Infinity in its Proper Place.
Projective Geometry works with just three primitive qualities (‘primitive’ in the sense of ‘prime’—i.e., ‘first’):
• The Point: this is just “place”, “location”, “somewhere”.
• The Line: this is just “extension”, “straightness”.
• The Plane: this is just “spread”, “flatness”.
None of these qualities can be made “more” or “less” what it is. For example, a point cannot be made more of a place than it is, or less.
Thus these three qualities are just that—qualities.
Accordingly, they are not quantified, or indeed quantifiable.
It follows that a line is not, as it is often held to be, a sum of contiguous points, because a sum is a quantity, not a quality. Moreover, because a point is, ‘just a place’, it cannot have a size as well, not even of zero. So a line cannot be a sum of sizeless points. In any case, size is an interval, so if a point is to have a size, it must be an interval. It is not an interval.
So a line may extend through places, but does not consist of those places. Line and point may be incident one with the other, but neither can be part, or a component, of the other.
a plane is one thing, not an interval between two things, so has no size, and cannot be a component.
a line is one thing, not an interval between two things, so has no size, and cannot be a component.
A sizeless plane cannot be a sum of sizeless points or lines, though it may spread through both and be incident with them.
The corollaries are that
geometric elements do not have parts,
and that incidence is non-analytic.
It also follows that
The latter of these two statements may seem laughably self-evident, and the former self-evidently laughable. But they are both true; each is simply the other's complement, and as such express the fundamental “Duality” of Line and Point.
From this we may perhaps already sense that common-sense and intuition are not the reliable guides we may have supposed them to be, and that there may be need to approach the self-evident with refreshed caution.
The Axioms of Incidence list the ways in which the three qualities interact. In particular—
It is by these axioms that we know that “parallel” lines meet. Because parallels certainly have a plane in common, by axiom 2 above, they must also have a point in common, and, perforce, meet.
If they do not meet, axiom number 3 in the list above applies: they have neither plane nor point in common and are in fact not parallel. They are skew.
However, Projective Geometry has no axiomatic means of “knowing” whether or not lines are in actual fact parallel, since that is a matter of measurement, of size, of units and of quantity. None of these is ‘sponsored’ by the axioms. The whereabouts of the meeting point of parallels (said to be the ‘infinite’) is likewise a quantitative issue of measurement.
Projective Geometry does not know what parallelity and infinity are.
Notwithstanding, it is widely assumed that a means to measure distance exists, and it is also widely supposed that parallel lines, such as straight railway tracks, cannot meet, because they will, by such measurement, always be found to have constant separation.
Now, this is mistaken, on all counts, but let us for the moment assume that absolute and universally valid measurement - of anything - is in fact possible.
The proper measure of
distance between points is a translation, which is reckoned
|Accordingly, there is never a lineal distance between lines, simply because lines are not points. So, the ‘constant separation’ statement above is wrong. It is an idea misapplied.|
The proper measure of distance between lines is
a rotation, and this is reckoned
By that rotational (angular) reckoning, lines are parallel iff ("if and only if") the rotational distance between them is zero.
That is, lines which, by some measure-system, have zero rotational separation, are parallel in that measure-system.
And, by (3) above, there can be no angle, of any value whatsoever, between lines which do not meet. There is no plane, and hence no angle, between skews.
In the hope of furthering insight, I give the diagram below, from H.S.M. Coxeter, “The Real Projective Plane”, 2nd Edition, Cambridge University Press, 1961, page 3.
The shadow of a circle is projected from point L on to a table top.
The line from L through A in the circle is supposedly parallel to the table top,
Euclidean Geometers would insist that there is no shadow. The famous “fifth postulate” (in John Playfair's version) states that no two distinct parallel lines can be drawn through a single point, which implies they do not meet, for the reason that such a meeting, if it occurs, must occur in just such a point.
But Projective Geometry, as we have seen above, says axiomatically that any pair of lines on a plane must meet, else they are not on the same plane and therefore must be skew.
We appear to have at least a paradox, if not an outright contradiction.
Now, as I read Mr. Coxeter's diagram, there are certainly shadows for points of the circle other than A, and these shadows are themselves points, but, for the sake of argument, assume that point A is indeed the exception, and has no shadow. There is then, presumably, a gap in the shadow. A gap left by a "missing" point is just a place. But a point (and, for that matter, the shadow of a point) is, "just a place". Ergo, the gap is a point—so A has a shadow, and is not in fact exceptional!
The lesson here is that points, being places, cannot be missing!
To see this more simply and directly, just suppose that some points have by some unspecified means actually been “removed”. Then ask, "What remains where these points were?"
‘Ideal-pointists’ attempt to reconcile the incompatibility. They do this by displacing the meeting point of parallels to numeric infinity—that is, to a position which is distant by an infinite number of unit distances. This precisely is the “ideal point at infinity” and the geometry arising from the use of this device is called “Affine Geometry”. The shadow of point A, above, is taken to be such a point.
Because, in affine geometry, lengths seem independent of the orientation of the lines to which they pertain,
seem also to be preserved.
But, for this to “work”—that is, to be consistent with the axioms of point, line and plane—we must have a plane to carry all the points in which variously directed sets of parallel lines will meet. This is because any three points, whether or not at infinity, form a triangle, and every triangle is incident with a plane. So these three points at infinity must lie in that plane—which must therefore lie at the infinite.
This is the so-called,
Ideal Plane at Infinity.
Recalling that Infinity is unknown to Projective Geometry, the ‘Infinity’ referred to here must depend entirely on non-geometric properties, possibly the numerical properties of the Measure System imported into, and grafted on to, the geometric context.
Physical bodies ( especially solid ones) are widely supposed to preserve properties of measure just as above: the units engraved on a ruler, for example, are expected to retain their sizes whatever the orientation of that ruler.
If such bodies do preserve measure in this way, then we must expect that for them the ideal plane at infinity will have particular significance, and much will depend on locating it, and knowing on its absolute whereabouts.. After all, calibration (measurement) and calculation—all forms of practical calculus, in fact, from co-ordinates to tensors, and all of analytic geometry—depend intimately, directly and completely upon it.
Less abstraction , please!
The location of the meeting point of a pair of parallels may very easily be found with quite reasonable precision merely by glancing along, rather than directly at, a school ruler, as in the image alongside, and estimating where the mentally-continued lines of the ruler's edges meet, unseen, though in plain sight. (Artists do this routinely, albeit often unconsciously, when drawing or painting in perspective.)
That place is the “point at infinity” pertaining to the ruler, if the ruler's edges are absolutely parallel. It is a real place. If points were lights, or reflected it, we would see (observe) this point. It would be an observable point, in every way actual, in no way “ideal”. It is not observable, because no point is, but is present nevertheless.
Establishing absolute parallelity of the edges is not a trivial business, however.