incidence geometry - eiu

Incidence Geometry

September 22, 2009

c© 2009 Charles Delman

What is Incidence Geometry?

v Incidence geometry is geometry involving only points, lines, and theincidence relation. It ignores the relations of betweenness and congruence.

v Incidence geometry based on the three neutral incidence axioms mightbe termed neutral incidence geometry.

v These axioms might be modified or expanded to obtain special types ofincidence geometry.

c© 2009 Charles Delman 1

DefinitionsRecall the following definitions based on the relation of incidence:

v A set of points is collinear if there is a common line incident with all ofthem. A set of points that is not collinear is called noncollinear.

v A set of lines is concurrent if there is a common point incident with allof them.

v Two lines are parallel if there is no point incident with both of them.(That is, they do not intersect; in this case, the word “intersect” isgiven a meaning distinct from its set-theoretic one, since lines are notnecessarily considered to be sets of points.) More generally, a set oflines is parallel if, for any two distinct lines in the set, there is no pointincident with both of them.


The Axioms of Neutral Incidence Geometry

Recall the three neutral incidence axioms:

v Axiom I-1: For every point P and for every point Q that is distinct fromP , there is a unique line l incident with P and Q. (That is, there is aline incident with both P and Q, and if lines l and m are each incidentwith both P and Q, then l = m.)

v Axiom I-2: For every line l there exist (at least) two points incidentwith l.

v Axiom I-3: There exist three distinct noncollinear points.


Recall also the following notation:

v The unique line incident with points P and Q is denoted by←→PQ.


Propositions of Neutral Incidence Geometry

It is instructive and useful to establish some theorems that can be provenusing only these three axioms. In fact, for the following five propositions, wewill not even need to use Axiom I-2! Of course, these theorems remain true(and proven) when more axioms are added. The propositions are numberedin accordance with the textbook.

v Proposition 2.1 If l and m are distinct lines that are not parallel, thenl and m have a unique point in common.

To get a feel for this proposition, let us consider an equivalent statement,its contrapositive.

v Proposition 2.1 If l and m do not have a unique point in common, thenl and m are parallel or l = m.


A Proof of Proposition 2.1

Proof. [This proof is based on the first statement: If l and m are distinctlines that are not parallel, then l and m have a unique point in common.]

Let l be a line, and let m 6= l be a line that is not parallel to l. [Here weassume the conditions of the theorem. Note that l can be any line, butthen restrictions depending on l apply to our choice of m.]

Claim 1. Lines l and m have a point in common. The claim followsimmediately from the definition of parallel.

Claim 2. Lines l and m do not have two distinct points in common. LetP be a point they have in common. [Choosing such a point P is justifiedby Claim 1.] By way of contradiction, suppose Q lies on both l and mand Q 6= P . [Note that this is really an existence statement: there existsa point Q lying on both l and m such that Q 6= P . Alternatively, wecould have let Q be a point on both l and m (which we know exists -


the set of such points is not empty, by Claim 1) and supposed only thatQ 6= P . Then we would be proving (still by contradiction) the equivalentuniversal statement that, for all points Q lying on both l and m, Q = P .In general, (6 ∃a ∈ A)p(a) is logically equivalent to (∀a ∈ A)¬p(a). ]Since Q 6= P , it follows by the uniqueness conclusion of Axiom I-1 thatl = m. This contradicts our initial assumption that l 6= m; therefore, weconclude there is no such point Q, so P is unique.


To summarize, without the commentary, we have:

Proof. Let l be a line, and let m 6= l be a line that is not parallel to l.

Claim 1. Lines l and m have a point in common. The claim followsimmediately from the definition of parallel.

Claim 2. Lines l and m do not have two distinct points in common. LetP be a point they have in common. By way of contradiction, supposeQ lies on both l and m and Q 6= P . Since Q 6= P , it follows by theuniqueness conclusion of Axiom I-1 that l = m. This contradicts ourinitial assumption that l 6= m; therefore, we conclude there is no suchpoint Q, so P is unique.


Another proof of Proposition 2.1

Proof. [This proof is based on the second statement: If l and m do nothave a unique point in common, then l and m are parallel or l = m.]

