This master thesis deals with conics in the real projective plane. The goal was to com- prehensibly introduce conics in the projective plane to high-school students and teachers. In order to fulfill this goal, the projective plane and homogenous coordinates were intro- duced, and harmonic set and priniple of duality were studied closely. The conics in the projective plane were approached from the perspective of history, and various definitions. Well-motivated introduction of a pole and a polar was emphasized.

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<p> We look at methods to generate uniformly distributed points from the classical matrix groups, spheres, projective spaces, and Grassmannians. We motivate the discussion with a number of applications ranging from number theory to wireless communications. The uniformity of the samples and the efficiency of the algorithms are compared. </p> / Thesis / Master of Science (MSc)

Yes / The Desargues property is well known in the context of projective geometry. An analogous property is presented in the context of both classical and Quantum Physics. In a classical context, the Desargues property implies that two logical circuits with the same input, show in their outputs selective correlations. In general their outputs are uncorrelated, but if the output of one has a particular value, then the output of the other has another particular value. In a quantum context, the Desargues property implies that two experiments each of which involves two successive projective measurements, have selective correlations. For a particular set of projectors, if in one experiment the second measurement does not change the output of the rst measurement, then the same is true in the other experiment.

[Truncated abstract] Image transition effects are commonly used on television and human computer interfaces. The transition between images creates a perception of continuity which has aesthetic value in special effects and practical value in visualisation. The work in this thesis demonstrates that better image transition effects are obtained by incorporating properties of projective geometry into image transition algorithms. Current state-of-the-art techniques can be classified into two main categories namely shape interpolation and warp generation. Many shape interpolation algorithms aim to preserve rigidity but none preserve it with perspective effects. Most warp generation techniques focus on smoothness and lack the rigidity of perspective mapping. The affine transformation, a commonly used mapping between triangular patches, is rigid but not able to model perspective effects. Image transition techniques from the view interpolation community are effective in creating transitions with the correct perspective effect, however, those techniques usually require more feature points and algorithms of higher complexity. The motivation of this thesis is to enable different views of a planar surface to be interpolated with an appropriate perspective effect. The projective geometric relationship which produces the perspective effect can be specified by two quadrilaterals. This problem is equivalent to finding a perspectively appropriate interpolation for projective transformation matrices. I present two algorithms that enable smooth perspective transition between planar surfaces. The algorithms only require four point correspondences on two input images. ...The second algorithm generates transitions between shapes that lie on the same plane which exhibits a strong perspective effect. It recovers the perspective transformation which produces the perspective effect and constrains the transition so that the in-between shapes also lie on the same plane. For general image pairs with multiple quadrilateral patches, I present a novel algorithm that is transitionally symmetrical and exhibits good rigidity. The use of quadrilaterals, rather than triangles, allows an image to be represented by a small number of primitives. This algorithm uses a closed form force equilibrium scheme to correct the misalignment of the multiple transitional quadrilaterals. I also present an application for my quadrilateral interpolation algorithm in Seitz and Dyer's view morphing technique. This application automates and improves the calculation of the reprojection homography in the postwarping stage of their technique. Finally I unify different image transition research areas into a common framework, this enables analysis and comparison of the techniques and the quality of their results. I highlight that quantitative measures can greatly facilitate the comparisons among different techniques and present a quantitative measure based on epipolar geometry. This novel quantitative measure enables the quality of transitions between images of a scene from different viewpoints to be quantified by its estimated camera path.

La thèse traite des arrêts non éclairés de cure psychothérapeutique, c'est-à-dire des interruptions définitives à l'initiative exclusive du patient ou de la patiente, sans discussion avec le clinicien ou la clinicienne. Pour la première fois, une telle étude est basée sur des témoignages de patient(e)s, recueillis lors d'entretiens cliniques après consentement éclairé. Au plan théorique, l'étude s'appuie sur une approche métapsychologique informationnelle originale de la psyché comme organe de modélisation du monde par la réalité psychique, inspirée par un historique de l'évolution des notions de projection et d'identification projective en psychanalyse. Pour l'ensemble du panel clinique de l'étude, les arrêts non éclairés de cure résultent systématiquement de la perte de confiance du patient en raison d'un trait disqualifiant imputé au clinicien par transfert objectal ou par identification projective, avec une conviction remarquablement forte, inébranlable et durable que le trait en question appartient effectivement à la personne du clinicien. Au terme de l'étude, il apparaît que les personnes ayant traversé des expériences traumatiques relationnelles dans leur enfance (rupture de continuité relationnelle avec l'environnement parental) pourraient être particulièrement exposées aux risques d'arrêts non éclairés lorsqu'elles entreprennent une psychanalyse ou une psychothérapie. En conséquence, l'étude propose des mesures de métacommunication préventive et curative visant à réduire un tel risque. Ces conclusions nécessiteraient d'être affinées par une recherche ultérieure plus large incluant notamment une comparaison avec une population témoin de patient(e)s n'ayant pas connu d'arrêts non éclairés dans leur(s) cure(s), afin de préciser davantage les spécificités des patient(e)s qui en provoquent et de mieux caractériser les traumatismes relationnels précoces dont ces dernier(e)s ont pu faire l'expérience. / This work is about unilateral treatment abortions by the patient, occurring without preliminary discussion with the therapist. For the first time, such a research is based upon testimonies from patients collected during clinical interviews with their informed consent. Theoretically, the study relies on an original informational approach of the psyche as an organ which models the outside world by the psychical reality, approach inspired by an historical survey of projection and projective identification psychoanalytical concepts evolutions. For the entire study sample, unilateral treatment abortion by the patient always derives from loss of confidence in the therapist caused by the attribution to him or her of a disqualifying feature through object transference or projective identification channel, with a remarkably strong, unwavering and long-lasting conviction that the disqualifying feature truly belongs to the therapist. At the end of the study, it appears that the persons who have had relational traumatic experiences during childhood (breaks in the continuity of relations with parental environment) would be particularly prone to unilateral treatment abortions during a psychoanalysis or a psychotherapy. Consequently, the study proposes preventive and curatives measures of metacommunication in order to reduce risk of treatment abortion. The conclusions of this study would require further wider research, involving particularly a control population of patients who never experimented unilateral treatment abortion during their psychoanalysis/psychotherapy, in order to more clearly identify the specificities of the patients who provoke such treatment abortions and the exact nature of the relational trauma which they may have experimented during childhood.

