Line geometry, as its name indicates, studies systems of straight lines in 3-dimensional space. Like with Euclidean Geometry, Line geometry gets a better understanding when
studied in the extended space, viz. in the real projective space of dimension 3.
The subject was born, one could say, in 1868, when Julius Plücker finished the first
volume of his Neue Geometrie des Raumes. It florished in the period around the turn
of the 19th century into the 20th and was very much embraced by physicists. Though
it never really disappeared, after WWI it was rather neglected until George Adams
wrote several studies in the years 1934-1939. In the 1970s Peter Gschwind wrote
about the Linear Complex and in 1981 Renatus Ziegler gave his first account on line
geometry. Two interesting books by Stoß followed and in 2012 Ziegler published an
extended (English) version of his 1981-book. Half of this last book is on general pro-
jective geometry, and Ziegler deliberately restricted to the synthetic treatment of his
subjects – as did Stoß. The analytic approach, however, is an important counterpart to the synthetic one, and it is a real joy to discover the differences in proofs between the two approaches.
By the way, above them reigns algebra. In this book geometric objects are treated as
algebraic ones, with the fundamental relation of containment (≺) or incidence, and
the basic operators meet (∧) and join (∨). In chapter 1 this is summarized, as are the
most important issues of elementary projective geometry. In the next two chapters
the parabolic strip and the regulus are treated, after which proper line geometry starts with the concept of linear dependency of lines, synthetically as well as analytically.
Chapter 5 is about linear congruences, chapter 6 about linear complexes. Only then
it is possible to finish the treatment of dependency of lines.