Talk:Euclidean planes in three-dimensional space

message: "Lines drawn on intersecting planes will either intersect or be skew, but will not be parallel. Intersecting planes may be perpendicular, or may form any number of other angles."
Imagine 2 intersecting planes \alpha and \beta. Intersection of two planes forms line k.
There is infinite amount of lines in \alpha e.g. l such that l||k. Same is true for plane \beta where e.g. m lies in \beta and m||k. If l||k and m||k then l||m.
Therefore 2 lines (m and l) drawn on intersecting planes can be parallel.
Can do a drawing if you need one. Could please check this conclusion and update article. --Cliff (talk) 13:32, 13 May 2008 (UTC)
thread_title: Lines in intersecting planes

message: I'm not done with this page by a long shot; I see a few logical overhauls in its future. Still, if anyone wants to chip in, like 63.162.240.46 has recently, I promise not to completely undo your edits! Melchoir 21:02, 15 November 2005 (UTC)
thread_title: Not yet done, but do chip in!

message: Intuitively, [a plane] may be visualized as a flat infinite sheet of paper.
What about planes that are not infinite but rather limited, therefore creating a fraction of a infinite plane? --Abdull 17:39, 10 July 2006 (UTC)
replies: text: In my experience, even if a plane is represented as a non-infinite area in space, it is still understood to extend to infinity.  If you restrict the area of a plane, it becomes a shape, like a polygon, only a (small) portion of the given plane on which the shape occurs.  -Kanogul (talk) 23:08, 5 May 2008 (UTC)
thread_title:  non-infinite planes 

message: We talk alot about planes in R3, would it be worth pointing out that in higher dimensions stranger things can happen, such as two planes that intersect in exactly one point. Thenub314 (talk) 16:47, 16 May 2008 (UTC)
thread_title:  Intersection 

message: In the discussion of using Cramer's Rule to define a plane given three points, what does the notation -d/D mean? Perhaps I'm missing the intent, but how does one divide a scalar by a matrix? --128.84.189.106 (talk) 16:58, 21 April 2009 (UTC)
thread_title:  Notation Question 

message: Is the phrase "In jargon" appropriate?
The so-called "In jargon" section is still in English.
Thus, if "In jargon" is to be kept shouldn't the "In English" be replaced by "In plain English"?  —Preceding unsigned comment added by Hedgehog0 (talk • contribs) 13:34, 28 May 2009 (UTC)
thread_title:  "In jargon" 

message: The article contains the following sentence:
"The plane is not directly given a definition, may be thought of as part of the common notions."
What is this sentence trying to say?  —Preceding unsigned comment added by Hedgehog0 (talk • contribs) 09:33, 29 May 2009 (UTC)
thread_title:  What? 

message: The article states:
"two intersecting but non-colinear (that is, not identical) lines"
I agree that 2 non-colinear lines are required to uniquely define a plane.
However, 2 lines can overlap and intersect and NOT be identical.
Thus, I suggest deleting the text in brackets:
"two non-colinear intersecting lines"  —Preceding unsigned comment added by Hedgehog0 (talk • contribs) 09:38, 29 May 2009 (UTC)
thread_title:  Colinear lines 

message: Plane which crosses point (0,0,0) is valid plane and with equation
ax + by + cz + d = 0
d in such equal to zero. But method 2 at the ends explains "setting d equal to non-zero...". Why I have to set it to non-zero when I would like to get the plane crossing point 0? The equations in such case get the form of:
0x + 0y + 0z = 0
which is "equation" of entire space, not a plane.  —Preceding unsigned comment added by Me-macias (talk • contribs) 13:11, 8 April 2010 (UTC)
replies: text: Method 2 as described won't work if one of the points is the origin, or if the plane contains the origin: in that case D would be zero, as would d. I've added something to that effect. There might be a way to use a method like this even for such planes, but I don't think it's worth adding as there are two other methods you can use instead.--JohnBlackburnewordsdeeds 13:53, 8 April 2010 (UTC)
replies: text: How do you find the distance from a point to a plane if you failed method 2 (D=0)? As in, find a, b, c, d?99.33.206.35 (talk) 05:44, 1 January 2013 (UTC)
replies: text: Use the formula given in the section with no modification. What is confusing here is that the D of this section refers to the distance between the point and the plane, while the D of method 2 in the earlier section refers to the determinant of a matrix. That is to say, they are not the same and in general will not have the same value, even though they are denoted by the same letter in this article. (This is a fixable problem) Bill Cherowitzo  (talk) 18:33, 1 January 2013 (UTC)
thread_title:  Plane defined by 3 points, method 2 -- what about planes crossing (0,0,0) 