Résumé : Dans cette thèse j'ai travaillé sur deux problèmes différents dans le domaine de la Géométrie Algébrique. La première partie de cette thèse consiste dans l'étude de la stabilité des images inverses du fibré tangent de l'espace projectif sur des variétés projectives. La stabilité de ces fibrés est équivalente à celle du noyau du morphisme d'évaluation M associé à un fibré en droites L engendré par ses sections globales. On obtient un résultat optimal dans le cas des courbes projectives et ensuite on utilise ce résultat pour en déduire la stabilité dans le cas des quelques surfaces projectives, notamment K3 et abéliennes. Un second problème que nous abordons est l'étude du lieu fixe d'une involution symplectique d'une variété irréductible holomorphe symplectique de dimension 4 telle que b2 = 23. On montre qu'il y a seulement trois cas possibles pour le nombre des points fixes isolés et des surfaces K3 fixées. On conjecture que seulement un cas soit possible, celui avec 28 points fixes isolés et une surface K3 fixée, et qu'une telle involution ne fixe jamais une surface abélienne. On vérifie cette conjecture dans quelques exemples.

Soit k un corps algébriquement clos de caractéristique 0 et F une suite de n polynômes en intersection complète sur k[X1,...,Xn]. Le Bezoutien de F fournit une forme dualisante sur k[X]/<F> appelée symbole de Kronecker, qui est un analogue algébrique du résidu. L'objet de ce travail est de construire et calculer le symbole de Kronecker dans le tore (C*)n relativement à une famille f de n polynômes de Laurent en n variables. La famille f possède un nombre fini de zéros et est régulière pour ses polytopes de Newton. La représentation du résidu global dans le tore à l'aide d'un résidu torique, donnée par Cattani et Dickenstein, suggère d'interpréter le symbole de Kronecker dans le tore dans la variété torique projective définie par le polytope P, somme de Minkowski des polytopes de Newton de f.Lorsque P est premier, Roy et Szpirglas ont défini le symbole de Kronecker dans le tore à partir des symboles de Kronecker définis sur les ouverts affines de la variété torique Xp relativement à une famille de n + 1 polynômes homogènes sans zéros communs dans la variété Xp. Nous montrons ici que le cas « P non premier » est réductible au cas précédent en explicitant les morphismes d'éclatement qui traduisent le raffinement de l’éventail de Xp en un éventail simplicial. / Let k be an algebraically closed field with char(k) = 0 and let be polynomials F1,..., Fn such that k[X1,...,Xn]/<F1,..., Fn> is a complete intersection k-algebra. The Bezoutian of F1,..., Fn gives a dualizing form acting on k[X1,...,Xn]/<F1,..., Fn> called Kronecker symbol. It is an algebraic analogue of residue. The aim of this work is to build and calculate the Kronecker symbol in the torus (C*)n for a system f of Laurent polynomials with a a finite set of zeroes and regular for its Newton polytopes. In the same way as Cattani and Dickenstein have done for the global residue in the torus, we consider the projective variety given by the Minkowski sum P of the Newton polytopes of f in order to build the Kronecker symbol in the torus.When P is prime, Roy and Szpirglas have defined the Kronecker symbol in the torus from Kronecker symbols on affine subsets of Xp for a system of n+1 homogeneous polynomials with no common zeroes in XP . We prove that the case "P no prime" can be reduced to the previous case by using simplicial refinements of the fan of Xp and making explicit the associated toric morphisms on the total coordinate spaces.