Projective Geometry - Lecture Notes Chapter 3

3.1 Central Collineations

Throughout this chapter we shall let P be a projective space of dimension d > 1.

Recall that a collineation of P is a bijective map from the points to the points and from the lines to the lines of P, which preserves incidence. It follows immediately from this definition that the line determined by points X and Y must be mapped to the line determined by the images of X and Y, i.e., if is a collineation of P then (XY) = (X)(Y). Another observation is that the set of all collineations of P form a group under the binary operation of composition of maps, called the collineation group of P (algebraists will sometimes refer to it as the automorphism group of P).

If is a collineation of P, a point X for which (X) = X is called a fixed point of . If l is a line and (l) = l, then l is called a fixed line of . It should be noted that the points of a fixed line need not be fixed points, the collineation may just shuffle the points of a fixed line amongst themselves. To emphasize this fact, we sometimes say that the points of a fixed line are stabilized by the collineation. The stronger condition, in which all the points of a fixed line are actually fixed points, can be described by saying that the the line is fixed pointwise. These modifying terms (stabilized, fixed pointwise) can be applied to any set of points, not just sets of points on a line.

We will start our examination of collineations by looking at a special type of collineation.

Def: A collineation of P is called a central collineation if there exists a hyperplane H (called the axis of the collineation) and a point C (called the center of the collineation) such that:

  1. Each point of H is a fixed point, and
  2. Each line through C is a fixed line (stabilized).
Examples: Consider the three fundamental types of collineations in the affine Euclidean plane, translations, rotations and reflections. As we wish to look at collineations of a projective space, we need to take the projective closure of this affine plane, the real projective plane, and extend these maps to the "points at infinity" by considering what they do to parallel classes of lines.

First consider a reflection with reflection axis l. The affine points of l are all fixed by the reflection. Since a line parallel to l is reflected to another line parallel to l, the parallel class that contains l is stabilized by the reflection. This means that the infinite point on l, which is this parallel class is a fixed point. So all the points on l (in the projective closure) are fixed points of this hyperplane. The only lines in the affine plane other than l which are fixed by the reflection are those which are perpendicular to l. These lines are all in one parallel class. The point at infinity corresponding to this parallel class is thus a point through which all the lines are fixed lines of the reflection (one easily sees that the line at infinity, which also goes through this point, is also a fixed line). Thus, a reflection extended to the projective closure has a center and an axis and is therefore a central collineation.

Now consider a rotation by an angle different from 0o or 180o with center C. The only fixed point in the affine plane is the point C, and there are no fixed lines. Parallel classes of lines are mapped to different parallel classes, so the line at infinity is a fixed line, but not fixed pointwise. This collineation is therefore not a central collineation.

Finally, consider a translation by the vector (a,b). No point of the affine plane is fixed by the translation. A line is mapped to a parallel line, so parallel classes are stabilized, thus the points at infinity are all fixed points. The only lines that are mapped to themselves are those that are parallel to the direction of (a,b). So, each line through the infinite point corresponding to this parallel class is a fixed line. Thus a translation is a central collineation with axis the line at infinity and center the point at infinity corresponding to direction of (a,b).

It is easy to see that two fixed lines of a collineation must intersect at a fixed point. Thus, the center of a central collineation is always a fixed point. As we shall now show, since central collineations have so many fixed points and lines, very little additional information beyond the fixed structure is needed to completely specify them.

Lemma 3.1.3: Let be a central collineation of P with axis H and center C. Let P ne C be a point not on H and let P' = (P). Then a is uniquely determined. In particular, the image of each point X that is neither on H nor on PP' (=PC) satisfies

(X) = CX FP',
where F = PX H.

Corollary 3.1.4: (The Uniqueness of Central Collineations). Let be a central collineation of P with axis H and center C that is not the identity. Then:

  1. If P is a point ne C and not on H then P is not fixed by .
  2. The central collineation is uniquely determined by one pair (P,(P)) with P (P).
From the above examples, we see that the center of a central collineation may lie on the axis (like translations) or not (like reflections). When the center of a central collineation lies on the axis, we call the collineation an elation, and when it doesn't we call the collineation a homology. The identity map is considered to be both an elation and a homology.

We now consider the question of the existence of central collineations.

Lemma 3.1.7: Let P be a projective space of dimension at least 2, and let g0 be a line of P. Let P' be the rank 2 geometry consisting of the points of P\g0 and the lines of P other than g0.
Let be a collineation of P'. Then can be uniquely extended to a collineation * of P which fixes the line g0.

Theorem 3.1.8: (Existence of Central Collineations [Baer 1942]). If P is a Desarguesian projective space then, if H is a hyperplane, C, P, P' distinct collinear points with P, P' H, then there is precisely one central collineation of P with axis H and center C mapping P onto P'.