Symmetries are crucial in studying geometry. In Euclidean geometry we have translations, rotations, reflections, dilations and also projections and perspectivities. This lecture introduces reflections into universal hyperbolic geometry. First we discuss the two different kinds of reflections (in a point or in a line) in Euclidean geometry. The hyperbolic version rests on some remarkable facts, also directly connected to the geometry of a circle. Somewhat surprisingly, reflection in a point is the same as reflection in the dual line, so the two notions agree in hyperbolic geometry.CONTENT SUMMARY: pg 1: @00:10 Reflections in hyperbolic? geometry,symmetries of a geometry, types of symmetries;pg 2: @04:57 Reflections in Euclidean geometry, multiplication of transformationspg 3: @08:37 Euclidean reflections in points and in linespg 4: @13:54 exercisespg 5: @16:35 Reflections in Universal Hyperbolic Geometry, simple and elegant generatorspg 6: @19:15 Reflections in a hyperbolic setting defined; reflection in a point a; pg 7: @22:10 reflection is well definedpg 8: @23:58 a reflection also acts on linespg 9: @26:29 example of reflections in point_a outside and its dualpg 10: @29:48 example of reflections in point_a inside the circle (THANKS to EmptySpaceEnterprise)
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