Incidence Relations in the Plane
and in Space

Incidence relations come in two distinct kinds. In the first kind we start with some simple geometric objects and we look for another which is determined by the given objects. An example of this kind is the most basic of the incidence axioms of planar geometry.

Given any two distinct points in the plane there is a unique line that passes through both of them.

The other kind specifies the intersection of geometric objects. The easiest example of this is the other basic incidence relation of planar geometry.

If two lines intersect, they meet in exactly one point. Lines which do not intersect are called parallel.

Incidence relations in space.

In space we have points and lines, just as in the plane, but we also have a lot of planes. As a result there are a lot more incidence relations. We will simply list them here. First the "determining" relations.

Given any two distinct points in space there is a unique line that passes through both of them.
Given three points which do not all lie on the same line, there is a unique plane that contains all three.
Given a line and a point not on the line, there is a unique plane that contains the line and the point.

Next the intersection relations.

If two planes intersect, they intersect in a line. Two planes that do not intersect are called parallel.
Given a plane and a line, there are three possibilities.
- The line lies totally in the plane.
- The line and the plane intersect in a single point.
- The line and the plane do not interesect. In this case they are called parallel.
If two distinct lines meet, they intersect in precisely one point. If they do not meet there are two possibilities.
- The two lines both lie in a plane. In this case the lines are parallel.
- There is no plane that contains both lines. In this case the lines are skew.

Return to Basic information about spheres.

Go to incidence on the sphere.

Table of Contents.

Return to Geometry of the Sphere.

url: http://math.rice.edu/~pcmi/sphere/linincidence.html

	Return to Basic information about spheres.
	Go to incidence on the sphere.
	Table of Contents.

Incidence Relations in the Plane and in Space

Incidence relations in space.

Incidence Relations in the Plane
and in Space