Projective Geometry
                        Alexander I. Bobenko
                       Draft from April 21, 2020
            Preliminary version. Partially extended and partially incomplete. Based
            on the lecture Geometry I (Winter Semester 2016 TU Berlin).
            Written by Thilo Rörig based on the Geometry I course and the Lecture
            Notes of Boris Springborn from WS 2007/08.
            Acknoledgements: We thank Alina Hinzmann and Jan Techter for the
            help with preparation of these notes.
                   Contents
                   1   Introduction                                                                                1
                   2   Projective geometry                                                                         3
                       2.1   Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      3
                       2.2   Projective spaces    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    5
                             2.2.1    Projective subspaces . . . . . . . . . . . . . . . . . . . . . . . . .       6
                             2.2.2    Homogeneousandaffinecoordinates . . . . . . . . . . . . . . . .                7
                             2.2.3    Models of real projective spaces . . . . . . . . . . . . . . . . . . .       8
                             2.2.4    Projection of two planes onto each other . . . . . . . . . . . . . . .     10
                             2.2.5    Points in general position . . . . . . . . . . . . . . . . . . . . . . .   12
                       2.3   Desargues’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        13
                       2.4   Projective transformations     . . . . . . . . . . . . . . . . . . . . . . . . . .  15
                             2.4.1    Central projections and Pappus’ Theorem . . . . . . . . . . . . . .        19
                       2.5   Thecross-ratio     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  21
                             2.5.1    Projective involutions of the real projective line . . . . . . . . . . .   25
                       2.6   Complete quadrilateral and quadrangle . . . . . . . . . . . . . . . . . . . .       25
                             2.6.1    Möbiustetrahedra and Koenigs cubes . . . . . . . . . . . . . . . .         29
                             2.6.2    Projective involutions of the real projective plane . . . . . . . . . .    31
                       2.7   Thefundamental theorem of real projective geometry . . . . . . . . . . . .          32
                       2.8   Duality    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  35
                       2.9   Conic sections – The Euclidean point of view . . . . . . . . . . . . . . . .        38
                             2.9.1    Optical properties of the conic sections    . . . . . . . . . . . . . . .  41
                       2.10 Conics – The projective point of view       . . . . . . . . . . . . . . . . . . . .  43
                       2.11 Pencils of conics     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  48
                       2.12 Rational parametrizations of conics       . . . . . . . . . . . . . . . . . . . . .  50
                       2.13 The pole-polar relationship, the dual conic and Brianchon’s theorem . . . .          52
                       2.14 Confocal conics and elliptic billiard     . . . . . . . . . . . . . . . . . . . . .  56
                             2.14.1 Circumscribable complete quadrilateral . . . . . . . . . . . . . . .         58
                       2.15 Quadrics. The Euclidean point of view.        . . . . . . . . . . . . . . . . . . .  62
                       2.16 Quadrics. The projective point of view . . . . . . . . . . . . . . . . . . . .       64
                             2.16.1 Orthogonal Transformations . . . . . . . . . . . . . . . . . . . . .         65
                             2.16.2 Lines in a quadric . . . . . . . . . . . . . . . . . . . . . . . . . . .     67
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                             2.16.3 Brianchon hexagon . . . . . . . . . . . . . . . . . . . . . . . . . .        68
                       2.17 Polarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     69
                       2.18 The synthetic approach to projective geometry . . . . . . . . . . . . . . . .        70
                       Problems     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  73