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Robotic manipulator based on CGA
Stodola, Marek ; Salač,, Tomáš (referee) ; Hrdina, Jaroslav (advisor)
Conformal geometric algebra is defined in the thesis. Representations of geometric objects and possibilities of their geometric transformations are presented. Conformal geometric algebra is applied to the calculation of forward kinematics of a robotic manipulator UR10 from Universal Robots. It is also applied to determine the position of the machine based on the location and rotation of two cameras. Then it is used in an inverse task, where based on records from the two cameras, dimensions of the UR10 manipulator and possibilities of its movement, the mutual position of these cameras is determined. And consequently the possibilities of their location in space. Finally, the derived procedures are implemented in a custom program created in the CluCalc environment, using which a sample example verifying the correctness of these procedures is calculated.
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Three-dimensional kinematics of eye movements
Stodola, Marek ; Velan, Petr (referee) ; Hrdina, Jaroslav (advisor)
The goal of this thesis is to describe eye movements and general eye position using apparatus of geometric algebra. The introduction covers the theory about the appropriate geometric algebra, followed by the classifications of the eye movements and the terms used to describe these movements. Following this, the calculations that describe eye position derived from a single observed point are listed, for distant and close points. In addition, the possible eye movements in respect to the axis in which an eye can rotate is described, for any general position. All the calculations are based on Donders' law and Listing's law.
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Application of Geometric Algebras in Quantum Computing
Michálek, Jan ; Eryganov, Ivan (referee) ; Vašík, Petr (advisor)
Tato práce se zabývá využitím geometrických algeber v oblasti kvantového počítání. Nejprve je definována obecná Cliffordova algebra a následně je odvozena specifická komplexní geometrická algebra, která je vhodná pro reprezentaci kvantových výpočtů. Tento přístup je porovnán s tradiční metodou použití klasické maticové reprezentace. Cílem práce je poskytnout poznatky o potenciálních výhodách použití geometrických algeber pro kvantové výpočty.
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Stein-Weiss gradients
Malý, Marek ; Lávička, Roman (advisor) ; Souček, Vladimír (referee)
In this bachelor thesis, we describe the construction of rotation invariant differential operators of first order on the Euklidean space Rn given by E. Stein and G. Weiss. For this construction we show how to find an irreducible decomposition of a tensor product of re- presentations of group Spin(n) into irreducible subrepresetations. We shall also prove the rotation invariance of the gradient operator. Then we apply the Stein-Weiss construction to produce some of well-known differential operators. Namely, we construct the Dirac operator in Rn and Hodge-de Rham system of differential equations using this method. 1
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Robotic manipulator based on CGA
Stodola, Marek ; Salač,, Tomáš (referee) ; Hrdina, Jaroslav (advisor)
Conformal geometric algebra is defined in the thesis. Representations of geometric objects and possibilities of their geometric transformations are presented. Conformal geometric algebra is applied to the calculation of forward kinematics of a robotic manipulator UR10 from Universal Robots. It is also applied to determine the position of the machine based on the location and rotation of two cameras. Then it is used in an inverse task, where based on records from the two cameras, dimensions of the UR10 manipulator and possibilities of its movement, the mutual position of these cameras is determined. And consequently the possibilities of their location in space. Finally, the derived procedures are implemented in a custom program created in the CluCalc environment, using which a sample example verifying the correctness of these procedures is calculated.
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Twistors in relativistic field theories
Nárožný, Jiří ; Scholtz, Martin (advisor) ; Souček, Vladimír (referee)
In this thesis, we are concerning about the Twistor theory, field originally motivated purely physically, although these days fully developed into the many fields of mathem- atics and physics. With its complexion Twistor theory influences algebraic geometry, Clifford analysis as well as the String theory or Theory of quantum gravity. In the thesis we describe the origin of twistors projective or not. Mathematical background to the twistor theory is covered in the first chapter, where we study Clifford algebras and their representations. In the first part of the second chapter we are describing non-projective twistors as representation elements of certain Spin-group, and we find the connection with the standard definition of non-projective twistors as a kernel of the twistor operator. In the last part of the second chapter, we create a space of pro- jective twistors and show its certain properties, especially its correspondence with the complexified compactified Minkowski spacetime.
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Three-dimensional kinematics of eye movements
Stodola, Marek ; Velan, Petr (referee) ; Hrdina, Jaroslav (advisor)
The goal of this thesis is to describe eye movements and general eye position using apparatus of geometric algebra. The introduction covers the theory about the appropriate geometric algebra, followed by the classifications of the eye movements and the terms used to describe these movements. Following this, the calculations that describe eye position derived from a single observed point are listed, for distant and close points. In addition, the possible eye movements in respect to the axis in which an eye can rotate is described, for any general position. All the calculations are based on Donders' law and Listing's law.
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