Published on: **Mar 4, 2016**

- 1. Chapter 2Geometric FundamentalsIn this chapter we lay the geometric foundations that will serve as a basis for thetopics that we shall meet later. The statements of projective geometry, in contrast tothose of afﬁne geometry, often allow a particularly simple formulation. The projec-tive equivalence of polytopes and pointed polyhedra (Theorem 3.36) and Bézout’sTheorem (Theorem 8.27) on the number of intersections of two algebraic curves inthe plane are good examples of this. We will also introduce the notion of convexity,which is an irreplaceable concept in linear computational geometry.2.1 Projective SpacesThe basic motivation behind the introduction of projective spaces comes from theexamination of two distinct lines in an arbitrary afﬁne plane, for example the Eu-clidean plane R2 . The lines either intersect or are parallel to one another. The funda-mental idea of projective geometry is to extend the afﬁne plane so that parallel lineshave an intersection point at “inﬁnity”. For the remainder of this text, let K be an arbitrary ﬁeld and for any subset A ofa vector space V , let lin A denote the linear hull of A. The cases K = R and K = Care of primary interest in this book.Deﬁnition 2.1 (i) Let V be a ﬁnite dimensional vector space over K. The projective space P (V ) induced by V is the set of one-dimensional subspaces of V . The dimension of P (V ) is deﬁned as dim P (V ) = dim V − 1. The function which maps a vector v ∈ V {0} to the one-dimensional linear subspace lin v is called the canonical projection.(ii) For any natural number n, the set P (K n+1 ) is called the n-dimensional projec- tive space over K. We denote it by Pn and remove the lower index K if the K coordinate ﬁeld is clear from the context.M. Joswig, T. Theobald, Polyhedral and Algebraic Methods in Computational Geometry, 9Universitext, DOI 10.1007/978-1-4471-4817-3_2,© Springer-Verlag London 2013
- 2. 10 2 Geometric FundamentalsFig. 2.1 Embedding the Euclidean plane R2 into the real projective plane P2 R A one-dimensional linear subspace U of V is generated by an arbitrary non-zerovector u ∈ U . Thus, we can identify the projective space with the set of equivalenceclasses of the equivalence relation ∼ on V {0}, where x ∼ y if and only if thereexists a λ ∈ K {0} such that x = λy.Deﬁnition 2.2 Let (x0 , . . . , xn )T ∈ K n+1 {0} be a vector. Then x := lin{(x0 ,. . . , xn )T } ∈ Pn . We call any element of x {0} homogeneous coordinates of xand write x = (x0 : · · · : xn )T , with (x0 : · · · : xn )T = (y0 : · · · : yn )T if and onlyif (x0 , . . . , xn )T ∼ (y0 , . . . , yn )T , i.e., if there exists a λ ∈ K {0} such that xi = λyifor 0 ≤ i ≤ n. We can embed the afﬁne space K n in the projective space Pn via the injection: K ι : K n → Pn , K (x1 , . . . , xn )T → (1 : x1 : · · · : xn )T . (2.1)Figure 2.1 illustrates the embedding of the Euclidean plane into the real projectiveplane. The set of ideal points of Pn is K Pn ι K n = (x0 : x1 : · · · : xn )T ∈ Pn : x0 = 0 .Deﬁnition 2.3 Every subspace U of a vector space V deﬁnes a projective subspaceP (U ) = {lin(u) : u ∈ U {0}}. Therefore, the set of (non-empty) projective subspaces of a projective spaceP (V ) is in one-to-one correspondence with the (non-zero) linear subspaces of V .The set of ideal points of Pn forms a subspace of dimension n − 1. Also, lin ∅ = {0} Kand P ({0}) = ∅. Projective subspaces of dimension 0, 1 and 2 are called points, lines and planes,as usual. Projective subspaces of dimension n − 1 (i.e., codimension 1) are called
- 3. 2.1 Projective Spaces 11hyperplanes. The embedding ι(U ) of a k-dimensional subspace U of K n producesa k-dimensional projective subspace called the projective closure of U .Example 2.4 Consider the projective plane P2 . The projective lines of this space Kcorrespond to the two-dimensional subspaces of K 3 . Since the intersection of anytwo distinct two-dimensional subspaces of K 3 is always one-dimensional, any twodistinct lines of the projective plane have a uniquely determined intersection point. Conversely, given any two distinct projective points there exists one unique pro-jective line incident with both. This follows directly from the fact that the linear hullof two distinct one-dimensional subspaces of a vector space is two-dimensional. The extension