Perspex machine

We introduce the perspex machine which unifies projective geometry and the Turing machine, resulting in a supra-Turing machine. Specifically, we show that a Universal Register Machine (URM) can be implemented as a conditional series of whole numbered projective transformations. This leads naturally to a suggestion that it might be possible to construct a perspex machine as a series of pin-holes and stops. A rough calculation shows that an ultraviolet perspex machine might operate up to the petahertz range of operations per second. Surprisingly, we find that perspex space is irreversible in time, which might make it a candidate for an anisotropic spacetime geometry in physical theories. We make a bold hypothesis that the apparent irreversibility of physical time is due to the random nature of quantum events, but suggest that a sum over histories might be achieved by sampling fluctuations in the direction of time flow. We propose an experiment, based on the Casimir apparatus, that should measure fluctuations of time flow with respect to time duration- if such fluctuations exist.

Robot free will

We introduce the perspex machine which unifies projective geometry and Turing computation and results in a supra-Turing machine. We show two ways in which the perspex machine unifies symbolic and non-symbolic AI. Firstly, we describe concrete geometrical models that map perspexes onto neural networks, some of which perform only symbolic operations. Secondly, we describe an abstract continuum of perspex logics that includes both symbolic logics and a new class of continuous logics. We argue that an axiom in symbolic logic can be the conclusion of a perspex theorem. That is, the atoms of symbolic logic can be the conclusions of sub-atomic theorems. We argue that perspex space can be mapped onto the spacetime of the universe we inhabit. This allows us to discuss how a robot might be conscious, feel, and have free will in a deterministic, or semi-deterministic, universe. We ground the reality of our universe in existence. On a theistic point, we argue that preordination and free will are compatible. On a theological point, we argue that it is not heretical for us to give robots free will. Finally, we give a pragmatic warning as to the double-edged risks of creating robots that do, or alternatively do not, have free will.

Construção de mosaicos georreferenciados usando imagens aéreas de pequeno formato

In this work we propose a technique that uses uncontrolled small format aerial images, or SFAI, and stereohotogrammetry techniques to construct georeferenced mosaics. Images are obtained using a simple digital camera coupled with a radio controlled (RC) helicopter. Techniques for removing common distortions are applied and the relative orientation of the models are recovered using projective geometry. Ground truth points are used to get absolute orientation, plus a definition of scale and a coordinate system which relates image measures to the ground. The mosaic is read into a GIS system, providing useful information to different types of users, such as researchers, governmental agencies, employees, fishermen and tourism enterprises. Results are reported, illustrating the applicability of the system. The main contribution is the generation of georeferenced mosaics using SFAIs, which have not yet broadly explored in cartography projects. The proposed architecture presents a viable and much less expensive solution, when compared to systems using controlled pictures

Geometrical wave equation and the cauchy-like theorem for octonions

Riemann surfaces, cohomology and homology groups, Cartan's spinors and triality, octonionic projective geometry, are all well supported by Complex Structures [1], [2], [3], [4]. Furthermore, in Theoretical Physics, mainly in General Relativity, Supersymmetry and Particle Physics, Complex Theory Plays a Key Role [5], [6], [7], [8]. In this context it is expected that generalizations of concepts and main results from the Classical Complex Theory, like conformal and quasiconformal mappings [9], [10] in both quaternionic and octonionic algebra, may be useful for other fields of research, as for graphical computing enviromment [11]. In this Note, following recent works by the autors [12], [13], the Cauchy Theorem will be extended for Octonions in an analogous way that it has recentely been made for quaternions [14]. Finally, will be given an octonionic treatment of the wave equation, which means a wave produced by a hyper-string with initial conditions similar to the one-dimensional case.

