UCLA Math Circle: Projective Geometry

On the real number line, a point is represented by a real number x if x is finite. Infinity in this setting could be represented by the symbol ∞. However, it is more difficult to calculate with infinity. One way to resolve this issue is through a different coordinate system call homogeneous coordinate. For the real line, this looks a lot like fractions. Instead of using one number x ∈ R, we will use two real numbers [x : y], with at least one nonzero, to represent a point on the number line. Furthermore, we define the following equivalent relationship ∼ [x : y] ∼ [x′ : y′]⇔ there exists ∈ R∖{0} s.t. x = x′ and y = y′. Now two points [x : y] and [x′ : y′] represent the same point if and only if they are equivalent. The set of coordinates {[x : y] ∣ x, y ∈ R not both zero} with the equivalence relation ∼ represents all the points on the real number line including ∞, i.e. [1 : 0]. Furthermore, the operation on the real number line, such as addition, multiplication, inverse, can still be done in this new coordinate system. The line described by this coordinate system is called the projective line, denoted by the symbol RP. Now it is not too difficult to extend this coordinate system to the plane. Instead of using two real numbers as coordinate, we will use three real numbers [x : y : z], not all zero, with similar equivalence relationship to represent points in the plane