message: Would simply not taking the absolute value of the point-plane distance calculation as given on the page result in a positive value if the result is on the forward facing side of the plane and negative otherwise? It would be a more thorough if it included this information in the section.  — Preceding unsigned comment added by 50.84.48.6 (talk) 19:22, 15 December 2011 (UTC)
thread_title:  Signed point-plane distance 

message: ====Method 1====
The distance between two parallel planes can be calculated by selecting any random point on one plane and measuring it's distance from the second plane using the method shown above. 
====Method 2====
Is this correct???
Given two planes, \Pi_1 : Ax + By + Cz + D1 = 0 and \Pi_2 : Ax + By + Cz + D2 = 0, the distance between them would be equal to d = \frac{\left | D1-D2 \right |}{\sqrt{A^2+B^2+C^2}}
85.64.51.13 (talk) 09:17, 16 April 2012 (UTC)
thread_title: Distance between two parallel planes

message: Is this correct???
Given two planes, \Pi_1 : Ax + By + Cz + D1 = 0 and \Pi_2 : Ax + By + Cz + D2 = 0, the distance between them would be equal to d = \frac{\left | D1-D2 \right |}{\sqrt{A^2+B^2+C^2}}
85.64.51.13 (talk) 09:17, 16 April 2012 (UTC)
thread_title: Method 2

message: Several editors have changed the wikilink in the first line from space to space. I have indicated that this is incorrect because space, in the context used here, is not just Euclidean space ... there are other 3 dimensional spaces which are not Euclidean (projective 3-space for instance). The problem seems to be that the Space (mathematics) article is about the concept of abstract space as used in modern mathematics with almost no mention of geometric space and a faulty hatnote there sends readers interested in geometric space to Euclidean space. Fixing the Space (mathematics) article is an option which needs doing but may take a while. I have another way to fix the problem in the current article. In the progression, point (0-dim), line (1-dim), plane (2-dim), ... the next term is really solid (3-dim). Using "space" for "solid" is a common error caused by thinking of Euclidean space as the 3-dimensional space in which we live and the demise of the "Solid Geometry" course in high school curricula. While it is not the job of an editor to correct all the usage mistakes of the general populace, in this particular instance it would help to remove us from a quandry which only a major revision of a different article could fix. I'll make the edit and leave this explanation here. Bill Cherowitzo  (talk) 18:50, 26 September 2012 (UTC)
thread_title:  Space in first line 

message: Hi,
Why is not there an article dealing with half-plan on the English wikipedia version? Does this notion exist in English?
Thank you for your answer.  — Preceding unsigned comment added by 82.224.165.208 (talk) 17:49, 30 October 2012 (UTC)
replies: text: Half-plane? —Tamfang (talk) 17:56, 30 October 2012 (UTC)
replies: text: So okey, this notion is pooling with the half-space, my mistake.--82.224.165.208 (talk) 00:16, 31 October 2012 (UTC)
thread_title:  Half-plan 

message: As featured in Brian Greene physics film.
The "snakes" make complete circles and do not fall off the tilted plane.
http://jeffcarlson.typepad.com/thought/2006/04/snakes_on_a_pla.html  — Preceding unsigned comment added by 24.114.22.20 (talk) 20:30, 27 January 2013 (UTC)
thread_title:  visual gravity explanation 