Tópicos da geometria projetiva

Pós-graduação em Matemática Universitária - IGCE

Matemática e Arte: interdisciplinaridade

The objective of this study is to perform a theoretical study of the possible relationship between mathematics and art. For some people, the association between these fields of knowledge may simply seem absurd, possibly because they do not know what is in common between an area considered rigorous, rational, objective, and a more emotional, intuitive, subjective. In this context, interdisciplinarity has an important role, in proposing the contextual integration between the subject content, such as projective geometry and the renaissance. Are also presented as examples of this union, the technique of Moses and the geometric patterns of indigenous crafts, as well as the work of artist Maurits Cornelis Escher, a legitimate representative of the relationship between art and mathematics

Path planning for image based visual servoing

This thesis deals with Visual Servoing and its strictly connected disciplines like projective geometry, image processing, robotics and non-linear control. More specifically the work addresses the problem to control a robotic manipulator through one of the largely used Visual Servoing techniques: the Image Based Visual Servoing (IBVS). In Image Based Visual Servoing the robot is driven by on-line performing a feedback control loop that is closed directly in the 2D space of the camera sensor. The work considers the case of a monocular system with the only camera mounted on the robot end effector (eye in hand configuration). Through IBVS the system can be positioned with respect to a 3D fixed target by minimizing the differences between its initial view and its goal view, corresponding respectively to the initial and the goal system configurations: the robot Cartesian Motion is thus generated only by means of visual informations. However, the execution of a positioning control task by IBVS is not straightforward because singularity problems may occur and local minima may be reached where the reached image is very close to the target one but the 3D positioning task is far from being fulfilled: this happens in particular for large camera displacements, when the the initial and the goal target views are noticeably different. To overcame singularity and local minima drawbacks, maintaining the good properties of IBVS robustness with respect to modeling and camera calibration errors, an opportune image path planning can be exploited. This work deals with the problem of generating opportune image plane trajectories for tracked points of the servoing control scheme (a trajectory is made of a path plus a time law). The generated image plane paths must be feasible i.e. they must be compliant with rigid body motion of the camera with respect to the object so as to avoid image jacobian singularities and local minima problems. In addition, the image planned trajectories must generate camera velocity screws which are smooth and within the allowed bounds of the robot. We will show that a scaled 3D motion planning algorithm can be devised in order to generate feasible image plane trajectories. Since the paths in the image are off-line generated it is also possible to tune the planning parameters so as to maintain the target inside the camera field of view even if, in some unfortunate cases, the feature target points would leave the camera images due to 3D robot motions. To test the validity of the proposed approach some both experiments and simulations results have been reported taking also into account the influence of noise in the path planning strategy. The experiments have been realized with a 6DOF anthropomorphic manipulator with a fire-wire camera installed on its end effector: the results demonstrate the good performances and the feasibility of the proposed approach.

Higher mathematics. A text-book for classical and engineering colleges.

"Bibliography ... general works on the history of mathematics in the nineteenth century": p. 568-570.

Lezioni di geometria metrico-proiettiva.

Available on demand as hard copy or computer file from Cornell University Library.

Die Gebilde ersten und zweiten Grades der liniengeometrie in synthetischer Behandlung.

Available on demand as hard copy or computer file from Cornell University Library.

Die darstellende Geometrie in organischer Verbindung mit der Geometrie der Lage /

T. 1. Die Methoden der darstellenden und die Elemente der projectivischen Geometrie.--T. 2. Die darstellende Geometrie der krummen Linien und Flächen.--T. 3. Die construierende und analytische Geometrie der Lage.

Esercizi di geometria analitica e proiettiva;

At head of title: L. Amoroso [ed] E. Bompiani.

Geometria proiettiva ed analitica, lezioni; generalità e geometria di forme reali ad una coordinata reale o complessa.

Mode of access: Internet.

Corso di geometria analitico-proiettiva per gli allievi della R. Accademia Militare.

Available on demand as hard copy or computer file from Cornell University Library.

Die darstellende Geometrie in organischer Verbindung mit der Geometrie der Lage,

Available on demand as hard copy or computer file from Cornell University Library.