Assume it is not the case that l and m have a unique point in common.Then either they have no point in common or there exist two distinctpoints that they have in common.

Case 1: l and m have no point in common. Then, by definition, they areparallel.

Case 2: There are two distinct points common to l and m. Then by theuniqueness conclusion of Axiom I-1, l = m.


v Proposition 2.2 There exist three distinct lines that are not concurrent.

v Proposition 2.3 For every line, there is at least one point not incidentwith it.

v Proposition 2.4 For every point, there is at least one line not incidentwith it.

v Proposition 2.5 For every point, there are at least two lines incidentwith it.


Parallelism Properties

v Now that we have seen some models of incidence geometry, we see thatdifferent properties with respect to parallel lines may apply. The sameproperty need not even apply to all the lines and points of a given model.However, geometries with uniform properties are generally of greaterinterest, and the following three possibilities are most important.

© If l is a line and P is a point not lying on l, then there is a unique linethrough P that is parallel to l. (Euclidean Parallel Property)

© If l is a line and P is a point not lying on l, then there exist twodistinct lines through P that are parallel to l. (Hyperbolic ParallelProperty)

© Any two distinct lines are incident with a common point. (That is,if l is a line and P is a point not lying on l, then there are no linesthrough P that are parallel to l. (Elliptic Parallel Property)


Affine and Projective Geometry

v Incidence geometry with the additional assumption of the EuclideanParallel Property is called affine geometry.

v A model of affine geometry is called an affine plane.

v There is a very important construction, inspired by visual perspective,that adds points to an affine plane in order that all lines intersect. Theresulting new model is called the projective completion of the affineplane. (The name “projective” comes from the fact that rays of lightproject from points in space to our eyes.)


The Plane in Perspective

When a horizontal plane is viewed in perspective, parallel lines appear tomeet at a point infinitely far away. All the points where parallel lines canmeet appear to lie on a line, which we call the horizon.


This is because the rays of light from the two lines to a fixed point,representing an eye, form two intersecting planes. (The rays from behindyour head would cause the lines to continue beyond the horizon if youcould see them.)


If you imagine the rays of light projecting onto a sphere with your eye asa point at the center (the visual sphere), the two lines would project togreat circles intersecting at a pair of antipodal points.

By analyzing the geometry of projected light, artists and mathematiciansduring the renaissance learned to produce realistic depictions of sceneson a planar surface, as if the surface of the painting or drawing were awindow. Their analysis gave rise to projective geometry.


The Projective Completion of an Affine Plane

v To create the projective completion of an affine plane, we define twolines of the affine plane to be equivalent if they are parallel or equal (thesame). It is obvious that this relation is reflexive and symmetric. Thatit is transitive requires a bit of thought.

v We define a point of the projective completion to be either a point ofthe original affine plane or an equivalence class of lines. Each of the newpoints is called a point at infinity. We also introduce a new line, l∞, theline at infinity or horizon, which is incident with each of the points atinfinity (but not any of the original points of the affine plane). A line ofthe original affine plane passes though a point on the horizon if it is inthe equivalence class that defines that point.


In summary:

v Let A denote the original affine plane. Let A∗ denote its projectivecompletion.

© If l is a line of A, then l is incident with [l]. ([l] denotes the equivalenceclass of l under the relation just defined.) No other points at infinity lieon l.

© If l is a line of A, then the point [l] lies on l∞. No other points lie on l∞.© All the original incidence relations of A apply in A∗.

v It turns out that if we forget about how the line and points at infinitywere created, they do not have any features that distinguish them fromany other lines or points, respectively.

v Example: The projective completion of the four-point affine plane is theseven-point projective plane.


Projective Geometry

v It is easy to check that A∗ satisfies the following four axioms. (Notethat you must check several cases, depending on whether or not eachpoint or line lies at infinity.)

© (Axiom I-1) If P and Q are distinct points, there is a unique line incidentwith both of them.

© (Axiom I-2+) Every line is incident with at least three distinct points.© (Axiom I-3) There exist three noncollinear points.© (Elliptic Parallel Property) Any two distinct lines intersect.

v These are the axioms of projective geometry. A model of projectivegeometry is called a projective plane.

v Projective geometry has many applications, from abstract algebra tocomputer graphics.


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