Talk:Cone

Polygonal cone?

Currently the lead says that the base of a cone is frequently, though not necessarily, circular. It then goes on to say

A cone with a polygonal base is called a pyramid.[2]

But reference [2] does not say that. It says

Conic solids have but one base. Pyramids have lateral edges which connect vertices of the base polygon with the vertex. In a cone, the lateral edge is any segment whose endpoints are the vertex and a point on the base circle. The triangular, non-base, faces of a pyramid are lateral faces. Pyramids and cones can also be....

clearly distinguishing between pyramids and cones.

Is there any source for the assertion that something with a polygonal base and an apex can be called a cone? I doubt it, and I think the assertion should be either sourced or deleted. If deleted, I think we should find a source that the base of a cone has to be a circle or ellipse, and mention that in the article. Loraof (talk) 21:11, 3 May 2016 (UTC)

Are you expressing doubt that people use the word "cone" to refer to certain polyhedra? Grünbaum's Convex Polytopes defines on p23 of the second edition (paraphrasing notation lightly)

a convex set C is a cone with apex 0 provided ax is in C whenever x is in c and a ≥ 0.

This certainly contains the polygonal case (in fact it is the case of most interest in the book).

On the other hand, I would not propose using this definition early in this article. --JBL (talk) 21:23, 3 May 2016 (UTC)

Okay, I'll put that reference in there. Could you take a look at the article Conical surface and see if it needs some corrections? It says

In three coordinates, x, y and z, the general equation for a cone with apex at origin is a homogeneous equation of degree 2 given by

$S(x,y,z)=ax^{2}+by^{2}+cz^{2}+2uxy+2vyz+2wzx=0$

which looks to me like it puts a restriction on the base which excludes polygons and other non-quadratic shapes. Thanks. Loraof (talk) 00:05, 4 May 2016 (UTC)

A few observations: first, I agree, that equation is certainly about the surfaces that form the boundaries of only a proper subset of the things called "cones" in this article. (Though exactly which subset is not obvious to me.) Second, the phrase "conical surface" is very much about 3 dimensions, and Grünbaum is certainly contemplating cones in arbitrary dimension (and this agrees with the usage that is common among combinatorialists who study polyhedra, in my experience), so the phrase "conical surface" wouldn't come up in that context (you'd expect "boundary" or something instead). Third, this article is really hard to write well, because these more general definitions of cone (different bases, possibly unbounded, possibly in other dimensions) are not the common one (circular, bounded, three-dimensional). I do not really have a conclusion here; I will try to look at conical surface tomorrow or the next day. --JBL (talk) 01:56, 4 May 2016 (UTC)

I think we should stick to textbook definitions. I feel uneasy about some of the wording that is now in this article and in Conic section. For example: "if the base is right circular" How can a base be right circular? "an infinite or doubly infinite cone" Aren't all cones, by definition, infinite in both directions? In the article about Conic section: "the surface of a double cone" Isn’t a cone, by definition, a surface? "the intersection of the boundary of a cone with a plane" How does a cone have a boundary? — Anita5192 (talk) 04:35, 4 May 2016 (UTC)

While I agree with Anita5192's comment about sticking to textbook definitions, this can be a bit trickier than we would like. I have come across texts that distinguish pyramids and cones (one has "flat" lateral surfaces and the other doesn't); ones that define a directrix to be a curve (so, in general, these cones are two dimensional) and others that permit it to be a region (giving solid cones); elementary treatments that seem to think that all cones are right circular cones and others that grudgingly permit more general quadratic cones (BTW that equation in conical surface is a general equation for a quadratic cone whose directrix is a conic section ... but only over fields whose characteristic is not two, so not all that general from my point of view). I would say that as soon as you loosen up what you allow a directrix to be, you'll have to call pyramids cones. Deciding which textbooks to follow will be difficult (and presenting all points of view would be massively confusing). My personal favorite definition of a cone is the union of lines joining the apex with a point on the directrix (with a discussion of possibilities for the directrix). This is even more general than Grünbaum's definition since he only takes rays instead of full lines. Bill Cherowitzo (talk) 05:32, 4 May 2016 (UTC)

@Anita5192: you say twice "by definition", but by Grünbaum's definition neither of these things is true. (Although I also note that the lead section of this article explicitly restricts to dimension 3, so in some sense the polytopal/convex geometry definition is about a different kind of cone.)

Tentatively, it seems to me like the article should have a high-level structure with at least the following subdivisions: three-dimensional circular cones; three-dimensional cones with arbitrary bases; and higher-dimensional cones. --JBL (talk) 14:42, 4 May 2016 (UTC)

Given that we need to be neutral in the face of a variety of definitions in the literature, I propose this as a draft of a new lead:

In three-dimensional space in mathematics, a cone is formed by a set of line segments, half-lines, or lines connecting a common point called the apex to all of the points on a base that is in a plane that does not contain the apex. Depending on the author, the base may be restricted to be a circle, any one-dimensional quadratic form in the plane, any closed one-dimensional figure, or any of the above plus all the enclosed points. If the enclosed points are included in the base, the cone is a solid object; otherwise it is a two-dimensional object in three-dimensional space.

In the case of line segments, the cone does not extend beyond the base, while in the case of half-lines, it extends infinitely far. In the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a double cone.

Cones can also be generalized to higher dimensions.

Loraof (talk) 16:07, 4 May 2016 (UTC)

Generating line

In Conic section article there is mentioned "Generating line". In cone article it is called "generatrix". Would it be correct adding (mentioning) "Generating line" as synonym? generating_line_anon_user 11:26, 17 Nov 2016 (UTC) — Preceding unsigned comment added by 193.109.235.158 (talk)

I like this suggestion and have implemented it. --JBL (talk) 14:41, 17 November 2016 (UTC)

Apex angle?

The article uses "Apex angle" but does not define it. Is it the aperture, half the aperture, the slant angle, or something else? -AndrewDressel (talk) 12:01, 18 April 2018 (UTC)

Circular sector angle obtained by unfolding the lateral surface

What is the angle of the circular sector obtained from a cone by unfolding the lateral surface? Is it equal to the axial section apex angle? —213.233.84.39 (talk) 15:57, 29 April 2018 (UTC)

I just inserted a new subsection, Circular sector, to derive this angle. I think the article needed this. Thank you for pointing out the deficiency.—Anita5192 (talk) 17:18, 29 April 2018 (UTC)

Corrections

in the right-top visual explanation, the surface area uses "l" as a variable which is not defined in the variable list. the image should be self-consistent, is the "l" supposed to be "c". — Preceding unsigned comment added by Goofyseeker311 (talk • contribs)