of the afﬁne space K n to the projective space Pn simpliﬁes many Kproofs by eliminating case distinctions. In the particularly interesting cases K = Rand K = C, the ﬁeld K has a locally compact (and connected) topology, inducingthe product topology on K n . This topology has a natural extension to the point setsPn and Pn as a compactiﬁcation. See Exercise 2.19. R C Every hyperplane H in Pn can be expressed as the kernel of a non-trivial linear Kform, that is, a K-linear map φ : K n+1 → K, x = (x0 : · · · : xn )T → u0 x0 + · · · + un xn (2.2)where the coefﬁcients u0 , . . . , un ∈ K are not all zero. The set of all K-linear formson K n+1 yields the dual space (K n+1 )∗ . Pointwise addition and scalar multiplica-tion turns the dual space into a vector space over K. The map φ deﬁned in (2.2) isidentiﬁed with the row vector u = (u0 , . . . , un ). Clearly, every hyperplane uniquelydeﬁnes the vector u = 0 up to a non-zero scalar and vice versa. In other words:hyperplanes can also be expressed in terms of homogeneous coordinates, and wesimply write H = ker φ = [u0 : · · · : un ]. The following proposition shows how hyperplanes can be expressed with thehelp of the inner product ·, · : K n+1 × K n+1 → K, x, y := x0 y0 + x1 y1 + · · · + xn yn (2.3)on K n+1 . For x ∈ K n+1 and u ∈ (K n+1 )∗ , we write u(x) = u · x = x, uTwhere “·” denotes standard matrix multiplication.Proposition 2.5 The projective point x = (x0 : · · · : xn )T lies in the projective hy-perplane u = [u0 : · · · : un ] if and only if x, uT = 0.Proof Notice that the condition x, uT = 0 makes sense in homogeneous coordi-nates since it is homogeneous itself. The claim follows from the equation (λx0 , . . . , λxn )T , (μu0 , . . . , μun )T = λμ(x0 u0 + · · · + xn un ) = λμ x, uTfor every λ, μ ∈ K.
- 4. 12 2 Geometric Fundamentals At the end of the book, in Theorem 12.24, we will prove a far-reaching general-ization of Proposition 2.5.Example 2.6 As in Example 2.4, consider the afﬁne plane K 2 and its projective clo-sure, the projective plane P2 . We can use the homogeneous coordinates to represent Ka projective line of P2 . For a, b, c ∈ K with (b, c) = (0, 0) let K x = ∈ K 2 : a + bx + cy = 0 ybe an arbitrary afﬁne line. Then the projective line [a : b : c] is the projective closureof . It contains exactly one extra projective point that is not the image of an afﬁnepoint of the embedding ι. This point is the ideal point of and has the homogeneouscoordinates (0 : c : −b). The homogeneous coordinates of every line of K 2 parallel to differ only in a,their ﬁrst coordinate (in the projective closure). Therefore, they share the same pointat inﬁnity. All ideal points lie on the unique projective line [1 : 0 : 0], which is notthe projective closure of any afﬁne line. This line is called the ideal line. Ideal points in the real projective plane P2 are often called points at inﬁnity in Rthe literature. The idea of two parallel lines “intersecting at inﬁnity” means that theprojective closures of two parallel lines in R2 intersect at the same ideal point of P2 . R2.2 Projective TransformationsA linear transformation is a vector space automorphism, i.e., a bijective linear mapfrom a vector space to itself. Since projective spaces are deﬁned in terms of vectorspace quotients, linear transformations induce maps between the associated projec-tive spaces. More precisely, let V be a ﬁnite dimensional K-vector space and f : V → V aK-linear transformation. For v ∈ V {0} and λ ∈ K we have f (λv) = λf (v) andtherefore f (lin(v)) = lin(f (v)). As f is bijective, non-zero vectors are mapped tonon-zero vectors. Hence f induces a projective transformation: P (f ) : P (V ) → P (V ), lin(v) → lin f (v) . For V = K n+1 , the map f is usually described by a matrix A ∈ GLn+1 K. Wewill therefore use the notation [A] := P (f ) for projective transformations. LetP (V ) be an n-dimensional projective space. A ﬂag of length k is a sequence ofprojective subspaces (U1 , . . . , Uk ) with U1 U2 · · · Uk . The maximal lengthof a ﬂag is n + 2. Every maximal ﬂag begins with the empty set and ends with theentire space P (V ).Theorem 2.7 Let P (V ) be a ﬁnite dimensional projective space with two maximalﬂags (U0 , . . . , Un+1 ) and (W0 , . . . , Wn+1 ). Then there exists a projective transfor-mation π : P (V ) → P (V ) with π(Ui ) = Wi .
