Talk:Oval (projective plane)

While I am no expert on ovals, I believe most results and definitions also apply to projective planes in general (even when you don't assume that the order must be a prime power which is still not proven). Of course Segre and the parametrization are not possible in the most general case. Any ideas? Evilbu 23:12, 2 March 2006 (UTC)
 * Do the finite non-desarguesian planes have ovals? If so, I think they'd be good to describe right after the "even q" case and before the "abstract oval" case. JackSchmidt (talk) 19:53, 15 December 2009 (UTC)
 * Page 242 of Hughes and Piper's Projective Planes textbook says that the Hughes plane has ovals that are not the absolute points of a polarity (and so in particular has them). A little bit more is probably needed for a whole section.  Someone more familiar with the area might find enough in that chapter, but most of it is phrased in terms of polarities, rather than just finding ovals.  Apparently though it is "easy" to find ovals (and hard to classify). JackSchmidt (talk) 20:04, 15 December 2009 (UTC)
 * Actually they are not very "easy" to find, I've spent my entire professional life trying to do this and even though I have found more of them than anyone else, I have only found a handful. Wcherowi (talk) 05:39, 13 August 2011 (UTC)

Ovals can be defined in any projective plane, both finite and infinite, Desarguesian or not. In the infinite case you can not use the size of the point set in the definition as this is essentially meaningless. The condition that you want is the uniqueness of tangent lines at each point. In the finite case, where you can count the number of lines through a point, having q+1 points (in a plane of order q) in the set will guarentee that there is a unique tangent at each point. We need to be a little careful with the term tangent line since in projective geometry it is not quite the same concept as a tangent to a curve in Euclidean geometry (tangents are allowed to intersect the curve more than once in Euclidean geometry, it is only locally that the "meets in only one point" condition holds). I have made the appropriate changes in the definition. While the article is almost exclusively concerned with the finite case, I think it is appropriate to have the right definition in case someone would like to add something about infinite ovals (not my cup of tea, but there is an example of an infinite B-oval which does not embed that I might write up at some point). Wcherowi (talk) 05:39, 13 August 2011 (UTC)

I feel an obligation to upgrade this page. To this end I have added a number of references and a description of the known hyperovals in the finite Desarguesian planes. In the future I will discuss the classification of these hyperovals in the small planes, expand the B-oval section, add a section of ovals and hyperovals in non-Desarguesian planes (pretty much examples only as there is not much theory), and add a section on applications (primarily to MDS codes). That should be enough to get the page out of the START class. Wcherowi (talk) 01:00, 14 August 2011 (UTC)

I've done a little reorganizing and put up a subsection on the (hyper)ovals in the small planes. With this organization it will be easy to extend to non-Desarguesian planes. It will take a little longer to put up some of the non-Desarguesian stuff ... the Desarguesian material I already had written up and it was just a matter of modifying it, this other material will be newly written. Wcherowi (talk) 21:14, 15 August 2011 (UTC)

Finite Oval - Oval
There are many examples of ovals in infinite projective planes and results (see Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes. (PDF; 891 kB), S. 43. The article name of this article should be changed to "finite Ovals" or material an ovals in general added.--Ag2gaeh (talk) 08:51, 24 May 2014 (UTC)

Is an oval in the real projective plane necessarily a closed curve?
Could somebody please add a citation to the definition of an oval in the projective plane, particularly for the real projective plane?

I realize that the given definition may be utterly standard for the field, but i still think it would be useful to have a definite source to consult.

For reference, the definition as it currently stands is that an oval is any set of points such that no line intersects it more than twice, and at each point of the set there is a unique tangent line through that point, and the tangent does not intersect the set elsewhere.

This definition allows the empty set, which is probably ok.

One question is whether an oval in the real projective plane is necessarily a (closed) curve (i.e., a continuous one-to-one image of the unit circle).

That sure seems plausible, but i think it would require a little analysis or topology, and i would like to look at a book where oval is defined to see if the point is addressed.

Son of eugene (talk) 19:07, 12 October 2019 (UTC)