Homogeneous coordinatesThe introduction of ideal points provides many advantages,but it means we can no longer represent a point by its cartesian coordinates.
To allow us to use algebraic methods to study the euclidean plane with ideal points,
Definition 1
In other words, an element of RP^{2} is a line through the origin O
in R^{3} (but with O deleted).
Definition 2 Then, in R^{3}, UV is the plane through O generated by u and v (but with O deleted).
Thus an RP^{2}line has equation
of the form ax + by + cz = 0. 
Now suppose that L and M are distinct RP^{2}lines. Then the defining planes in R^{3} intersect (since each is through O), and so meet in a line through O, i.e. in an RP^{2}point. Thus, the incidence rules for the geometry RP^{2} are,
two RP^{2}points define an RP^{2}line,
These are exactly the same as those of the euclidean plane with ideal points! 
Let Ê denote the euclidean plane E, together with its ideal points. As usual, we introduce x,y coordinates on E.
Now the clever bit  we think of E as lying on the plane z=1 in R^{3},
Let P be the point (a,b) on E.
Then P lies on the RP^{2}point [(a,b,1)]. Thus,
Now suppose that the lines L and M lie on E and are parallel. Thus, the points of RP^{2} correspond to those of Ê. Lines in RP^{2}.Let L be an RP^{2}line with equation ax + by + cz = 0.

Conics revisited.Let C be the solution set of the equation ax^{2} + bxy + cy^{2} +fxz +gyz +hz^{2} = 0.If (x,y,z) is a solution, then so is (kx,ky,kz) for any nonzero k. Thus, C is an object in RP^{2}, i.e. a set of RP^{2}points.
We are interested cases where C does not contain an RP^{2}line. 

As in the case of an RP^{2}line, we shall look at how C meets Ê by considering
It is not difficult to see that these correspond to an ellipse, a parabola and a hyperbola.

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