Editing Real projective plane (section)

=== Ideal points ===
[[Image:prj geom.svg|right]]

In '''P'''<sup>2</sup> the equation of a line is {{nowrap|''ax'' + ''by'' + ''cz'' {{=}} 0}} and this equation can represent a line on any plane parallel to the ''x'', ''y'' plane by multiplying the equation by ''k''.

If {{nowrap|''z'' {{=}} 1}} we have a normalized homogeneous coordinate. All points that have ''z'' = 1 create a plane. Let's pretend we are looking at that plane (from a position further out along the ''z'' axis and looking back towards the origin) and there are two parallel lines drawn on the plane. From where we are standing (given our visual capabilities) we can see only so much of the plane, which we represent as the area outlined in red in the diagram. If we walk away from the plane along the ''z'' axis, (still looking backwards towards the origin), we can see more of the plane. In our field of view original points have moved. We can reflect this movement by dividing the homogeneous coordinate by a constant. In the adjacent image we have divided by 2 so the ''z'' value now becomes 0.5. If we walk far enough away what we are looking at becomes a point in the distance. As we walk away we see more and more of the parallel lines. The lines will meet at a line at infinity (a line that goes through zero on the plane at {{nowrap|''z'' {{=}} 0}}). Lines on the plane when {{nowrap|''z'' {{=}} 0}} are ideal points. The plane at {{nowrap|''z'' {{=}} 0}} is the line at infinity.

The homogeneous point {{nowrap|(0, 0, 0)}} is where all the real points go when you're looking at the plane from an infinite distance, a line on the {{nowrap|''z'' {{=}} 0}} plane is where parallel lines intersect.
{{Clear}}