6.1: Axioms for Projective Geometry

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6.1.1 Motivating Illustration

Consider the following illustration as a motivation of this geometry. Consider standing in the middle of Kansas looking down a perfectly straight road that extends all the way to the horizon.

Presuming perfect construction, the two sides of the road are lines with what geometric property?
As you look toward the horizon, what do the sides of the road appear to do?
Two lines always intersect in one what?
Consider all the lane line markings (there are more than two). All of these lines are what compared to each other and appear to do what?
If you are in an intersection of two roads (not in the same direction), will the lane markings all converge together?
How many different convergent locations are there?

Definition: Ideal Point

A point is an ideal point if and only if it is the intersection of parallel lines. These are sometimes called "points at infinity."

Definition: Ideal Line

A line is the ideal line if and only if it consists of solely ideal points.

6.1.2 Axioms for Projective Geometry

Axiom: Projective Geometry

A line lies on at least two points.
Any two distinct points have exactly one line in common.
Any two distinct lines have at least one point in common.
There is a set of four distinct points no three of which are colinear.
All but one point of every line can be put in one-to-one correspondence with the real numbers.

The first four axioms above are the definition of a finite projective geometry. The fifth axiom is added for infinite projective geometries and may not be used for proofs of finite projective geometries.

Theorem

A line lies on at least three points.

Theorem

Any two, distinct lines have exactly one point in common.

Lemma

For any two distinct lines there exists a point not on either line.

Theorem

There exists a one-to-one correspondence between the points of any two lines.

Theorem

Every point lies on the same number of lines.

Corollary

A projective plane in which each line lies on exactly k+1 points has a total of k^2+k+1 points and k^2+k+1 lines.

6.1.3 Duality

Definition: Projective Duality

A statement is the projective dual of another statement if and only if one statement is obtained from the other by switching the roles of "point" and "line."

Theorem

Each point is incident with at least three lines.

Theorem

There exist four lines no three of which are coincident in a point.

Theorem

There is a one-to-one correspondence between the real numbers and all but one of the lines incident with a point.

Theorem: Projective Duality

The projective dual of every projective theorem is also true.

Theorem

Every line consists of the same number of points.

Theorem

There exists a one-to-one correspondence between the lines thru any two points.