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11	Metrical Problems in Minkowski Geometry Fankhänel, Andreas 19 October 2012 (has links) (PDF) In this dissertation we study basic metrical properties of 2-dimensional normed linear spaces, so-called (Minkowski or) normed planes. In the first chapter we introduce a notion of angular measure, and we investigate under what conditions certain angular measures in a Minkowski plane exist. We show that only the Euclidean angular measure has the property that in an isosceles triangle the base angles are of equal size. However, angular measures with the property that the angle between orthogonal vectors has a value of pi/2, i.e, a quarter of the full circle, exist in a wider variety of normed planes, depending on the type of orthogonality. Due to this we have a closer look at isosceles and Birkhoff orthogonality. Finally, we present results concerning angular bisectors. In the second chapter we pay attention to convex quadrilaterals. We give definitions of different types of rectangles and rhombi and analyse under what conditions they coincide. Combinations of defining properties of rectangles and rhombi will yield squares, and we will see that any two types of squares are equal if and only if the plane is Euclidean. Additionally, we define a ``new\'\' type of quadrilaterals, the so-called codises. Since codises and rectangles coincide in Radon planes, we will explain why it makes sense to distinguish these two notions. For this purpose we introduce the concept of associated parallelograms. Finally we will deal with metrically defined conics, i.e., with analogues of conic sections in normed planes. We define metric ellipses (hyperbolas) as loci of points that have constant sum (difference) of distances to two given points, the so-called foci. Also we define metric parabolas as loci of points whose distance to a given point equals the distance to a fixed line. We present connections between the shape of the unit ball B and the shape of conics. More precisely, we will see that straight segments and corner points of B cause, under certain conditions, that conics have straight segments and corner points, too. Afterwards we consider intersecting ellipses and hyperbolas with identical foci. We prove that in special Minkowski planes, namely in the subfamily of polygonal planes, confocal ellipses and hyperbolas intersect in a way called Birkhoff orthogonal, whenever the respective ellipse is large enough. Winkelmaß Birkhoff-Orthogonalität Codis Kegelschnitt gleichschenklig-rechtwinklig metrische Ellipse metrische Hyperbel metrische Parabel metrisches Parallelogramm Minkowski-Ebene normierte Ebene Parallelogramm Radon-Ebene Rechteck Rhombus glatte Ebene Quadrat streng konvexe Ebene angular measure Birkhoff orthogonality codis conic section isosceles orthogonality metric ellipse metric hyperbola metric parabola metric parallelogram Minkowski plane normed plane parallelogram Radon plane rectangle rhombus smooth plane square strictly convex plane ddc:516 Minkowski-Geometrie Kegelschnitt Viereck Winkel
12	Metrical Problems in Minkowski Geometry Fankhänel, Andreas 07 June 2012 (has links) In this dissertation we study basic metrical properties of 2-dimensional normed linear spaces, so-called (Minkowski or) normed planes. In the first chapter we introduce a notion of angular measure, and we investigate under what conditions certain angular measures in a Minkowski plane exist. We show that only the Euclidean angular measure has the property that in an isosceles triangle the base angles are of equal size. However, angular measures with the property that the angle between orthogonal vectors has a value of pi/2, i.e, a quarter of the full circle, exist in a wider variety of normed planes, depending on the type of orthogonality. Due to this we have a closer look at isosceles and Birkhoff orthogonality. Finally, we present results concerning angular bisectors. In the second chapter we pay attention to convex quadrilaterals. We give definitions of different types of rectangles and rhombi and analyse under what conditions they coincide. Combinations of defining properties of rectangles and rhombi will yield squares, and we will see that any two types of squares are equal if and only if the plane is Euclidean. Additionally, we define a ``new\'\' type of quadrilaterals, the so-called codises. Since codises and rectangles coincide in Radon planes, we will explain why it makes sense to distinguish these two notions. For this purpose we introduce the concept of associated parallelograms. Finally we will deal with metrically defined conics, i.e., with analogues of conic sections in normed planes. We define metric ellipses (hyperbolas) as loci of points that have constant sum (difference) of distances to two given points, the so-called foci. Also we define metric parabolas as loci of points whose distance to a given point equals the distance to a fixed line. We present connections between the shape of the unit ball B and the shape of conics. More precisely, we will see that straight segments and corner points of B cause, under certain conditions, that conics have straight segments and corner points, too. Afterwards we consider intersecting ellipses and hyperbolas with identical foci. We prove that in special Minkowski planes, namely in the subfamily of polygonal planes, confocal ellipses and hyperbolas intersect in a way called Birkhoff orthogonal, whenever the respective ellipse is large enough.:1 Introduction 2 On angular measures 3 Types of convex quadrilaterals 4 On conic sections info:eu-repo/classification/ddc/516 ddc:516
13	Trigonometry: Applications of Laws of Sines and Cosines Su, Yen-hao 02 July 2010 (has links) Chapter 1 presents the definitions and basic properties of trigonometric functions including: Sum Identities, Difference Identities, Product-Sum Identities and Sum-Product Identities. These formulas provide effective tools to solve the problems in trigonometry. Chapter 2 handles the most important two theorems in trigonometry: The laws of sines and cosines and show how they can be applied to derive many well known theorems including: Ptolemy¡¦s theorem, Euler Triangle Formula, Ceva¡¦s theorem, Menelaus¡¦s Theorem, Parallelogram Law, Stewart¡¦s theorem and Brahmagupta¡¦s Formula. Moreover, the formulas of computing a triangle area like Heron¡¦s formula and Pick¡¦s theorem are also discussed. Chapter 3 deals with the method of superposition, inverse trigonometric functions, polar forms and De Moivre¡¦s Theorem. Sum Identities Difference Identities Double-Angle Identities Product-Sum Identities Sum-Product Identities Ceva's Theorem Menelaus's Theorem Ptolemy's Theorem Law of Sines Law of Cosine Euler Triangle Formula Inverse Trigonometric Functions Euler's Formula De Moivre's Theorem Pick's Theorem Weitzenbock's Inequality Heron' Formula Polar Coordinates Angle Bisector Formula Brahmagupta's Formula Median Formula Stewart's Theorem Parallelogram Law
14	Geometrické pojmy kolmost a rovnoběžnost ve 4. a 5. ročníku ZŠ / Geometrical concepts of perpendicularity and parallelism in 4th and 5th grade of primary school Kopecká, Alena January 2021 (has links) The diploma thesis is focused on pupils attending 4th and 5th grade in primary school. These children are observed for their knowledge of perpendicularity and parallelism and their ability to use it in practice. The theoretical part describes two geometrical terms - perpendicularity and parallelism. Other chapters describe different theories of learning, the process of knowledge construction and also the stages of language development in geometry. Furthermore the thesis explores the topic in state curriculum documents. The geometrical terms and their occurance, frequency and preparative tasks are searched in different textbooks. The practical part includes experiments based on qualitative research. The survey is conducted through a worksheet with different types of tasks which were chosen to test the participants' knowledge of above-mentioned terms. These experiments were recorded and analyzed by means of phenomena that appeared. The phenomena allowed me to answer the questions that were posed in the beginning of the research. Key words: perpendicularity, parallelism, preparative tasks, depth of knowledge, experiment

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