- 5. 2.3 Convexity 13Proof Since the subspace Ui is strictly larger than Ui−1 , we can pick vectors u(i) ∈Ui Ui−1 for i ∈ {1, . . . , n + 1}. By construction u(i) is linearly independent ofu(1) , . . . , u(i−1) , and therefore (u(1) , . . . , u(n+1) ) is a basis of V . Similarly we obtaina second basis (w (1) , . . . , w (n+1) ) from the second maximal ﬂag (W0 , . . . , Wn+1 ). From linear algebra we know that there exists a unique invertible linear mapf : V → V that maps u(i) to w (i) for all i ∈ {1, . . . , n + 1}. Therefore π := P (f ) isa projective transformation with the properties stated in the theorem. An equivalent formulation of the above statement is: The group of invertiblelinear maps GL(V ) operates transitively on the maximal ﬂags of P (V ). For a not necessarily maximal ﬂag F = (V1 , . . . , Vk ) we call the strictly mono-tone sequence of natural numbers (dimK V1 , . . . , dimK Vk ) the type of F .Corollary 2.8 Let (U1 , . . . , Uk ) and (W1 , . . . , Wk ) be two ﬂags of P (V ) withthe same types. Then there exists a projective transformation π on P (V ) withπ(Ui ) = Wi .Proof Both (U1 , . . . , Uk ) and (W1 , . . . , Wk ) can be extended to maximal ﬂags. Thusthe statement follows from Theorem 2.7. One may think that the uniqueness of the linear transformation f in the proofof Theorem 2.7 implies the uniqueness of π = P (f ). Showing that this is generallynot true is the goal of the exercise below. First we clarify some terminology: A pointset M ⊆ Pn is called collinear if there exists a projective line that contains all pointsof M. A quadruple (a (1) , a (2) , a (3) , a (4) ) of points of P2 is called a quadrangle if nosubset of three points is collinear.Exercise 2.9 For any two quadrangles (a (1) , a (2) , a (3) , a (4) ) and (b(1) , b(2) ,b(3) , b(4) ) there exists a projective transformation π of P2 with π(a (i) ) = b(i) for1 ≤ i ≤ 4. An afﬁne transformation is a projective transformation that maps ideal points toideal points.Exercise 2.10 For every afﬁne transformation π of Pn there exists a linear trans- Kformation A ∈ GLn (K) and a vector v ∈ K n such that π(ι(x)) = ι(Ax + v) for allx ∈ Kn.2.3 ConvexityWe begin by summarizing some notation from linear algebra to clarify the terminol-ogy and concepts that we will use. As before, let K denote a ﬁeld.
