Today, we give an overview of Projective Geometry. We define affine -space and projective -space , and spend time discussing the geometry of the projective line . In particular, we discuss the differences when and .
Let and be sets. We write “” to mean “ is an element of ”; and we write “” to mean “every element of is also an element of .” Types of sets are , the integers; , the rational numbers; , the real numbers; and , the complex numbers. We write because zahlen is German for “number”; and we write for quotient since each rational number is the ratio of two integers. We will denote the end of the proof by “”, which is called a tombstone. More notation can be found here.
Let be a map. If as an element in for in , we write . We say that is well-defined if in whenever in . For example, consider the map defined by ; this is a well-defined map. As another example, consider the map defined by ; this is not a well-defined map. Indeed, we have so that .
Given a well-defined map , we say is injective if in whenever in . We say is surjective if given there exists such that . We say is bijective if is both injective and surjective. For example, the trivial map defined by is a bijective. As another example, the map defined by is neither injective ( yet ) nor surjective (there does not exist a real number such that ).
Let be a set. Any relation among the elements of which satisfies the following three axioms is said to be an equivalence relation:
- Reflexivity: For all we have .
- Symmetry: For all , if then .
- Transitivity: For all , if and , then .
If , we say “ is equivalent to .” The most natural equivalence relation is equality: if and then . As another example, let be a well-defined map. We define an relation on as follows: the notation means . To see why this is an equivalence relation, note (Reflexivity) is satisfied because ; (Symmetry) is satisfied because if and only if ; and (Transitivity) is satisfied because if and then .
Given an equivalence relation on , the set is called the equivalence class of the element . Note that by property (Reflexivity). We denote by the collection of all equivalence classes. As an example, consider the map defined by . This induces an equivalence on by saying precisely when . Hence is the equivalence class of . The collection of equivalence classes may be identified with the nonnegative real numbers.
Affine and Projective -Space
Fix a positive integer , and let denote either , , or . We define affine -space as . Sometimes this is denoted as , especially in the sense of or as 2-space and 3-space, respectively. For example, . In general, elements are called -rational affine points.
Let denote either , , or . Denote as affine -space minus the origin . Two -rational affine points and in are said to be equivalent if for some nonzero number . This defines an equivalence relation on .
Proof: We must show properties (Reflexivity), (Symmetry), and (Transitivity) are satisfied. The relation means for some . This is reflexive, that is, because as . We now show it is symmetric. Say . Then , and so because in invertible in . Hence . Finally, we show it is transitive. Say and . Then and for some . Then where is also nonzero, so as desired.
Given any -rational point as above, denote its equivalence class as . We define projective -space as the collection of these equivalence classes: . For example, when we have
Note that the first set is essentially , while the second set consists of one equivalence class, which we call the “point at infinity.” In general, elements are called -rational projective points.
We give a geometric interpretation of when is either , , or .
Let denote either or , and fix a nonzero . Define the collection of -rational points on the circle of radius as the set ; and consider the map defined by which sends . This map is a bijection, that is, .
Proof: First, we show that is well-defined. Say that . Then and for some , so that
Next, we show that is injective. Say that we have two points and which map to the same image. Then we have
This forces , so that . Finally, we show that is surjective. Let be a point on the circle. Choose a point such that . The equation implies and .
To make this bijection a bit more concrete, say that . Then the -rational affine point is on the circle. However, the “north pole” is not in the image of this map; we need , that is, . This explains why this point is called the “point at infinity.”
Let , and fix a nonzero . Define the collection of real points on the sphere of radius as the set ; and consider the map defined by . This map is a bijection, that is, .
We call the Riemann Sphere.
Sketch of Proof: It is easy to check that this map is well-defined. To show this map is surjective, let be a point on the sphere and choose a point such that . The equation implies , , and . (Here is the complex conjugate of the complex number .) To show that this map in injective, the proof is essentially the same as above.