Two Models of Projective Geometry Model 1: the Fano Plane Http

Model 1: The Fano Plane

http://directrelativity.blogspot.com/2012/11/fano-plane.html The Fano plane is a projective geometry with 7 points and 7 lines. In the picture, the points are the nodes labeled 1–7, and the lines are the sides, the bisectors, and the incircle of the triangle.

What matters is the abstract pattern—which points lie on which lines, and whether the axioms of a projective geometry are satisﬁed (through any two distinct points there is exactly one line, any two distinct lines intersect in exactly one point, and there are 4 points no 3 of which lie on a line).

What does not matter is that one of the 7 lines isn’t a line in the usual sense of Euclidean geometry, and that there are 3 intersections without labels. A point or line is what we deﬁne it to be.

Sylvester-Gallai implies that there is no way to represent the Fano plane using only straight lines. So the fact that we used a circle to represent one of the lines of the Fano plane is not a coincidence.

Model 2: The Real Projective Plane

Projective geometry was originally introduced to repair a defect of Euclidean geometry: that parallel lines in the Euclidean plane do not intersect even though they appear to intersect at inﬁnity (e.g. the 1 point of intersection of the lines y = 0 and y = 1 − x is ( , 0), which moves to inﬁnity as → 0).

You should check that the following deﬁnitions of points and lines satisfy the axioms of a projective geometry. This model of projective geometry is called the real projective plane.

There are two kinds of lines. First, take an arbitrary straight line L in the Euclidean plane and extend it by adding a point ∞L to it (here ∞L is just a formal symbol or label, not an actual physical point in the plane or in space or anywhere else, and we deﬁne ∞L to be equal to ∞M if and only if L, M are parallel). Secondly, the set consisting of all of these “new” points ∞L (where L now ranges over all possible straight lines in the Euclidean plane) is itself considered to be a line.

Points in the real projective plane are points in the usual sense, or are of the form ∞L for some L.

To check that the axioms are satisﬁed requires several steps. For example, if L, M are straight lines in the Euclidean plane, we need to show that their extensions (augmentations by ∞L, ∞M ) do intersect as lines in the real projective plane. Here is how. If L, M are not parallel, then they have exactly one point of intersection in the Euclidean plane, and this is by deﬁnition considered to be a point in the real projective plane as well. But if L, M are parallel, then ∞L = ∞M , so the extended lines do intersect, even though their point of intersection, ∞L (or ∞M ), is not a conventional point.