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Write down all of the results of Universal Hyperbolic Geometry. Organize them and interpret them in terms of
Affine geometry Lines are parallel:
Affine combinations Vector is the relation between parallel lines. Projective geometry Space is divided into "infinity" and "finity". Conformal geometry Lines are perpendicular: Pythagorean theorem, inner product Quadratic interpolations: law of cosines A2  2ABcos(theta) + B2 = C2 interpolation of (AB)2 = C2, A2 + B2 = C2, (A+B)2 = C2. Consider triangles more generally. Or consider the function: A2 + B2  C2 which is 0 when A and B are perpendicular, and nonzero 2ABcos(theta) otherwise. Inner product defines the equation of a line and whether points lie on it. (a1, a2, a3)(x1, x2, 1) = 0. Symplectic geometry Oriented area, volume given by determinant.
What if we interpret the line as ax + by + cz = 0 where z=1 ? Why is the s1s2s3 term of the third power and not the second power? 
UniversalHyperbolicGeometryNaujausi pakeitimai 
Puslapis paskutinį kartą pakeistas 2018 vasario 04 d., 11:34
