Richter J. Perspectives on Projective Geometry...Real and Complex Geometry 2011 torrent

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It is relatively easy to transform geometric objects into algebraic ones. For instance, points in the plane may be easily represented by their xy-coordinates. However, these “naive” approaches to representing geometric objects are very often not the ones that lead to far-reaching conclusions. Often it is useful to introduce more sophisticated algebraic methods that may seem more abstract at first sight but are ultimately more powerful and elegant. Guided by these more abstract and elegant structures, it may even happen that one is willing to modify the original concept of the geometric first-class citizens (say points or lines) and for instance add some new type of (more abstract) objects. When we talk about homogeneous coordinates, one of the most fundamental concepts of this book, exactly this will happen. We will first see that in the plane very elementary operations such as computing the line through two points and computing the intersection of two lines can be very elegantly expressed if lines as well as points are represented by three-dimensional coordinates(where nonzero scalar multiples are identified). Taking a closer look at the relation of planar points and their three dimensional representing vectors, we will observe that certain vectors do not represent points in the real Euclidean plane. This motivates the search for a geometric interpretation of these nonexistent points. It turns out that they may be interpreted as “points
that are infinitely far away.” We will then extend the usual two-dimensional plane by these new points at infinity and obtain a richer and more elegant geometric system: the system of projective geometry