- 6. 14 2 Geometric FundamentalsFig. 2.2 Afﬁnely independent points (left) and afﬁnely dependent points (middle and right) in theEuclidean plane R2Deﬁnition 2.11 Let A ⊆ K n . An afﬁne combination of points in A is a linearcombination m λ(i) a (i) with m ≥ 1, λ(1) , . . . , λ(m) ∈ K, a (1) , . . . , a (m) ∈ A and i=1 m i=1 λ = 1. The set of all afﬁne combinations of A is called the afﬁne hull of A (i)or simply aff A. We call the points a (1) , . . . , a (m) ∈ K n afﬁnely independent if theygenerate an afﬁne subspace of dimension m − 1. For example, the three points in the picture on the left hand side of Fig. 2.2 areafﬁnely independent and each set of four or more points in the real plane (as inthe middle and on the right hand side of Fig. 2.2) are afﬁnely dependent. We setaff ∅ = ∅ and dim ∅ = −1. The language of projective geometry allows us to describe linear algebra overan arbitrary ﬁeld in geometric terms. In the case of an ordered ﬁeld like the realnumbers (and unlike C) we can further exploit the geometry to obtain results. Forthe remaining part of this chapter, let K be the ﬁeld R of real numbers.Deﬁnition 2.12 Let A ⊆ Rn . A convex combination of A is an afﬁne combination m (i) (i) which additionally satisﬁes λ(1) , . . . , λ(m) ≥ 0. The set conv A of all i=1 λ aconvex combinations of A is called the convex hull of A. A set C ⊆ Rn is calledconvex if it contains all convex combinations that can be obtained from it. The di-mension of a convex set is the dimension of its afﬁne hull. The empty set is convex by deﬁnition. The simplest non-trivial example of aconvex set is the closed interval [a, b] ⊆ R. It is one-dimensional and is the convexhull of its end points. Analogously, for a, b ∈ Rn we deﬁne: [a, b] := λa + (1 − λ)b : 0 ≤ λ ≤ 1 = conv{a, b}. See Fig. 2.3 for some examples.Exercise 2.13 A set C ⊆ Rn is convex if and only if for every two points x, y ∈ C,the segment [x, y] is contained in C.2.3.1 Orientation of Afﬁne HyperplanesFor real numbers a0 , a1 , . . . , an with (a1 , . . . , an ) = 0 consider the afﬁne hyper-plane H = {x ∈ Rn : a0 + a1 x1 + · · · + an xn = 0}. Then [a0 : a1 : · · · : an ] are the
- 7. 2.3 Convexity 15Fig. 2.3 Convex hulls of the points from Fig. 2.2homogeneous coordinates of its projective closure. The complement Rn H has twoconnected components, + H◦ := x ∈ Rn : a0 + a1 x1 + · · · + an xn > 0 and (2.4) − H◦ := x ∈ R : a0 + a1 x1 + · · · + an xn < 0 . n (2.5)These components are called the open afﬁne half-spaces deﬁned by H , with H◦ + −and H◦ attributed as positive and negative, respectively. The (closed) positive half-space H + := x ∈ Rn : a0 + a1 x1 + · · · + an xn ≥ 0satisﬁes H + = H ∪ H◦ = Rn H◦ . The opposite half-space H − is analogously + −deﬁned. The vector (λa0 , λa1 , . . . , λan ) deﬁnes the same afﬁne hyperplane H forany λ = 0, however the roles of H + and H − are reversed when λ is negative. Wewill let [a0 : a1 : · · · : an ]+ := x ∈ Rn : a0 + a1 x1 + · · · + an xn ≥ 0and analogously deﬁne [a0 : a1 : · · · : an ]− . When we wish to distinguish which ofthe two half-spaces deﬁned by H is positive or negative, we will call [a0 : a1 : · · · :an ] the oriented homogeneous coordinates of H . We often consider a given afﬁne hyperplane H in Rn and use the notation H +and H − without having ﬁrst ﬁxed a coordinate representation of H . This is simply anotational device which enables us to differentiate between the two half-spaces; thecoordinates for H can always be chosen so that the notation is in accordance withthe above deﬁnition. The inner product introduced in (2.3) is the Euclidean scalar product on Rn . Asin Proposition 2.5 the sign of the scalar product (1, x1 , . . . , xn )T , (a0 , a1 , . . . , an )Tdenotes the half-space for [a0 : a1 : · · · : an ] in which the point (1, x1 , . . . , xn )T lies.2.3.2 Separation TheoremsFor M ⊆ Rn , we let int M denote the interior of M. That is, the set of points p ∈ Mfor which there exists an -ball centered at p, completely contained in M. A set