message: I added a new section at the top (called simply "Definition") where I try to give the most general definition of a plane. I felt this was needed as the "Definition with point and a normal vector" section referred to things like "the familiar equation" (which was actually not shown before in that article!). This may hopefully also mitigate some confusion between the ax + bx + cx - d = 0 and the a(x-x_0) + b(y-y_0) + c(z-z_0) = 0 forms (although it's quite easy to see that they are equivalent). Let me know if you disagree Vegard (talk) 23:36, 25 January 2014 (UTC)
replies: text: While definitions can vary, it is more common these days to see planes described in the point-normal form and then prove the equivalence with linear equations. This has pedagogical advantages in that it makes the normal to the plane a natural object that can be easily identified. Your treatment of the Hessian form is a little off (you must use a unit normal vector) and comes far too early in the article. You are correct in pointing out that there are problems with the "Definition with point and normal vector" section, it seems to have been lifted from somewhere else, without regard to the things it referenced. I will make a stab at rewriting these two sections. Bill Cherowitzo  (talk) 04:49, 26 January 2014 (UTC)
replies: text: Thanks! Vegard (talk) 10:15, 26 January 2014 (UTC)
thread_title:  Definition/forms 

message: Substituted at 03:06, 30 April 2016 (UTC)
thread_title: Assessment comment

message: Wcherowi Hi, Did you add that sentence "as with a room's walls extended infinitely far" ? First of all, rooms of a wall are a poor analogy for a two dimensional space.  How can 4 walls of a room become a two dimensional space.  Secondly "infinitely far" and "infinitely" imply the same thing. Not interested in entering into a dispute. If you are willing to rethink what you wrote and correct it yourself, fine. Else, let it stand as is. I don't care either way --Sahir 08:43, 24 October 2020 (UTC)
replies: text: No, that awkward sentence was not one of mine, although I am sure that I have added some awkward sentences in my time. The analogy is good, although poorly expressed; any one of a room's walls is a model for a plane, when infinitely extended. I have used the word "infinitely" properly in that last sentence. It is an adverb, and so it must modify something. In this case, it is modifying the extension process. In English, we generally place modifiers before the thing they are modifying. Whoever wrote that sentence was uneasy with the dangling modifier and so put the "far" in to give it something to modify. A poor choice, but acceptable if one isn't too stringent about the rules of grammar. I will correct the sentence in accordance with what I have written here.--Bill Cherowitzo  (talk) 20:35, 24 October 2020 (UTC)
thread_title:  Extending wall infinitely far 

message: There is material in, e.g., #Line of intersection between two planes, that assumes the space to be three dimensional. In higher dimensions, planes may intersect in a single point or even be skew. There should be a section for planes in 3D space, with the remaining sections being generic. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:55, 9 September 2021 (UTC)
thread_title:  Implicit assumption of only three dimensions 

message: @Fgnievinski just created a new article Plane (mathematics) (previously redirected here) and started moving some of the material from here to there. This seems like a problematic distinction to me. In my opinion there is not a very good reason to have separate plane (geometry), plane (mathematics), and Euclidean plane alongside a wide variety of other articles about 2-dimensional "planes" and surfaces of various flavors.
I don't really understand what the intended scope of these new articles is supposed to be, and I think we should discuss it before settling on a strategy. I do agree that the content and organization of these articles can be much better, especially to be more legible to non-technical readers.
If we want to figure out what needs to be covered, which parts most readers are likely to be interested/curious about, how it can be split into articles and then organized under those, etc. may also want to take a look at (just grabbing everything I can think of; perhaps other editors have more):
: Euclidean geometry, Euclidean plane, Trigonometry, Triangle, Circle, Euclidean plane isometry, Rotations and reflections in two dimensions, Glide reflection, Wallpaper group, Affine plane, Shoelace formula, Green's theorem, Double integral, Surface integral,  (redirects from Vector plane), Displacement (geometry), Euclidean vector, Bivector, , Polar coordinate system, , Real projective plane, Homogeneous coordinates, Barycentric coordinates, Smooth projective plane, Non-Desarguesian plane, Translation plane, Non-Euclidean geometry, Elliptic geometry, Hyperbolic geometry (redirects from Hyperbolic plane; also cf. Hyperboloid model, Poincaré_disk model, Poincaré half-plane model, Beltrami–Klein model, Band model), Minkowski plane, Möbius plane, Laguerre plane, Inversive geometry, Möbius transformation, Cayley–Klein metric, Spherical geometry, Sphere, Spherical trigonometry, Stereographic projection, Torus, Quadric, Surface (mathematics), Parametric surface, Implicit surface, Algebraic surface, Differential geometry of surfaces, Gauss–Bonnet theorem, Cayley plane, Complex plane, Riemann sphere, Directed infinity, Complex projective plane, Line coordinates, Laguerre transformations, Plane of rotation, ...
There is a ton of material that could plausibly fit into any of these titles (including much that is nowhere yet on Wikipedia). I think it's worth trying to think through which topics are most important, how to clearly direct readers to what they are looking for, and how to accessibly explain what we decide to cover. –jacobolus (t) 05:39, 20 February 2023 (UTC)
replies: text: @Jacobolus I'd suggest proposing that template ban in WP Talk:SS. fgnievinski (talk) 02:05, 21 February 2023 (UTC)
replies: text: I didn’t say it should be banned. I only am asking people to please try to avoid it, because the results are invariably mediocre. For example, in the excerpts you posted to this page, we have basic concepts repeated and repeatedly wikilinked, well established context repeated, wildly inconsistent length/style between sections, essential information left out, weird information about potential applications that makes no sense in the context of this page, etc. The result is a hodgepodge that very poorly serves our readers. –jacobolus (t) 03:45, 21 February 2023 (UTC)

