be observable at large distances from the star for typical thermal relaxation timescales and low turbulence levels.

Molecules with ALMA at Planet-forming Scales (MAPS). IV. Emission Surfaces and Vertical Distribution of Molecules

Abstract The Molecules with ALMA at Planet-forming Scales (MAPS) Large Program provides a unique opportunity to study the vertical distribution of gas, chemistry, and temperature in the protoplanetary disks around IM Lup, GM Aur, AS 209, HD 163296, and MWC 480. By using the asymmetry of molecular line emission relative to the disk major axis, we infer the emission height (z) above the midplane as a function of radius (r). Using this method, we measure emitting surfaces for a suite of CO isotopologues, HCN, and C2H. We find that 12CO emission traces the most elevated regions with z / r > 0.3 , while emission from the less abundant 13CO and C18O probes deeper into the disk at altitudes of z / r ≲ 0.2 . C2H and HCN have lower opacities and signal-to-noise ratios, making surface fitting more difficult, and could only be reliably constrained in AS 209, HD 163296, and MWC 480, with z / r ≲ 0.1 , i.e., relatively close to the planet-forming midplanes. We determine peak brightness temperatures of the optically thick CO isotopologues and use these to trace 2D disk temperature structures. Several CO temperature profiles and emission surfaces show dips in temperature or vertical height, some of which are associated with gaps and rings in line and/or continuum emission. These substructures may be due to local changes in CO column density, gas surface density, or gas temperatures, and detailed thermochemical models are necessary to better constrain their origins and relate the chemical compositions of elevated disk layers with those of planet-forming material in disk midplanes. This paper is part of the MAPS special issue of the Astrophysical Journal Supplement.

The Ponzano–Regge cylinder and propagator for 3d quantum gravity

We investigate the propagator of 3d quantum gravity, formulated as a discrete topological path integral. We define it as the Ponzano–Regge amplitude of the solid cylinder swept by a 2d disk evolving in time. Quantum states for a 2d disk live in the tensor products of N spins, where N is the number of holonomy insertions connecting to the disk boundary. We formulate the cylindric amplitude in terms of a transfer matrix and identify its eigen-modes in terms of spin recoupling. We show that the propagator distinguishes subspaces with different total recoupled spin. This may select the vanishing overall spin sector at late time depending on the chosen cylinder boundary data, leading to an emergent symmetry scenario in the continuum limit. We discuss applications to quantum circuits and the possibility of experimental simulations of this 3d quantum gravity propagator.

Adsorption on Fractal Surfaces: A Non Linear Modeling Approach of a Fractional Behavior

This article deals with the random sequential adsorption (RSA) of 2D disks of the same size on fractal surfaces with a Hausdorff dimension 1<d<2. According to the literature and confirmed by numerical simulations in the paper, the high coverage regime exhibits fractional dynamics, i.e., dynamics in t−1/d where d is the fractal dimension of the surface. The main contribution this paper is that it proposes to capture this behavior with a particular class of nonlinear model: a driftless control affine model.