Introduction E9F5FC

Questions FFFFC0

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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