Chapter: Basic Geometric Concepts in Projective Geometry

Projective geometry builds on some simple mathematical concepts also known in the Euclidean geometry but with some subtle differences. In the following, we want to refer to the concepts known from the Euclidean geometry and explain how the concepts differ in projective geometry.


Naturally, we think of points as infinitesimal dots with a "position" but "without any magnitude". This concept of a point is also used in projective geometry, however, "position" only has a relative meaning, i.e. in relation to other geometrical objects like lines or planes, not in the absolute meaning of distances or coordinates. For instance, in projective geometry it perfectly makes sense to speak about "points lying on a line", but sentences like "The distance of two points is $2$.", or a "The point in the plane has the coordinates $(1,2).$" do not make any sense!

In the course of the next text we will extend the Euclidean definition to allow points at infinity - i.e. infinitely distant points on a straight line.

Notation: In the following, we will consistently denote points with Latin capital letters, e.g. $A,B,C,\ldots$

Straight Lines

Since distances and lengths do not play any role, projective geometry does not study segments. Even rays are not objects of study in projective geometry. Therefore, in projective geometry, straight lines always have unlimited extend in both directions.

In the course of the text to follow we will extend the Euclidean definition of a straight line to allow a straight line at infinity - i.e. a straight line connecting two distinct points at infinity.

Notation: In the following, we will consistently denote straight lines with small Latin letters, e.g. $k,l,m\ldots$


The concept of a plane is the same as defined in the Euclidean geometry, with the difference that also contains all points and lines at infinity.

Notation: In the following, we will consistently denote planes with small Greek letters, e.g. $\alpha,\beta,\gamma\ldots$

The principle of duality in a plane asserts that every definition in the projective geometry remains meaningful and every theorem remains true, when we interchange the words "points" and "straight line", and consequently also certain other pairs of words such as "collinear" and "join" and "meet", Other simple geometrical concepts, which are specific to projective geometry, will be provided in the following text. These concepts include

  1. Definition: Incidence
  2. Definition: Collinear Points
  3. Definition: Concurrent Straight Lines
  4. Definition: Coplanar Points and Straight Lines
  5. Definition: Projectivities, Ranges and Pencils
  6. Definition: Perspectivities

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