Geometric Fundamentals
In this chapter we lay the geometric foundations that will serve as a basis for the topics that we shall meet later. The statements of projective geometry, in contrast to those of affine geometry, often allow a particularly simple formulation. The project
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Geometric Fundamentals
In this chapter we lay the geometric foundations that will serve as a basis for the topics that we shall meet later. The statements of projective geometry, in contrast to those of affine geometry, often allow a particularly simple formulation. The projective equivalence of polytopes and pointed polyhedra (Theorem 3.36) and Bézout’s Theorem (Theorem 8.27) on the number of intersections of two algebraic curves in the 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 Spaces The basic motivation behind the introduction of projective spaces comes from the examination of two distinct lines in an arbitrary affine plane, for example the Euclidean plane R2 . The lines either intersect or are parallel to one another. The fundamental idea of projective geometry is to extend the affine plane so that parallel lines have an intersection point at “infinity”. For the remainder of this text, let K be an arbitrary field and for any subset A of a vector space V , let lin A denote the linear hull of A. The cases K = R and K = C are 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 projective space over K. We denote it by PnK and remove the lower index K if the coordinate field is clear from the context. M. Joswig, T. Theobald, Polyhedral and Algebraic Methods in Computational Geometry, Universitext, DOI 10.1007/978-1-4471-4817-3_2, © Springer-Verlag London 2013
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Geometric Fundamentals
Fig. 2.1 Embedding the Euclidean plane R2 into the real projective plane P2R
A one-dimensional linear subspace U of V is generated by an arbitrary non-zero vector u ∈ U . Thus, we can identify the projective space with the set of equivalence classes of the equivalence relation ∼ on V \ {0}, where x ∼ y if and only if there exists 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 x and write x = (x0 : · · · : xn )T , with (x0 : · · · : xn )T = (y0 : · · · : yn )T if and only if (x0 , . . . , xn )T ∼ (y0 , . . . , yn )T , i.e., if there exists a λ ∈ K \ {0} such that xi = λyi for 0 ≤ i ≤ n. We can embed the affine space K n in the projective space PnK via the injection: ι : K n → PnK ,
(x1 , . . . , xn )T → (1 : x1 : · · · : xn )T .
(2.1)
Figure 2.1 illustrates the embedding of the Euclidean plane into the real projective plane. The set of ideal points of PnK is Pn \ ι K n = (x0 : x1 : · · · : xn )T ∈ Pn : x0 = 0 . Definition 2.3 Every subspace U of
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