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Published on Oct 12, 2011
To really understand the fundamental concept of quadrance between points in universal hyperbolic geometry, which replaces the more familiar notion of distance, it is useful to think about circles. Circles are conics, defined in terms of quadrance, and in our usual two dimensional picture they can appear as ellipses, parabolas or hyperbolas. We illustrate three different families, with three different centers. A careful study of these examples will give the student a good understanding of this crucial concept in geometry.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
This is part of the UnivHypGeom series, a new treatment of hyperbolic geometry, purely algebraic, and much prettier.
CONTENT SUMMARY: pg 1: @00:11 Visualizing quadrance with circles pg 2: @03:58 circles in the hyperbolic plane; note - remark on letter c @05:18 dletter x @08:17 ; conics introduced; choice of center @09:52 pg 3: @10:08 example 1; of pictures of circles centered at 0; recap @15:35; pg 4: @16:21 example 2; c=[1:0:2] ; Exercise 24.1 ; remark of no quadrance between zero and one @22:07 pg 5: @24:11 example 3; circles with centers outside the null circle; c=[2:0:1];Exercise 24.2 ; How these curves appear in classical hyperbolic geometry @32:02 (THANKS to EmptySpaceEnterprise)