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...Projective geometry alexander i bobenko draft from april preliminary version partially extended and incomplete based on the lecture winter semester tu berlin written by thilo rorig course notes of boris springborn ws acknoledgements we thank alina hinzmann jan techter for help with preparation these contents introduction spaces subspaces homogeneousandanecoordinates models real projection two planes onto each other points in general position desargues theorem transformations central projections pappus thecross ratio involutions line complete quadrilateral quadrangle mobiustetrahedra koenigs cubes plane thefundamental duality conic sections euclidean point view optical properties conics pencils rational parametrizations pole polar relationship dual brianchon s confocal elliptic billiard circumscribable quadrics orthogonal lines a quadric ii hexagon polarity synthetic approach to problems...