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Everything here follows from the Fundamental Theorem of Projective Geometry.
Ranges of randomly-located points in two lines, m and n, may always be projected by points M and N respectively (not in m and n) into two ranges in a third line, i,
but,
no more than three randomly-located points in each of the first two lines may be projected into the third line, if the two ranges in the third line, i, are to be the same range.
animation
Given our three lines and two Projection Points, and a point P, say, in line m, say, we can always find the corresponding point, P', in line n, and it is of great interest to follow what happens to the line joining these two points as they are moved in their lines.
It envelops a conic!
Drag point I to see it happen.
In this case, the conic is an ellipse.
[Another construction from Nick Thomas' Site]
The Principle of Duality gives us the Dual Form at once.
- Where we had ranges of points in lines, m and n, we will have pencils in points, M and N.
- Where we had projection points, M and N, we will have projecting lines, m and n.
- Where we had a line tangent to and enveloping a conic, we will have a point moving in a conic.
Drag the bold red line to make it happen.