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UnivHypGeom7b: The circle and projective homogeneous coordinates (cont.)

Universal hyperbolic geometry is based on projective geometry. This video introduces this important subject, which these days is sadly absent from most undergrad/college curriculums. We adopt the 19th century view of a projective space as the space of one-dimensional subspaces of an affine space of one higher dimension. Thus the projective line is viewed as the space of lines through the origin in two dimensional space, while the projective plane deals with one dimensional and two dimensional subspaces of a 3 dimensional affine xyz space (called respectively projective points and projective lines). This relates to Renassiance artists attempts to render perspectives correctly; we illustrate by looking at a parabola in a somewhat novel way.The usual two dimensional view of the projective plane emerges by intersecting with the plane z=1 in the ambient x,y,z space. This way the circle is the two dimensional representation of a cone: a view relating back to the ancient Greeks.CONTENT SUMMARY: 2 dimensional geometry the arena of hyperbolic geometry @00:12 page change: adopting a viewing plane @08:07 The circle in? projective homogeneous coords.(very important picture in mathematics and physics: special relativity) at the heart of hyperbolic geometry @16:14
Length: 24:17

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