Epistemology m a t h 4 w i s d o m - g m a i l +370 607 27 665 My work is in the Public Domain for all to share freely. 读物 书 影片 维基百科 Introduction E9F5FC Questions FFFFC0 Software Understand what conformal geometry is all about. 保形几何 Angle geometry A metric yields distance, an inner product and angles. A (quadratic) metric relates two different dimensions. In a single dimension we would simply use a linear metric, subtraction. Conformal geometry In conformal geometry (Euclidean geometry), we have inversions. The (infinite) horizon line is a circle that we are within. Reflection takes us in and out of this circle. An example of conformal geometry is (universal conformal) stereographic projection. The infinite line (of the horizon) is reduced to a point (the top of the sphere). Algebraic geometry presumes orthogonal basis elements, thus, perpendicularity and angles. Thus affine geometry and projective geometry should be restricted to not using algebraic geometry. Universal hyperbolic geometry (projective geometry with a distinguished circle) is perhaps conformal geometry. It relates two different spaces, the inside and the outside of the circle. Moebius transformations revealed. Hyperbolic geometry: projective plane (empty space) + distinguished circle + tools: straightedge = projective relativistic geometry perpendicularity via Appolonius pole-polar duality: dual of point is line and vice versa orthocenter - exists in Universal Hyperbolic Geometry but not in Classical Hyperbolic Geometry - need to think outside of the disk. most important theorem: Pythagoras q=q1+q2 - q1q2 second most important theorem: triple quad formula (q1+q2+q3)2 = 2(q1^2 + q2^2 + q2^3) + 4q1q2q3 Compare to: Beltrami-Klein model of hyperbolic geometry Books Geometrinė algebra Conformal geometric algebra includes a description of seven transformations: reflections, translations, rotations, general rotations, screws, inversions, dilations Versor and sandwiching. Chiral - antichiral. 2-dimensional conformal theory <-> 3 dimensional topological theory
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