         Projective geometry is concerned with incidences, that is, where elements such as lines planes and points either coincide or not. The diagram illustrates DESARGUES THEOREM, which says that if corresponding sides of two triangles meet in three points lying on a straight line, then corresponding vertices lie on three concurrent lines. The converse is true i.e. if corresponding vertices lie on concurrent lines then corresponding sides meet in collinear points. This illustrates a fact about incidences and has nothing to say about measurements. This is characteristic of pure projective geometry.

It also illustrates the PRINCIPLE OF DUALITY, for there is a symmetry between the statements about lines and points. If all the words 'point' and 'line' are exchanged in the statement about the sides, and then we replace 'side' with 'vertex', we get the dual statement about the vertices.

The most fundamental fact is that there is one and only one line joining two distinct points in a plane, and dually two lines meet in one and only one point. But what, you may ask, about parallel lines? Projective geometry regards them as meeting in an IDEAL POINT at infinity. There is just one ideal point associated with each direction in the plane, in which all parallel lines in such a direction meet. The sum total of all such ideal points form the IDEAL LINE AT INFINITY.

The next figure shows the process of projecting a RANGE of points on a yellow line into another range on a distinct (blue) line. The set of (green) projecting lines in the point of projection is called a PENCIL of lines. The points are indicated by the centre points of white crosses. The two ranges are called PERSPECTIVE ranges. The process of intersection of a pencil by a line to produce a range is called SECTION. Projection and section are dual processes. The above procedure may be repeated for a sequence of projections and sections. The first and last range are then referred to as PROJECTIVE RANGES. If corresponding points of two projective ranges are joined the resulting lines do not form a pencil, but instead very beautifully envelope a CONIC SECTION, that is an ellipse, hyperbola or parabola. These are the shapes arising if a plane cuts a cone, and in fact include a pair of straight lines and also, of course, the circle. Using the dual process a conic can be constructed by points using projective pencils.

There are many theorems that there is no space to explain here. An example is given on the home page showing Pascal's theorem, and illustrations of others are listed below.

A particularly important subject for counter space is that of polarity, which is related to the principle of duality. If the tangents to a conic through a point are drawn, the line joining the two points of tangency is called the POLAR LINE of the point, and dually the point is called the POLE of that line. This is illustrated below. The fact to note here is that the polars of the points on a line form a pencil in a point, which is the polar of that line. The situation is self-dual.

In three dimensions we illustrate the same principle but with a sphere and a point. The cone with its apex in that point, and which is tangential to the sphere, determines a plane (red) containing the circle of contact. That plane is the POLAR PLANE of the point, and the point is the POLE of the plane. Similarly to the two-dimensional case, if we take the polar planes of all the points in a plane, they all contain a common point which is the pole of that plane. Lines are now self-polar.

When counter space is studied this property of points and planes is used to conceptualise the idea of a negative space, as we reverse the roles of centre and infinity. AFFINE AND METRIC GEOMETRY

Infinity is not invariant for projective geometry, in the sense that ideal points may be transformed by it into other points. In a plane the ideal points form an ideal line, and in space they form an ideal plane or plane at infinity. A special case of projective geometry can be defined which leaves the plane at infinity invariant (as a whole) i.e. ideal elements are never transformed into ones that are not at infinity. This is known as affine geometry. A further special case is possible where the volume of objects remains invariant, which is known as special affine geometry. Finally a further specialisation ensures that lengths and angles are invariant, which is metric geometry, so called because measurements remain unaltered by its transformations. OTHER THEOREMS Cross Ratio. The cross ratio of four points is the only numerical invariant of projective geometry (if it can be related to Euclidean space). Flat line pencils and axial pencils of planes containing a common line also have cross ratios. Quadrangle Theorem. If two quadrangles have 5 pairs of corresponding sides meeting in collinear points, the sixth pair meet on the same line. Proof indicated using Desargue's Theorem. Harmonic Range. Construction of two pairs of points harmonically separated, which have a cross ratio of -1. Homology. A basic projective transformation in which corresponding sides meet on a fixed line called the axis, and corresponding points lie on lines through the centre. Pappus' Theorem. This was one of the earliest discoveries, and can be regarded as a special case of Pascal's Theorem. Brianchon's Theorem. This is the dual of Pascal's Theorem although it was discovered independently. MEASURES AND TRANSFORMATIONS

It is best to view the first item before those later in the list. They show repeated transformations of the points on a line. Breathing (or hyperbolic) Measure. A point is shown moving along a line between two invariant points (with construction). Growth Measure with one invariant point at infinity. The ratios of the distances of successive points from the other invariant point are constant. Step (or parabolic) Measure, in which the two invariant points coincide. This is how equal steps appear in counter space for our ordinary consciousness. Step Measure with both invariant points at infinity, which yields equal steps. The proof follows from the fact that triangles on the same base and between the same parallels are equal in area. Circling (or elliptic) Measure in which there are no invariant points. The two auxiliary lines used in the above constructions may be regarded as special cases of a conic.

If you attempt to impose three invariant points on a line (e.g. in the first construction by taking the first corresponding pair as coincident) you will find all points are self-corresponding. This is the Fixed Point Theorem of projective geometry.

The following animations show the application of the above to transformation of a plane, in these examples lines being transformed by means of two measures on two sides of the invariant triangle. Projective transformation in which it is demonstrated that parallelism is not conserved. Affine transformation where two red parallel lines are transformed into two parallel lines (one green and one blue). This is affine because one side of the invariant triangle is at infinity since each measure has an invariant point at infinity.
References 6 and 8 and 9 give a good introduction to projective geometry, where the above facts are proved. 