Projective Geometry Lecture Notes
                                                          Thomas Baird
                                                         March 26, 2013
                  Contents
                  1 Introduction                                                                                    2
                  2 Vector Spaces and Projective Spaces                                                             4
                      2.1   Vector spaces and their duals      . . . . . . . . . . . . . . . . . . . . . . . . .    4
                            2.1.1   Fields   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    4
                            2.1.2   Vector spaces and subspaces . . . . . . . . . . . . . . . . . . . . . .         5
                            2.1.3   Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      7
                            2.1.4   Dual vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . .        7
                      2.2   Projective spaces and homogeneous coordinates . . . . . . . . . . . . . . .             8
                            2.2.1   Visualizing projective space . . . . . . . . . . . . . . . . . . . . . .        8
                            2.2.2   Homogeneous coordinates . . . . . . . . . . . . . . . . . . . . . . .          12
                      2.3   Linear subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       13
                            2.3.1   Two points determine a line . . . . . . . . . . . . . . . . . . . . . .        13
                            2.3.2   Two planar lines intersect at a point      . . . . . . . . . . . . . . . . .   14
                      2.4   Projective transformations and the Erlangen Program . . . . . . . . . . . .            14
                            2.4.1   Erlangen Program . . . . . . . . . . . . . . . . . . . . . . . . . . .         15
                            2.4.2   Projective versus linear . . . . . . . . . . . . . . . . . . . . . . . . .     16
                            2.4.3   Examples of projective transformations . . . . . . . . . . . . . . . .         17
                            2.4.4   Direct sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      18
                            2.4.5   General position     . . . . . . . . . . . . . . . . . . . . . . . . . . . .   19
                            2.4.6   The Cross-Ratio      . . . . . . . . . . . . . . . . . . . . . . . . . . . .   21
                      2.5   Classical Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       23
                            2.5.1   Desargues’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . .         23
                            2.5.2   Pappus’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .        24
                      2.6   Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    25
                  3 Quadrics and Conics                                                                           28
                      3.1   Affine algebraic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      28
                      3.2   Projective algebraic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . .      29
                      3.3   Bilinear and quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . .       30
                            3.3.1   Quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      32
                            3.3.2   Change of basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      33
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                            3.3.3   Digression on the Hessian      . . . . . . . . . . . . . . . . . . . . . . .   36
                      3.4   Quadrics and conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      36
                      3.5   The rational parametrization of the circle . . . . . . . . . . . . . . . . . . .       39
                            3.5.1   Rational parametrization of the circle . . . . . . . . . . . . . . . . .       40
                      3.6   Linear subspaces of quadrics and ruled surfaces        . . . . . . . . . . . . . . .   42
                      3.7   Polars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     43
                      3.8   Dual quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      46
                      3.9   Pencils of quadrics and degeneration       . . . . . . . . . . . . . . . . . . . . .   47
                  4 Exterior Algebras                                                                             50
                      4.1   Multilinear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      50
                      4.2   The exterior algebra     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   52
                      4.3   The Grassmanian and the Pluc¨ ker embedding . . . . . . . . . . . . . . . .            56
                      4.4   Decomposability of 2-vectors . . . . . . . . . . . . . . . . . . . . . . . . . .       57
                      4.5   The set of lines in projective space     . . . . . . . . . . . . . . . . . . . . . .   58
                  5 Ideas for Projects                                                                            61
                  1 Introduction
                  Projective geometry has its origins in Renaissance Italy, in the development of perspective
                  in painting: the problem of capturing a 3-dimensional image on a 2-dimensional canvas.
                  It is a familiar fact that objects appear smaller as the they get farther away, and that the
                  apparent angle between straight lines depends on the vantage point of the observer. The
                  more familiar Euclidean geometry is not well equipped to make sense of this, because in
                  Euclidean geometry length and angle are well-defined, measurable quantities independent
                  of the observer. Projective geometry provides a better framework for understanding how
                  shapes change as perspective varies.
                      The projective geometry most relevant to painting is called the real projective plane,
                                       2         3
                  and is denoted RP or P(R ).
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                  Definition 1. The real projective plane, RP = P(R ) is the set of 1-dimensional sub-
                  spaces of R3.
                      This definition is best motivated by a picture. Imagine an observer sitting at the origin
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                  in R looking out into 3-dimensional space. The 1-dimensional subspaces of R can be
                  understood as lines of sight. If we now situate a (Euclidean) plane P that doesn’t contain
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                  the origin, then each point in P determines unique sight line. Objects in (subsets of) R
                  can now be “projected” onto the plane P, enabling us to translate a 3-dimensional scene
                  onto a 2-dimensional scene. Since 1-dimensional lines translate into points on the plane,
                  we call the 1-dimensional lines projective points.
                      Of course, not every projective point corresponds to a point in P, because some 1-
                  dimensional subspaces are parallel to P. Such points are called points at infinity. To
                  motivate this terminology, consider a family of projective points that rotate from pro-
                  jective points that intersect P to one that is parallel. The projection onto P becomes a
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                  Figure 1: A 2D representation of a cube
                   Figure 2: Projection onto a plane
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               family of points that diverges to infinity and then disappears. But as projective points
               they converge to a point at infinity. It is important to note that a projective point is only
               at infinity with respect to some Euclidean plane P.
                  One of the characteristic features of projective geometry is that every distinct pair
               of projective lines in the projective plane intersect. This runs contrary to the parallel
               postulate in Euclidean geometry, which says that lines in the plane intersect except when
               they are parallel. We will see that two lines that appear parallel in a Euclidean plane will
               intersect at a point at infinity when they are considered as projective lines. As a general
               rule, theorems about intersections between geometric sets are easier to prove, and require
               fewer exceptions when considered in projective geometry.
                  According to Dieudonn´e’s History of Algebraic Geometry, projective geometry was
               among the most popular fields of mathematical research in the late 19th century. Pro-
               jective geometry today is a fundamental part of algebraic geometry, arguably the richest
               and deepest field in mathematics, of which we will gain a glimpse in this course.
               2 Vector Spaces and Projective Spaces
               2.1    Vector spaces and their duals
               2.1.1  Fields
               Therational numbers Q, the real numbers R and the complex numbers C are examples of
               fields. A field is a set F equipped with two binary operations F ×F → F called addition
               and multiplication, denoted + and · respectively, satisfying the following axioms for all
               a,b,c ∈ R.
                 1. (commutativity) a+b = b+a, and ab = ba.
                 2. (associativity) (a +b)+c = a+(b+c) and a(bc) = (ab)c
                 3. (identities) There exists 0,1 ∈ F such that a + 0 = a and a · 1 = a
                 4. (distribution) a(b+c) = ab+ac and (a+b)c = ac+bc
                 5. (additive inverse) There exists −a ∈ F such that a+(−a) = 0
                 6. (multiplicative inverse) If a 6= 0, there exists 1 ∈ F such that a · 1 = 1.
                                                                 a                  a
               Two other well know operations are best defined in terms of the above properties. For
               example
                                                a−b=a+(−b)
               and if b 6= 0 then                          1
                                                   a/b = a· b.
                  We usually illustrate the ideas in this course using the fields R and C, but it is
               worth remarking that most of the results are independent of the base field F. A perhaps
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