Talk:Imaginary point
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This concept appears to be nonsense
editI may be over my head here, but I'll say it anyway: This article makes no geometric sense, and should be deleted without further ado. In particular, a complex projective space possesses symmetries that place every point on exactly the same footing as every other point (its transforms are transitive). It is therefore nonsense to claim that its points can be classified without first imposing additional structure, for example an identification of several special points. Or, stated another way, it is always possible to choose a basis so that any chosen point of the complex projective plane is an "imaginary point" or not, according to whim. —Quondum 01:56, 8 February 2015 (UTC)
- It's also possible to choose a basis for the Euclidean plane such that any given point of it is real or imaginary, as you please; and yet, the distinction between real and imaginary complex numbers is still meaningful and helpful. My guess is that these points are considered with respect to some specific algebraic curve, so that one can talk about the real and imaginary points of the curve. The transformation you describe might not make sense e.g. if you want to keep the coefficients of the defining polynomial of the curve real. And indeed, searching Google books for "imaginary point" "algebraic curve" finds several relevant references that (because of the metaphorical uses of the phrase) would have been hard to find using "imaginary point" by itself. —David Eppstein (talk) 02:37, 8 February 2015 (UTC)
- Given sufficient context (and structure, as I said), the distinction may be meaningful. This would suggest that the article should include more by way of context than it does. For example, if it was framed "In algebraic geometry ..." (of which I know nothing) I would have made no comment. But your suggestion indicates that some work is to be done to make this article meaningful, and I should retract my suggestion about immediate deletion. Surely you will agree that with no qualification other than working in a complex projective space, the distinction as defined here between real and imaginary points cannot be made. For example, if one makes the identification of the (geometric, i.e. as a homogeneous space) complex projective line with the Riemann sphere, one is adding structure (which identifies specific points including 0, 1 and ∞ that are not identified in the purely geometric context). It is no surprise that one can identify real and not-real ("imaginary"?) points once such an identification has been made. I am having difficulty seeing enough of the books on Google to find a definition.
- A related observation is that if one is studying a real projective space as embedded in its complexification (a fairly natural thing to do, as Kaplansky appears to do, even describing points as imaginary [Linear Algebra and Geometry, p. 113: "Our ancestors showed considerable courage in introducing the circular points at infinity, points that are ghostly for two reasons: they are at infinity and they are imaginary"],
though my guess is that he meant pure imaginary, not simply complex), the concept probably makes perfect sense. I would understand it as any point not in the embedded real (sub)geometry :) —Quondum 03:52, 8 February 2015 (UTC)- Okay, I get it. The term imaginary point (along with real point, conjugate imaginary point, real and imaginary line, plane etc.) occurs in projective geometry with great frequency (e.g. [1]). This appears to correspond to my "related observation" above, namely the study of a real projective geometry embedded in its complexification. Once the real subspace has been identified (thus adding the structure of a privileged subspace of the geometry), the description in this article makes sense. I think it would make sense to merge all these terms into one article. I'm not sure what it should be called. It seems to me to be a complex projective geometry in which (in the Kleinian sense) the group of symmetries has been restricted from GL(n+1,C)/C× to its subgroup GL(n+1,R)/R× (or whatever it is), along with the orbit of any single point being selected as the "real" subspace. —Quondum 07:54, 8 February 2015 (UTC)
- I agree that it makes more sense to merge with real point than to keep separate articles, and also to describe it as something that makes sense for a real geometry embedded in a complex one rather than for a single geometry without added structure (as it is now). —David Eppstein (talk) 08:12, 8 February 2015 (UTC)
- Okay, I get it. The term imaginary point (along with real point, conjugate imaginary point, real and imaginary line, plane etc.) occurs in projective geometry with great frequency (e.g. [1]). This appears to correspond to my "related observation" above, namely the study of a real projective geometry embedded in its complexification. Once the real subspace has been identified (thus adding the structure of a privileged subspace of the geometry), the description in this article makes sense. I think it would make sense to merge all these terms into one article. I'm not sure what it should be called. It seems to me to be a complex projective geometry in which (in the Kleinian sense) the group of symmetries has been restricted from GL(n+1,C)/C× to its subgroup GL(n+1,R)/R× (or whatever it is), along with the orbit of any single point being selected as the "real" subspace. —Quondum 07:54, 8 February 2015 (UTC)
Geometry with imaginary elements
editI'm trying to save here some of the concepts that might be pulled together to expand Imaginary point, Real point, Circular points at infinity, Isotropic line etc. into a full topic (name uncertain for now). For now this might not be well-organized, and anyone who wants to add or comment is welcome to do so, or even to take over the initiative. (I'm not known for developing whole articles...)
It seems to me that the development of geometry historically went through a phase of generalization: addition of points at infinity allowed exceptions to be removed from some statements, e.g. "two distinct coplanar lines [that are not parallel] intersect at one point". Addition of complex points allowed further exceptions to be removed, e.g. "there are two [or no] lines from a point not on a conic tangent to the conic". A key point here is that for the application of the concept of imaginary elements, the real points are a privileged set, the most general framework appears to be a specialization of complex projective geometry, which accommodates the same concepts (imaginary points etc.) within complexification of the Euclidean, elliptic etc. contexts. We should find a reference confirming this fundamental point.
The concepts are often expressed within the framework of complex projective geometry using homogeneous coordinates, but with specializations, e.g. (largely guessing/speculation on my part for now)
- A geometry of this type may be described as real geometry (any of the specializations of projective real geometry, such as conformal, affine, Euclidean, elliptic) embedded in a complex projective geometry.
- A real point is one with homogeneous coordinates [λx,λy,...,λz] with complex λ and real x,y,...,z; every other point is an imaginary point.
- A real line is a line defined by two distinct real points (but also contains imaginary points); every other line is an imaginary line.
- A real plane is a plane defined by three noncollinear real points; every other plane is an imaginary plane.
- And so on for all flats. In homogeneous coordinates, a real flat is the set of (complex) solutions to a set of homogeneous linear equations with real coefficients.
- Real and imaginary conics can be defined.
- A circle is a conic through the two circular points at infinity (Kaplansky, Linear Algebra an Geometry, p.115) – presumably in the context of the (complexified) Euclidean plane, and conceivably also conformal geometry.