Let $\Pi$ be an affine plane (An affine plane is a set of points and subsets called lines Satisfying (I1), (I2), (I3), (The axioms of Incidence) and the following stronger form of Playfair's axiom.

$P'$: For every line $l$, and every point A, there exists a unique line m containing A and parallel to $l$.

A pencil of parallel lines is the set of all the lines parallel to a given line (including that line itself). We call each pencil of paralleI lines an "ideal point," or a "point at infinity," and we say that an ideal point "lies on" each of the lines in the pencil. Now let $\Pi'$ be the enlarged set consisting of $\Pi$ together with all these new ideal points.

A line of $\Pi'$ will be the subset consisting of a line of $\Pi$ plus its unique ideal point, or a new line, called the "line at infinity," consisting of all the ideal points.

Question: I am asked to show that this new set $\Pi'$ with subsets of lines as just defined forms a projective plane.

This feels like when you draw something in $3D$ on a piece of paper one uses the concepts of line at infinity for things like tileings going off into the "horizon"

A projective plane is a set of points and subsets called lines that satisfy the following four axioms:

Pl. Any two distinct points lie on a unique line.

P2. Any two lines meet in at least one point.

P3. Every line contains at least three points.

P4. There exist three noncollinear points.

I believe we get P4 for free and P3 follows almost immediately from every line having 2 points and its ideal point laying on the same line which implys at least 3 points lay on every line .

Proving that each ideal point lies on a unique line im not sure and how to form P2 from playfairs axiom i am also not sure.


1 Answer 1

  1. Any two ordinary points lie on an ordinary line. Any two points at infinity lie on the line at infinity. If $A$ is an ordinary point and $B$ is a point at infinity, then we can show that $A$ and $B$ lie on a line as follows. First, let $\ell$ be one of the lines with $B$ as its ideal point. We know that an ordinary line passes through $B$ if and only if it is parallel to $\ell$. So, we need a line $m$ which is containing $A$ and parallel to $\ell$. Can you see what to do at this point?

  2. Any two non-parallel ordinary lines meet at a point on $\Pi$. An ordinary line $\ell$ meets the line at infinity, at the ideal point of $\ell$.

  3. You've given a reason why every ordinary line has at least three points. What about the line at infinity? Are there at least three ideal points? To show this, you might want to show that there are at least three non-parallel lines in $\Pi$.


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