Description of the thesis topic

Abstract :
The aim of the project is to propose a new methodology for increasing understanding of non-Euclidean geometries through virtual reality exploration. The novelty of our project will be to set up a programme of multi-sensory interactions in these geometries (perception of sounds, haptic interactions, displacements, movements, etc.) via a headset or an immersive room such as the Immersia platform (www.irisa.fr/immersia). This project is a way of taking a fresh look at ancient mathematics, leading to original and exploratory research problems.

The goal of the project is to propose a new methodology for increasing understanding of understanding of non-Euclidean geometries through exploration in virtual reality (real-time, 3D). The foundations of Euclidean geometry date back to the ancient Greeks. In his work compiling the knowledge of his time, Euclid formulated five postulates from which all the rest follows. For a long time, this foundation was seen as unshakeable. Very early on, however, the fifth postulate many mathematicians sought, in vain, to demonstrate that it could be deduced from the other four. In the 19th century, Gauß, Bolyai and Lobachevski observed that there were other possible geometries, known as non-Euclidean, like so many 'parallel worlds', in which Euclid's fifth postulate was not always true. This discovery fundamentally changed our approach to geometry.
Since then, mathematicians have sought to 'classify' all possible geometries. Thurston's geometrisation conjecture - proved by Perelman - states that any closed, orientable and indecomposable variety of dimension three can be cut along toroids, so that the interior of each subvariety can be given one of eight geometric structures, called Thurston geometries: E3, S3 and H3 isotropic geometries, H2 × E and S2 × E product geometries, and the Lie groups Nil, Sol and SL(2,R).
Inspired by the work of Weeks [12] and Berger [9], an international team of four
mathematicians, including Rémi Coulon (IRMAR), developed software in the form of a web application (http://www.3-dimensional.space) to simulate what an inhabitant would see in each of them [1]. The difficulty lies in the fact that light does not generally move in a straight line. The final graphic rendering is very confusing, even for experts. This shows the extent to which certain aspects of these geometries remain mysterious. The novelty of our project will be to set up a multi-sensory interaction programme in these geometries, in virtual reality via a headset or an immersive room such as the Immersia platform.
IRISA's Seamless team specialises in Virtual Reality and multisensory interaction [2]. Immersia (https://www.irisa.fr/immersia) is a virtual reality research platform based at the INRIA Rennes Bretagne-Atlantique centre and the IRISA laboratory. It consists of large-scale immersive equipment that immerses users in a high-quality visual, auditory and haptic world. This technological tool is 10m wide, 3m high and 3m deep, enabling interaction with a virtual world. It also has a unique two-handed haptic interaction device capable of transporting one a user.
Immersia is first and foremost a research tool, open to the scientific community, whether regional, national or European, academic or industrial. As a member of a national research infrastructure grouping together major visualisation and interaction facilities, it is associated with several PIA projects (DemoES AIR, EUR Digisport, Equipex+ Continuum) and participates in numerous international projects (INTEREG, H2020, JPI, etc.).
Preliminary work has been carried out in 2020/2021 by INSA students who have implemented five of Thurston's eight geometries on the Immersia platform.
The proposed methodology will give mathematicians new keys to developing their intuition. Even more than the final result, it's the road ahead that will be a source of progress.
The problems that need to be solved to bring these geometries to life are often far removed from the usual concerns of mathematicians. Here's a very simple example. To calculate realistic 'shading' of objects in a scene, you need to be able to solve the following problem: describe all the geodesics connecting two given points. Often, the surveyor will content with knowing that there are one or more geodesics connecting his points, or with a 'large-scale' understanding of these geodesics. The computer, on the other hand, needs an explicit and easily calculable description of these paths. In Sol geometry, for example, this problem has not yet been solved. For the moment, we are content with approximate shading.
This project is therefore a way of taking a fresh look at ancient mathematics. It raises original research problems. Here are a few examples of avenues to explore.
Like light, sound waves behave very differently from one geometry from one geometry to another. We want to explore models for the propagation and rendering of sound waves, as well as their 'spatialised' perception.
as well as their 'spatialised' perception in non-Euclidean universes. First, we first step is to analyse the wave equation in Thurston geometries.
The second stage, which involves signal processing, consists of creating the necessary filters, so that when the user emits a sound, the loudspeakers send it back to the user in the form of a signal. so that when the user emits a sound, the loudspeakers return a signal simulating the signal simulating the sound rendition in each of the geometries.
The graphical rendering of Thurston's geometries has already revealed some very surprising behaviours. surprising behaviours that had to be analysed [10,11]. We are extremely curious to see what this feature will contribute to our understanding of non-Euclidean non-Euclidean universes.
When the geometry is not isotropic, all the movements we are used to do not necessarily correspond to isometries. For example, the stabiliser of a point in Sol is a finite group. The result is that something as simple as turning your head cannot be done without deformation. An inhabitant of Sol should therefore feel constraints exerted on them. Certain movements will require more effort than others.
To understand this phenomenon, we need to study the laws of physics on non-Euclidean worlds. Immersia's equipment will make it possible to experiment with haptic navigation, i.e. with a perception of the physical effort required to move.
From the moment we place a user in interaction with an environment, non-Euclidean in our case, a performance problem arises so that this interaction can be carried out in interactive real time (very low latency for the calculation of all the mathematical and/or physical phenomena). Fundamental optimisation work is therefore required to enhance the tool and improve rendering quality, while maintaining the performance required for real-time use. Indeed, if the image refresh rate is not high enough, users could be subject to unpleasant effects (headaches, nausea, etc.) known as cyberkinetosis.
In particular, this means revisiting the mathematical tools used for 3D rendering of geometry, calculating sound propagation, movement in the environment, etc. environment, etc.
This project requires both mathematical and computer skills. The computer is a tool devoid of initiative that simply executes the instructions it is given! So each function we plan to develop will require upstream work at the frontier between geometry, topology, numerical analysis and optimisation, to model the mathematical objects to be implemented and identify the algorithms for calculating them. Rémi Coulon has gained experience in these areas through his previous work.
The Seamless team specialises in virtual reality, augmented reality and multimodal interaction in these environments [3]. As part of this research, the members of the Hybrid team are interested in simulation performance [4], interaction modalities [5], visual and sound perception issues [6], the quality of the user experience [7] and, finally, increasing the productivity of AR/VR applications by proposing software abstractions [8].
The concrete constraints (linked to hardware, performance problems, etc.) will be as many challenges to be solved mathematically.
As part of this project, we are applying for a 3-year PhD contract. This thesis will be directed by Rémi Coulon and Valérie Gouranton. The emphasis will be on the mathematical aspects of the project. It is important to clarify this part before embarking on the machine implementation. The PhD student's task will be to develop the multi-sensory perception and interaction part of the application.
Even though their involvement is not strictly necessary, our American collaborators with whom this project started will be invited to participate. Their expertise in physical interactions in Thurston geometries could be useful.