- 8. 16 2 Geometric Fundamentalsis called open when int M = M and is closed if it is the complement of an openset. The closure M of M is the smallest closed set in Rn containing M. The set∂M := M int M is the boundary of M. All of these terms are deﬁned with respectto the ambient space Rn . Some concepts from analysis are essential for the structure theory of convex sets.The following statements rely on two core results which are proved in Appendix B.Here, an afﬁne hyperplane H is called a supporting hyperplane for a convex setC ⊆ Rn if H ∩ C = ∅ and C is entirely contained in one of the closed afﬁne half-spaces determined by H .Theorem 2.14 Let C be a closed and convex subset of Rn and p ∈ Rn C anexterior point. Then there exists an afﬁne hyperplane H with C ⊆ H + and p ∈ H − ,that meets neither C nor p. The next statement is a direct consequence of Theorem 2.14.Corollary 2.15 Let C be a closed and convex subset of Rn . Then every point of theboundary ∂C is contained in a supporting hyperplane. A convex set C ⊆ Rn is called full-dimensional if dim C = n. When C is not fulldimensional, it is often useful to use these topological concepts with respect to theafﬁne hull. The relative interior relint C of a convex set C consists of the interiorpoints of C interpreted as a subset of aff C. Analogously, the relative boundary ofC is the boundary of C as a subset of aff C.2.4 ExercisesExercise 2.16 Let P (V ) be a projective space. For every set S ⊆ V the set T = lin{x} : x ∈ S {0}is a subset of P (V ) and for the subspace lin S generated by S, P (lin S) is a projectivesubspace which we denote by T . Prove the dimension formula dim U + dim W = dim U ∪ W + dim(U ∩ W )for two arbitrary projective subspaces U and W of P (V ).Exercise 2.17 Let K be any ﬁeld, and let A = (aij ) ∈ GLn+1 K. Show:(a) If H is a projective hyperplane with homogeneous coordinates (h0 : h1 : · · · : hn ) then the image [A]H under the projective transformation [A] is the kernel of the linear form with coefﬁcients (h0 , h1 , . . . , hn )A−1 .(b) The projective transformation [A] acting on Pn is afﬁne if and only if a12 = K a13 = · · · = a1,n+1 = 0.
- 9. 2.5 Remarks 17Exercise 2.18(a) Every projective transformation on the real projective line P1 (apart from the R identity) has at most two ﬁxed points.(b) Every projective transformation on the complex projective line P1 (apart from C the identity) has at least one and at most two ﬁxed points. (Explain why it is natural to talk about a double ﬁxed point in the ﬁrst case.) A projective space over a topological ﬁeld has a natural topology that will bediscussed in the following exercise.Exercise 2.19 Let K ∈ {R, C}. Show:(a) The point set of a projective space Pn = Kn+1 /∼ is compact with respect to the K quotient topology.(b) Every projective subspace of Pn , interpreted as a subset of the points of Pn , is K K compact.Exercise 2.20 Let K be a ﬁnite ﬁeld with q elements.(a) Show that the projective plane P2 has exactly N := q 2 + q + 1 points and K equally many lines.(b) Denote by p (1) , . . . , p (N ) the points and by 1 , . . . , N the lines of P2 . Further- K more, let A ∈ RN×N be the incidence matrix deﬁned by 1 if p (i) lies on j, aij = 0 otherwise. Compute the absolute value of the determinant of A. [Hint: Study the matrix A · AT .]Exercise 2.21 (Carathéodory’s Theorem) If A ⊆ Rn and x ∈ conv A, then x can bewritten as a convex combination of at most n + 1 points in A. [Hint: Since m ≥ n + 2points are afﬁnely dependent, every convex combination of m points in A can bewritten as a convex combination of m − 1 points.]2.5 RemarksFor further material on projective geometry, refer to the books of Beutelspacher andRosenbaum [13] and Richter-Gebert [88]. More detailed descriptions of convexitycan be found in Grünbaum [56, §2], Webster [98] or Gruber [55]. For basic topo-logical concepts, see the books of Crossley [30] and Hatcher [58]. Although ourprojective transformations are by deﬁnition always linearly induced, in other texts itis common to extend this notion to include collineations induced by ﬁeld automor-phisms.
- 10. http://www.springer.com/978-1-4471-4816-6