The paper [Kötter's synthetic geometry of algebraic curves](http://journals.cambridge.org/action/displayFulltext?type=1&fid=3146652&jid=PEM&volumeId=7&issueId=-1&aid=3131388&bodyId=&membershipNumber=&societyETOCSession=), (N. Fraser, Proceedings of the Edinburgh Mathematical Society 7, 46–61, 1888) opens with a sketch of what appears to be a synthetic construction of the complex projective plane from the real one. "Imaginary points" are defined to be projective involutions on real lines without real fixed points. This looks very cool, but there are a lot of details left out; is it written down carefully anywhere?


**Edit:** Here is a quote to give more of the idea:



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> An elliptic involution has, we know, no real double elements. Imaginary points in a line are defined as the double points of the various elliptic involutions which may be formed of the real points of the line. Each involution thus yields a pair, an imaginary point and its conjugate, which, like the involution itself, are completely determined when two real pairs of the involution, say $AA'$ and $BB'$, are given. These pairs will divide one another, $B$ and $B'$ lying the one in the finite, and the other in the infinite line $AA'$. We denote the two imaginary points which they determine by writing them down in order, as $ABA'B'$. It remains to distinguish between the two imaginary points thus denoted... by the order in which we name the four points which determine the involution; and if we denote the one by $ABA'B'$, we denote the other by $AB'A'B$...
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> In exactly the same way imaginary lines, through a real point, are defined as the double rays of an elliptic involution in a pencil, and are denoted by $aba'b'$, $ab'a'b$, respectively.
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> An imaginary line is said to contain an imaginary point, if it be possible to represent the one by $aba'b'$ and the other by $ABA'B'$, where $a$, $b$, $a'$ and $b'$ are the lines joining $A$, $B$, $A'$ and $B'$ to $O$, the real point of the imaginary line.
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A geometric construction of the complex projective plane?
Perhaps you are looking for an exposition of von Staudt's theory of complex elements in synthetic projective geometry.


A bit of history: Poncelet, in his work of the early 19th century, which launched the explosion of projective geometry, reasoned synthetically, but somewhat mysteriously, with imaginary points. At mid-century, von Staudt attempted to create a purely synthetic (and entirely non-metrical) foundation of projective geometry, which would include the imaginary objects that Poncelet and others found convenient to reason with. The new synthetic theory would, for example, make precise the sense in which a line and a conic are either tangent or meet in two distinct points, which may be a pair of conjugate imaginary points. The theory of von Staudt was successful as far as it went, but synthetic methods in general fell out of fashion as the analytic techniques of invariant theory and the study of higher-order curves and surfaces became more popular. One can, however, find discussions of von Staudt's theory in old texts written in the synthetic spirit.


For an overview of the subject in English, one might look at Coolidge's *A History of Geometrical Methods*. The part of that long book addressing this chapter in the history of projective geometry is essentially the same as Coolidge's lecture published as "The Rise and Fall of Projective Geometry" in *The American Mathematical Monthly*, Vol. 41, No. 4 (Apr., 1934), pp. 217-228. (That lecture contains a few comments on Kötter specifically as well, which you may find amusing, if you are interested in his work.) A search in Google books will yield many of the old textbooks discussing von Staudt's approach. One might read, for example, Chapter XX of *Projective Geometry* by G. B. Mathews. If you read German, you can look at von Staudt's original treatment in his *Beiträge zur Geometrie der Lage*, various editions of which are freely available online. All of these treatments are missing a satisfying treatment of completeness and order ("sense," which the old authors tried to use to distinguish the two complex points corresponding to an elliptic involution) for real projective geometry. To fill in the missing details, one can see volume 2 of Veblen and Young's comprehensive *Projective Geometry*, which, I believe, was written by Veblen alone. (Some hints on the theory of order--based the four-term separation predicate rather than the three-term betweenness predicate of formalized Euclidean geometry--can also be found in more recent texts such as Coxeter's *Non-Euclidean Geometry*.)


