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 .

### Preliminaries

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 ).

### Equivalence

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.

Proposition.

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.

### Projective Line

We give a geometric interpretation of when is either , , or .

Proposition.

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.”

Proposition.

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.

This proof is very closely related to Stereographic Projection. While is the sphere, note that is a Great Circle on the sphere. This gives a geometric way to view as sitting inside .

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