text: I don't understand the reason for the split or the difference between the two topics. I do note with pleasure that the new article drops the assumption in the first sentence that a plane is flat; the original is disconcerting in an article that briefly discusses elliptic or hyperbolic geometry, where planes are not flat. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 17:19, 20 February 2023 (UTC)
replies: text: @Chatul that's exactly the distinguishing feature: Wikipedia needs an article about the everyday meaning of geometrical plane: a flat surface. fgnievinski (talk) 17:26, 20 February 2023 (UTC)
replies: text: How is not Euclidean plane the requisite article for the ordinary meaning? I can see the need for two articles, one generic and one specific, but not for three, one generic and two specific to the Euclidean plane. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 17:42, 20 February 2023 (UTC)
replies: text: @Chatul the hatnote in Plane (geometry) already explains its scope well: it is about geometrical planes embedded in three-dimensional Euclidean space. by contrast, Euclidean plane is about life in Flatland, so to speak. fgnievinski (talk) 01:50, 21 February 2023 (UTC)
replies: text: Pages like Euclidean plane and affine plane should include the topic of planes embedded in larger spaces as one section. If you think it makes the article too long though, it can be split out as a separate article with a (significant) summary left at Euclidean plane. –jacobolus (t) 03:44, 21 February 2023 (UTC)

text: Nothing in the hatnote clarifies why there are separate Plane (geometry) and Euclidean plane articles. Also, ithe ordinary meaning of Plane Geometry does not assume any imbedding. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:14, 26 February 2023 (UTC)
replies: text: I've responded further down below. fgnievinski (talk) 05:50, 5 March 2023 (UTC)

text: There was no need to create a new article. It would be sufficient to refocus the article plane (geometry) (or if you prefer, plane (mathematics) but the parenthetical disambiguation in the title makes no real difference) to discuss a broader collection of topics and move some of the material to Euclidean plane, affine plane, projective plane, or whatever. –jacobolus (t) 22:34, 20 February 2023 (UTC)
replies: text: @Jacobolus Wikipedia needs a separate article about the broader concept of non-flat "planes". It should not pollute the explanation about the lay concept of geometrical planes. fgnievinski (talk) 02:03, 21 February 2023 (UTC)
replies: text: In that case plane (geometry) should be moved to planes in three-dimensional space or planes in solid geometry or similar. The title plane (geometry) does not at all reflect the scope you are imagining for this page. –jacobolus (t) 03:38, 21 February 2023 (UTC)
replies: text: Thank you for your suggestion! fgnievinski (talk) 05:40, 5 March 2023 (UTC)

text: I've also redirected plane (geometry) to the more concrete Euclidean plane, because that's what most people would expect (WP:PLA). A dab hatnote was left for mathematicians, who might prefer the more abstract concept, plane (mathematics). fgnievinski (talk) 05:47, 5 March 2023 (UTC)
replies: text: I've added instances of occurrence in nature, to make it more accessible for non mathematicians. fgnievinski (talk) 07:29, 5 March 2023 (UTC)
thread_title:  Split of [[Plane (mathematics)]]