On Real and Imaginary (1)
Projective Geometry is concerned with incidence—with meetings.
Two real lines in a real plane must meet, at some real, locatable place, so that meeting is real, and with it, they share a real point.
elements in this case
Strictly speaking, planes and lines never have locations, as location (place, somewhere) is uniquely the quality of a point.
They are, however, findable by the locatable virtue of the (real) points with which they are incident.
Conics, being closed, divide the plane into two regions; both are enclosed by the conic.
This makes it possible for a line to be at the same instant entirely in one of the two regions and not at all in the other,
and this means that a line and a conic need not intersect.
If a Real Line and a Real Conic in a Real Plane intersect, they share two, locatable points—the so-called real points.
These points exist, and it is in principle always possible (though it might not be practical) to visit such points.
If a Real Line and a Real Conic do not intersect, they share two, non-locatable points—the so-called imaginary points.
These points exist, but it is in principle never possible to visit such points.
But real and imaginary elements have a
If line and conic do intersect, all the ways in which these bridging lines can connect points is the ensemble of real relationships.
If line and conic do not intersect, all the ways in which these bridging lines can connect points is the ensemble of imaginary relationships.
It is often asked what Imaginary elements, "actually are."
There seems to be an ontological issue!
It is, admittedly, rather hard to fathom how a point, whose only property is location, can be “non-locatable”.
Simplicity is the key.
Though a point may stand alone,
a point of the kind now in question exists by virtue of a relationship.
Whether real or imaginary, it is
found as an intersection
(i.e., a relationship) of
(lines, or a line and a curve),
and is traceable/detectable only by virtue of that relationship.
It is probably obvious that there are relationships – as opposed to their members – (such as marriage, or friendship, or family) without whereabouts.
It may not be so obvious that all relationships are without whereabouts
—they are all ‘non-locatable’.
is between at least
while a place is just
Two things simply cannot be a place.
If the above seems dubious, consider instead that, if a line is not a point, it is still not a point on addition of a second line, or of a curve. Moreover, neither is the added line, or curve, a point, nor can it become one.
(See http://mathworld.wolfram.com/ContinuityPrinciple.html for a very formal statement of the position!)
On the conics(3) page, we see the conic defined algebraically, this being done there, and historically, in response to the realisations —
To meet the case (and nevertheless stick as closely as possible to the axioms), metrics and co-ordinates were derived from the projective ground up, and used to direct the combination through the ordered numbers of algebra, and first and second order functions.
Straight lines are first order. Conics are the second order item. Algebraically, this means the conics function has terms in x raised to the second power - that is, x2, while straight-line functions have terms in x to the first power only.
Graphically, being second order corresponds to being the kind of curve cut exactly twice by a straight line. Working algebraically, on the other hand, means taking a square root to find the co-ordinates of the cuts, and there are always two answers to this, one for each of the cuts. (All this of course assumes that numbers and points-in-curves invariably go one-to-one. They don't, not invariably, but ignore this—at least until the foot of this panel.)
And it was in this business of extracting square roots that the imaginary was originally detected, because when there is no intersection, the numbers under the square root sign turn negative—and you can't get a real answer extracting roots from a number in that state!
But that negative square is real (it is calculable), so the inference is that there must be a root – actually, two of them – of some sort (they must be in some way calculable, too!), for all that we can't write them down in cold blood, and, in the nature of the case, they must refer to points.
The lesson for us now is that, from this, at any rate, the imaginary is numeric, not geometric. We seem here to be concerned about the nature of certain kinds of number, not about certain sorts of line, or point, or plane.
It hereby becomes clear, if it had not been already clear, that the connection of numbers to elements of geometry is not immediate, or indeed elementary, and we are left with a curious apposition—