Geometry :: Notes :: Incidence Geometry

1 Incidence Geometry

For the incidence geometry we will have three undefined terms: point, line, and incident. We may think of incident as lies on. So if \(P\) is a point and \(l\) a line, we can say \(P\) is incident with \(l\) or \(P\) lies on \(l\) to mean the same thing. Incident is preferred since we can say either \(P\) is incident with \(l\) or \(l\) is incident with \(P\).

For every pair of distinct points \(P\) and \(Q\), there exists a unique line \(l\) incident with \(P\) and \(Q\). We say that "\(l\) is the line joining \(P\) and \(Q\)", and denote it by \(\overleftrightarrow{PQ}\).

For every line \(l\), there exist at least two distinct points incident with \(l\).

There exist three distinct points with the property that no line is incident with all three of them.

Three or more points \(A,B,C, \ldots\) are collinear if there exists a line incident with all of them. The points are noncollinear if no such line exists.

Three or more lines \(l,m,n,\ldots\) are concurrent if there exists a point incident with all of them. The lines are nonconcurrent if no such point exists.

Lines \(l\) and \(m\) are parallel if they are distinct and no point is incident with both of them. If \(l\) and \(m\) are parallel we write \(l \parallel m\).

Interpret point to mean one of the letters \(A,B,C\), and interpret line to mean a set of two points. Finally interpret incident to mean "is an element of" (or "contains" if going the other direction). List all lines in this interpretation and show that this interpretation is a model for incidence geometry.

Interpret point to mean one of the letters \(A,B,C\), and interpret line to mean the set of all points. Finally interpret incident to mean "is an element of" as in the Three Point Geometry Example. List all lines in this interpretation. Is this interpretation a model for incidence geometry?

Interpret point to mean one of the letters \(A,B,C,D\), line to mean a set of two points, and incident to mean "is an element of". How many lines are there? Is this interpretation a model for incidence geometry?

Interpret point to mean one of the letters \(A,B,C,D,E\), line to mean a set of two points, and incident to mean "is an element of". How many lines are there? Is this interpretation a model for incidence geometry?

Let \(0 < N \in \mathbb{Z}\). Interpret point to mean one of the numbers \(1,2,\ldots N\), line to mean a set of two points, and incident to mean "is an element of". How many lines are there? Is this interpretation a model for incidence geometry?

Interpret point to be any ordered pair \((x,y)\) of real numbers. Interpret line as the collection of points which satisfy the equation \(ax+by+c = 0\), where \(a,b\) and \(c\) are real numbers and \(a\) and \(b\) are not both 0. Interpret incident as \((x_0,y_0)\) lies on a line if that point satisfies the lines equation. Show that this is a model for incidence geometry. We use \(\bbR^2\) to denote the set of points in this model.

Interpret point to mean an ordered triple \((x,y,z)\) of real numbers so that \(x^2+y^2+z^2 = 1\). This set of points is called the 2-sphere and is denoted by \(\mathbb{S}^2\). A great circle is a circle on the sphere whose radius is equal to that of the sphere. Interpret line to mean a great circle on the sphere and incident to mean "is an element of". Is this interpretation a model for incidence geometry?

If \(l\) and \(m\) are distinct lines that are not parallel, then \(l\) and \(m\) have a unique point in common.

There exist three distinct lines that are not concurrent.

For every line, there is at least one point not incident to it.

For every point, there is at least one line not passing through it.

For every point \(P\), there exists at least two distinct lines through \(P\).

Interpret point to mean a point in the Cartesian Plane and inside the unit circle, that is a pair \((x,y)\) so that \(x^2 + y^2 < 1\). Interpret line to mean the part of a Euclidean line which lies inside the unit circle. Finally interpret lies on (incident) to be the Euclidean meaning, see the Cartesian Plane Example. Show this is a model for the incidence geometry.

Interpret point to mean an element of the set \(\{A,B,C,D,E,F,G\}\) and line to mean an element of the set

\begin{equation*} \{ \{A,B,C\}, \{C,D,E\}, \{E,F,A\}, \{A,G,D\}, \{C,G,F\}, \{E,G,B\}, \{B,D,F\} \}. \end{equation*}

Lies on means element of. Show this is a model for the incidence geometry. Draw a picture of this model.

Two models for the incidence geometry are isomorphic if:

there is a one to one correspondence \(P \leftrightarrow P'\) between the points in the models;
there is a one to one correspondence \(l \leftrightarrow l'\) between the lines in the models;
and \(P\) is incident with \(l\) if and only if \(P'\) is incident with \(l'\).

Such a correspondence between the models is called an isomorphism.

Let the set of lines to be \(\mathcal{L} = \{a,b,c\}\) and the set of points to be the two element subsets of \(\mathcal{L}\). Let incidence be set membership; so the point \(\{a,b\}\) is incident with lines \(a\) and \(b\). Show this is a model for the incidence geometry. It is the dual of the model given in Three Point Geometry Example.

Give an isomorphism between the model in Three Point Geometry and the model in Dual Three Point Geometry.

Give the "dual" of the four point model given in Four Point Example. Is this a model? Does an isomorphism exist between the four point model and its dual?

For every line \(l\) and for every point \(P\) that does not lie on \(l\), there is no line \(m\) such that \(P\) lies on \(m\) and \(m \parallel l\).

For every line \(l\) and for every point \(P\) that does not lie on \(l\), there is exactly one line \(m\) such that \(P\) lies on \(m\) and \(m \parallel l\).

For every line \(l\) and for every point \(P\) that does not lie on \(l\), there are at least two lines \(m\) and \(n\) such that \(P\) lies on both \(m\) and \(n\), \(m \parallel l\) and \(n \parallel l\).

Classify each of the three, four, and five point geometries given in Three Point Geometry Example, Four Point Geometry Example, and Five Point Geometry Example respectively as having either the Elliptic, Euclidean, or Hyperbolic Parallel Property (or none of the properties).

Classify each of the Cartesian Plane, the Sphere, and the Klein Disk as having either the Elliptic, Euclidean, or Hyperbolic Parallel Property (or none of the properties).