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CHAPTER III CONSTRUCTION OF PROJECTIVE SP ACE In this chapter we describe the classical process of adding points at infinity to an affine plane or a 3-dimensio nal affine space. The ob ject s obtai ned in this manner are called  projective planes  or  pro-  jective spaces, and predictably they are one of the main objects of attention in projective geometry. 1. Ideal points and lin es Extending the space ... [is a] fruitful method for extracting understandable results from the bewildering chaos of special cases. —  J. Dieudo nn´ e (1906 –199 2) In calculus — particularly in the study of limits — it is frequently convenient to add one or two numbers at infinity  to the real number system. 1 Among the reasons for this are the following: (i) It all ows one to for mulat e otherwise complicated notions mor e understandable (fo r example, infinite limits). (ii) It emphasizes the similarities between the infinite limit concept and the ordinary limit concept. (iii) It allows one to perform formal manipulations with limits much more easily. For example, suppose we add a single point at infinity (called  ∞ as usual) to the real numbers. If f  is a real-valued rational function of the form f (t) =  p(t)/q (t), where p  and  q  are polynomials with no common factors and q  is not identically zero, then strictly speaking f  is not definable at the roots of  q . However, an inspection of the graph of  f  suggests defining its value at these points to be  ∞, and if this is done the function is also continuous at the roots of  q  (in an appropriate sense ). Procee ding further alon g these lines , one can even define f (∞) in such a way that  f  is continuous at ∞; the limit value may be a finite number or ∞, depending upon whether or not the degree of  p  is less than or equal to the degree of  q  (in which case the limit value is finite) or the degree of  p  is greater than the degree of  q  (in which case the limit value is infinite). The discussion above illustrates the ideas presented in the following quotation from previously cited the book by R. Winger. 2 1 Of course, if this is done then one must also recognize that the numbers at infinity  do not necessarily have all the usef ul propertie s of ordinary real numbers . The existence of suc h difficulties has been reco gnize d since ancient times, and in particular this is implicit in the celebrated paradoxes which are attributed to Zeno of Elea (c. 490 B.C.E. –  c. 425 B.C.E.). 2 Winger, Introduction to Projective Geometry , pp. 31–32. 43

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