A Bit of Projective Geometry

The horizontal rules on this page are breaks of continuity in the presentation.

As Parasitic on Euclid

Euclid’s lines were what we now call line segments; that’s why Euclid spoke of ‘extending a line’. Euclid took points and lines as primitive where the modern idea is to take sets as primitive and explain a line as a set of points. Euclid took the notion of a point on a line as a primitive relation.

Here our lines are infinite but primitive. Points are primitive and so is the relation of a point being on a line. Euclid’s lines intersected (suitably extended) except when parallel. We introduce 2D projective geometry by constructing a model. The model is a collection of points and a collection of lines, along with a specification of exactly which points lie on which lines. Projective geometry adds to Euclid’s lines and points one more line and a bunch of new points thereon. We call the new line “I” or the “line at infinity”. An old point lies on an old line exactly as Euclid had it. None of the old points lie on the new line. Each new point lies on the new line. Each new point lies on some old lines which are parallel to each other in the old sense. For each old line there is just one new point that lies on it and also lies on the new line. Any two old lines are parallel iff there is a point on the new line that is also on both old lines.

Two lines intersect exactly when there is just one point that lies on both lines. In the new model two lines always intersect.

We digress a moment to note that when we say “two lines” or “two points” we mean two distinct lines or points. To say that two lines intersect in a point in formal logic we must say ∀x∀y(Line(x) ∧ Line(y) ∧ x≠y → ∃z(Point(z) ∧ On(z, x) ∧ On(z, y))). Note the explicit qualification that x and y be distinct. We take the English construct “two points” to mean two distinct points. In most other mathematical contexts this may not be so as when we say in group theory that any two group elements can be multiplied by the group operator. If we were to optimize formal logic to suit projective geometry we might decree that distinct individual identifiers never denote the same object. With this modification the equality relation is useless for ∀x∀y¬(x=y) is a theorem. This would shorten each of a long list of projective geometry theorems that I once examined. A further substantial shortening would be to adopt two sorts of individual identifier for the two sorts of thing that this geometry speaks of: points and lines. The Line and Point predicates would then disappear. Our theorem would become ∀l∀m∃z(zl ∧ zm). We will mainly avoid formality here however.

A meta theorem is that any Euclidean theorem expressed solely in terms of points, lines, incident, intersect, concurrent and collinear is true in the new model, and furthermore the old qualifications on lines not being parallel falls away. This could be made precise by describing a formal geometry sub-language including points lines and incidence, but excluding distance, congruence, angle. Then the meta theorem would say that any proposition, expressed in the sub-language, that is true in Euclid’s model, is true in the new model.

Euclid’s notion of points and lines in a plane can be extended a bit to speak and reason about some important new geometry. We add to Euclid’s lines one more called the line at infinity and a new set of points thereon. To save typing we write lai for line at infinity. By ‘ordinary’ point or line, we mean one that Euclid taught and henceforth ‘line’ and ‘point’ are taken to include those at infinity.

Euclid explained everything about ordinary points being on ordinary lines and so we must explain how this notion extends to our new elements. Any two ordinary lines intersect at just one point if they are not parallel. Any ordinary line intersects the lai at just one point. Two ordinary parallel lines intersect the lai at the same point and the set of ordinary lines that meet at some point at the lai. Two parallel lines do not meet for Euclid but they intersect at some point at our lai. Note that in newpseak we can say that two lines intersect in just one point whereas Euclid had to exempt parallel lines.

Euclid had a notion of distance between points. That notion must be left behind here for we cannot ascribe a distance between points on our line at infinity. The only relation of the new geometry to the old is that whenever an ordinary point is on an ordinary line, the same relationship pertains to the same point and line in the new geometry, and conversely.

In the new geometry the lai becomes like any other line if we also let go of the parallel notion. More precisely we note that the line at infinity is ‘congruent’ with any other line. “Congruent” is in quotes because we are limited here to a weakened equivalence notion that omits distance. More precisely we can find a mapping of the points of our new geometry onto themselves that carries lines (conceived as point sets) onto lines. (This is an automorphism.)

Concrete Projective Plane

Here is concrete model of the projective plane, restricted to points with rational coordinates. Projections might move rational points to irrational points, but not if we restrict our projections to rational ones. Rational projections are dense among all projections.

Consider the set of triples of rational numbers excluding (0, 0, 0). We immediately assume an equivalence relation that says that if r is rational and not zero then (x, y, z) is equivalent to (rx, ry, rz). Each triple is thus equivalent to some triple of integers.

Our style of model requires a set of points and a set of lines. Rational triples serve for both sets in our model but the point (2, 3, 4) has absolutely no relationship with the line (2, 3, 4) and thus we write P(2, 3, 4) for the point and L(2, 3, 4) for the line respectively. Here we write “P(x, y, z) = P(x', y', z')” to mean that there exists a non-zero rational such that rx=x', ry=y' & rz=z'. Ditto for the lines.

P(x, y, z) lies on L(a, b, c) just if ax + by + cz = 0. And that completes the model. If we select a coordinate system in Euclid’s plane and relate a point with rational coordinates (x, y) there with P(x, y, 1) we have an isomorphism with our earlier model. The line at infinity that we required in our earlier model are points here of the form P(x, y, 0) where x and y are not both 0.

This model extends easily to arbitrary fields but for 2D projective geometries. Classic 3D projective geometry deals with points, lines and planes. It seems that the algebraic properties of the Grassmannian is exactly what we need here. Topological properties of the Grassmannian are generally absent but the algebra is just right.

We show that this model fits these axioms.

∀l∃x∃y∃z(xl ∧ yl ∧ zl)
∀x∀y∃l(xl ∧ yl ∧ ¬∃m(xm ∧ ym))