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but it means we can no longer represent a point by its cartesian coordinates.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,
In other words, an element of RP2 is a line through the origin O
in R3 (but with O deleted).
Then, in R3, UV is the plane through O generated by u and v (but with O deleted).
Thus an RP2-line has equation
of the form ax + by + cz = 0.
Now suppose that L and M are distinct RP2-lines.
Then the defining planes in R3 intersect (since each is through O),
and so meet in a line through O, i.e. in an RP2-point.
Thus, the incidence rules for the geometry RP2 are,
two RP2-points define an RP2-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 R3,
Let P be the point (a,b) on E.
Then P lies on the RP2-point [(a,b,1)]. Thus,
Now suppose that the lines L and M lie on E and are parallel.
Thus, the points of RP2 correspond to those of Ê.
Lines in RP2.Let L be an RP2-line with equation ax + by + cz = 0.
Conics revisited.Let C be the solution set of the equation ax2 + bxy + cy2 +fxz +gyz +hz2 = 0.
If (x,y,z) is a solution, then so is (kx,ky,kz) for any non-zero k.
Thus, C is an object in RP2, i.e. a set of RP2-points.
We are interested cases where C does not contain an RP2-line.
As in the case of an RP2-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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