Скачать или смотреть Reflections in hyperbolic geometry | Universal Hyperbolic Geometry 14 | NJ Wildberger

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.

Video Content:
00:00 Introduction, symmetries of a geometry, types of symmetries
4:57 Reflections in Euclidean geometry: 1 dimension
08:37 Reflections in Euclidean geometry: 2 dimensions
16:35 Reflections in Universal Hyperbolic Geometry
19:14 Reflections in a point
23:58 Reflections also act on lines
26:29 example of reflections in point_a outside and its dual
29:48 example of reflections in point_a inside the circle

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