In the last two decades, qualitative spatial representation and reasoning, and in particular spatial logics, have been the subject of an increased interest from the Artificial Intelligence community. By a spatial logic, we understand a formal language whose variables range over subsets of a fixed topological space, called regions, and whose non-logical primitives have fixed geometric meanings. A spatial logic for reasoning about regions in a Euclidean space is called a Euclidean spatial logic. We consider first-order and quantifier-free Euclidean spatial logics with primitives for topological relations and operations, the property of convexity and the ternary relation of being closer-than. We mainly focus on the computational properties of such logics, but we also obtain interesting model-theoretic results. We provide a systematic overview of the computational properties of firstorder Euclidean spatial logics and fill in some of the gaps left by the literature. We establish upper complexity bounds for the (undecidable) theories of logics based on Euclidean spaces of dimension greater than one, which yields tight complexity bounds for all but two of these theories. In contrast with these undecidability results, we show that the topological theories based on one-dimensional Euclidean space are decidable, but non-elementary. We also study the computational properties of quantifier-free Euclidean spatial logics, and in particular those able to express the property of connectedness. It is known that when variables range over regions in the Euclidean plane, one can find formulas in these languages satisfiable only by regions with infinitely many connected components. Using this result, we show that the corresponding logics are undecidable. Further, we show that there exist formulas that are satisfiable in higher-dimensional Euclidean space, but only by regions with infinitely many connected components. We finish by outlining how the insights gained from this result were used (by another author) to show the undecidability of certain quantifier-free Euclidean spatial logics in higher dimensions.

The ordinary cross product in R3 is a widespread tool in mathematics and other sciences. It has applications in many areas such as several variable calculus, abstract algebra, geometry, and physics. In this thesis, we investigate in which Euclidean spaces R𝑛 there exist cross products. Based on the properties of the cross product in R3, we introduce two different notions of a cross product in R𝑛. Our first definition is based on the Pythagorean property and the perpendicular property of the cross product in R3. By direct calculation, we show that there is exactly one cross product in R1, no cross product in R2, and exactly two cross products in R3. We also show that if R𝑛 has a cross product, then 𝑛 = 1, 3, or 7. Our second definition uses the following self-selected properties of the cross product in  R3: the triple property, and the nondegeneracy property, leading to the notion of a semi-crossproduct. By direct computation, we discover that R3 has exactly two semi-cross products, which coincide with its cross products, moreover, there does not exist any semi-cross product in R1 or R2. The main result of the thesis is that there are no semi-cross products in R𝑛 for 𝑛 ≥ 4. As far as we know, the results of this chapter are new.

The main goal of this master's thesis is to research different approaches of rendering geometries and spaces in virtual reality. Learn more about the terms, non-Euclidean geometry and non-Euclidean spaces, their origin and different principles used in video game industry to simulate such geometries or spaces. Based on the research, a selection of an optimal API is needed for the implementation of such application. Application is designed to run on desktop computers with Microsoft Windows operating system. Application, in it's core, is a video game and the main goal of the player is to successfully complete each and every level of the game. These levels are designed in a specific way so that they each individually represent some form of non-Euclidean geometry or space.

In this work, we generalize three famous results obtained by Schoenberg: I) the characterization of the continuous positive definite isotropic kernels defined on a real sphere; II) the characterization of the continuous positive definite radial kernels defined on an Euclidean space; III) the characterization of the continuous conditionally negative radial kernels defined on an Euclidean space. From this new approach, we reobtain several results in the literature and obtain some new ones as well. With the exception of S1 and R , we obtain necessary and sufficient conditions in order that these kernels be strictly positive definite and strictly conditionally negative definite. / Neste trabalho, nós generalizamos três resultados famosos obtidos por Schoenberg: I) a caracterização dos núcleos contínuos isotrópicos positivos definidos em esferas reais; II) a caracterização dos núcleos contínuos radiais positivos definidos em espaços Euclidianos; III) a caracterização dos núcleos contínuos radiais condicionalmente negativos definidos em espaços Euclidianos. A partir destas novas abordagens, reobtemos vários resultados da literatura assim como obtemos novos. Com a exceção de S1 e R, obtemos condições necessárias e suficientes para que estes núcleos sejam estritamente positivos definidos e estritamente condicionalmente negativos definidos.