Talk:Plane curve

Algebraic curve of degree 2

@D.Lazard: You recently added this to the section Plane curve#Examples:

an algebraic curve of degree less than 3 is always contained in a plane

I don't understand why this is true. For example, what about the curve represented by the two second-degree polynomial equations

$x^{2}+y^{2}+z^{2}=a$

$x^{2}+bxy+cy^{2}+dz^{2}=e$

It's an algebraic curve according to Algebraic curve#Non-plane algebraic curves, it has degree 2, but it is not contained in a plane. Loraof (talk) 19:14, 25 September 2017 (UTC)

To editor Loraof: The intersection of two surfaces of degree 2 is a curve of degree 4, except if the surfaces have a common component (in this case the intersection is not a curve) or if the intersection has a component at infinity. A simple proof of the fact that a curve of degree 2 is a plane curve is the following: the intersection of an irreducible curve of degree 2 with a plane either consists of at most two points, or contains the curve. Thus any plane containing three points of the curve contains the curve, and the curve is a plane curve.

It is funny that you ask this question when I was writing Hilbert series and Hilbert polynomial#Complete intersection, which contains a proof that your example has degree 4. D.Lazard (talk) 20:17, 25 September 2017 (UTC)

Merge with algebraic curve?

Much of the content in algebraic curve on plane curves is duplicated here. Either most of that content should be moved here or this article should be merged into algebraic curve. — MarkH21 (talk) 19:00, 14 April 2019 (UTC)

I have not a clear opinion. We have also curve, which has also a large part devoted to algebraic curves. In fact, there are essentially four cases (plane or not, differentiable or not). The distinction between plane or not is not very meaningful in the differential case, as one generally work with a parametrization. In the algebraic case, a plane curve is defined by a single equation, while, in the non-plane case, it may be difficult to distinguishing between a curve and a variety of higher (or lower) dimension. Nevertheless, the study of non-plane curves passes generally by the study of their projections as plane curves. So, an article on algebraic curves can naturally be structured as a first part on plane curves and a second part on non-plane curves, which uses some results of the first part.

A solution may be to transform curve into a dab page (or a short article explaining the two cases) linking to Differentiable curve and Algebraic curve. This would be a major restructuring, so further advices are needed. D.Lazard (talk) 19:50, 14 April 2019 (UTC)

Ah, I do like that idea. Even before looking at curve, it's strange that the almost the entire lead & the first half of the article on algebraic curve is about plane algebraic curves, whereas this article is very short and stubby. Most of Differentiable curve (a very short section) is really discussed at Differential geometry of curves.

My proposal for the details of the restructuring is:

1. differentiable curve being made into a standalone article combining the smooth plane curve section here, differential geometry of curves, and the relevant short parts of curve, with a section on differentiable plane curves, smooth plane curves, etc.
2. algebraic plane curve being made into a standalone article combining the algebraic plane curve section here and a good chunk of the article on algebraic curve, and the relevant parts of curve
3. curve being a DAB / short list article that links to differentiable curve and algebraic curve as well as algebraic plane curve
4. plane curve redirecting to curve

The algebraic curve article can then be shortened and refer to the main article on algebraic plane curves. The existing material on differentiable plane curves seems quite short so it can be a section in the differential curve article for now. — MarkH21 (talk) 04:55, 15 April 2019 (UTC)

— Notified WikiProject Mathematics in case anyone else wants to provide input. — MarkH21 (talk) 05:08, 15 April 2019 (UTC)

If we're talking about how our various articles on curves are to be structured, shouldn't we also include Jordan curve? In any case, we certainly need an article on curves that are not assumed to be algebraic or even smooth (e.g. to support topics like curve-shortening flow and inscribed square problem), so merging this article into an article on algebraic curves seems like a non-starter to me. —David Eppstein (talk) 06:19, 15 April 2019 (UTC)