References
[1] R. Coulon, E. A. Matsumoto, H. Segerman, and S. J. Trettel. Ray-marching Thurston geometries. ArXiv 2010.15801, 2020. à paraître dans Experimental Math.
[2] T. Nicolas, R. Gaugne, C. Tavernier, Q. Petit, V. Gouranton, and B. Arnaldi. Touching and interacting with inaccessible cultural heritage. Presence : Teleoperators and Virtual Environments, 24(3) :265–277, 2015.
[3] Bruno Arnaldi, Pascal Guitton, Guillaume Moreau. Réalité virtuelle et réalité augmentée : mythes et réalités. ISTE éditions, pp.324, 2018
[4] François Lehericey, Valérie Gouranton, Bruno Arnaldi. GPU Ray-Traced Collision Detection: Fine Pipeline Reorganization. Proceedings of 10th International Conference on Computer Graphics Theory and Applications (GRAPP'15), Mar 2015, Berlin, Germany
[5] Morgan Le Chénéchal, Thierry Duval, Valérie Gouranton, Jérôme Royan, Bruno Arnaldi. The Stretchable Arms for Collaborative Remote Guiding. Proceedings of International Conference on Artificial Reality and Telexistence Eurographics Symposium on Virtual Environments, 2015, Kyoto, Japan
[6] Francesco Grani, Ferran Argelaguet Sanz, Valérie Gouranton, Marwan Badawi, Ronan Gaugne, et al.. Audio-Visual Attractors for Capturing Attention to the Screens When Walking in CAVE Systems. Sonic interaction with a virtual orchestra of factory machinery, IEEE VR, Mar 2014, Minneapolis, United States.
[7] Vailland, G., Grzeskowiak, F., Devigne, L., Gaffary, Y., Fraudet, B., Leblong, E., ... & Babel, M. (2019, June). User-centered design of a multisensory power wheelchair simulator: towards training and rehabilitation applications. In 2019 IEEE 16th International Conference on Rehabilitation Robotics (ICORR) (pp. 77-82). IEEE VR
[8] Guillaume Claude, Valérie Gouranton, Rozenn Bouville Berthelot, Bruno Arnaldi, #SEVEN, a Sensor Effector Based Scenarios Model for Driving Collaborative Virtual Environment, ICAT-EGVE, International Conference on Artificial Reality and Telexistence, Eurographics Symposium on Virtual Environments, Dec 2014, Bremen, Germany.
[9] Pierre Berger, Espaces Imaginaires, http://espaces-imaginaires. fr, 2015.
[10] Rémi Coulon, Elisabetta Matsumoto, Henry Segerman, and Steve Trettel, Non-euclidean virtual reality III: Nil, Proceedings of Bridges 2020: Mathematics, Art, Music, Architecture, Education, Culture (Phoenix, Arizona), Tessellations Publishing, 2020, pp. 153–160.
[11] Rémi Coulon, Elisabetta Matsumoto, Henry Segerman, and Steve Trettel, Non-euclidean virtual reality IV: Sol, Proceedings of Bridges 2020: Mathematics, Art, Music, Architecture, Education, Culture (Phoenix, Arizona), Tessellations Publishing, 2020, pp. 161–168.
[12] Jeffrey Weeks, Curved Spaces, a flight simulator for multiconnected universes, available from http://www.geometrygames.org/ CurvedSpaces/

Work Context

Rennes

Constraints and risks

The candidate should be familiar with non-Euclidean geometries and have a thorough knowledge of virtual reality.