Talk:Euclidean planes in three-dimensional space

Lines in intersecting planes

"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)

Not yet done, but do chip in!

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)

algebraic geometry

Someone should write something about the affine plane in algebraic geometry, Spec k[x,y].

Euclidean definition needs clarification or correction

The statement defining a plane "...a plane is a surface such that, given any two distinct points on the surface, the surface also contains the unique straight line that passes through those points." is also satisfied by a cylinder. I'd defer to the common page maintainers whether they'd rather mention this or change the defining statement. —Preceding unsigned comment added by Poppafuze (talk • contribs) 17:17, 17 April 2008 (UTC)

I'm not sure what is meant by this. The cylinder (x, y, z)(s, t) = (cos t, sin t, s) goes through (1, 0, 0) and (-1, 0, 0). A line through (1, 0, 0) and (-1, 0, 0) goes through (2, 0, 0). The point (2, 0, 0) does not lie on the cylinder, so the cylinder does not contain the line.

Many surfaces contain at least one line through two points on that surface, not limited to planes or cylinders. But that is not what the definition says. 96.26.243.241 (talk) 04:10, 17 May 2008 (UTC)

Poppafuze is correct. There is a notion of straight line that lie on surfaces, and on the cylinder also contains the straight line that contains them. This type of distinction is important if one wants to think about non-Euclidean Geometries. But it certainly is not a definition. Thenub314 (talk) 06:39, 18 September 2008 (UTC)

Is this article accessible to enough people?

I don't think it is. Terms like R³ and Euclidean Geometry, and objects like determinates and normal vectors need knowledge of degree level mathematics to understand.

I intend to restructure parts of the article to bring some concepts down to a simpler level where possible, e.g. using terms like 3 dimentional space instead of R³ where possible. And, I think, giving a warning and some guidance when the concepts do get compliacted, like links to web pages and books that will give a relevent introcuction to the theory.

I am for keeping all of the results - and am impressed by the expertise and effort shown by contributers - perhaps I may ask up to what level you have studied maths Melchoir?

So while I believe that the technical jargon is needed for some of the results, it is important to keep things simple. Fuzzyslob 12:16, 16 November 2005 (UTC)

Well, I carry a BA in pure math and a BA in physics, so I'm torn in several directions. I still remember, from years ago, slogging through Stewart's presentation of the material here and thinking that it was scatterbrained and inelegant. That's why I'm trying to emphasize vector notation, and why I want to write about the relationships between all the concrete descriptions of planes in the future.

As for accessibility, among other reasons, I wonder if this article should be split in two. Currently it's dominated by boring calculations in 3D; maybe we should split that material into a separate, more technical page, and try to expand on conceptual stuff here. What do you think?

Anyway, I'm interested in seeing what you have in mind, so fire away! Melchoir 14:37, 16 November 2005 (UTC)

OK, well I came here because I'm writing a game in OpenGL, and I wanted to use some clipping planes. After slowly piecing together what I think is meant in the section "Define a plane with a point and a normal vector", I added a couple of expanded, English-heavy versions to the bottom of that section, for the benefit of other barely-mathematical people like me. I hope I got them right - I recognise that removing the symbols and jargon from mathematics is very likely to produce inaccuracies and half-truths, which is why plain language isn't used in the first place. I'm not sure if my use of the term "dot product" is quite correct. I linked to the article on it, but that seems rather sadistic since I can't understand a word (or symbol) of that article, and if it was the only page on the web explaining dot products I would never be able to make use of one. What I have in mind is just as explained here: http://freespace.virgin.net/hugo.elias/routines/r_dot.htm ... I'd approve in general of separate pages for geometrical concepts addressed on a simpler level, rather than single monolithic pages for everybody that start with the deepest, most abstract versions of the concepts, and that bury the popular usage of the concepts somewhere in the middle in hard-to-recognise form, as often seems to happen. For instance, a page for "Euclidean dot product of unit vectors". It would be boring, but useful.

Edit: actually on consideration the article for dot product has the explanation I'm used to right at the top, but it took me twenty minutes to recognise it because it's generalised for any number of dimensions. I can never remember how the symbol for "sum" works, and seeing a1, a2, etc being used instead of ax, ay and az meant I didn't understand them as coordinates 81.131.29.228 (talk) 16:59, 27 January 2009 (UTC)

Today I got to grips with ax+by+cz+d, and finished the job by adding another paragraph that probably makes some highly dubious assertions ("All points with the same cosine of their angle as measured from the normal vector, and scaled by the distance to the point, are on the same plane", etc.) I hope what I've written is correct and helpful. It's helpful to me, anyway. 213.122.56.14 (talk) 20:15, 28 January 2009 (UTC)

Coming back a year later, I see that the section I edited is now incredibly short, and ends with "...which is the familiar equation for a plane." Whoever wrote this, please bear in mind that some of us are not familiar with any equation for a plane, or any equation at all, and nevertheless have reasons to want to understand and make use of such things. I'd like to add something like this: "the first three variables in the equation of the plane are the direction in which the front of the plane faces, as a 3D vector, and the final variable is the distance from the origin to the nearest point on the plane"... if that's right. 81.131.14.190 (talk) 15:24, 4 February 2010 (UTC)

non-infinite planes

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)

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)

rendering graphics

I can't read the R-cubed symbol on my browser. It just comes out as a blob!

Requested move

The following discussion is an archived discussion of the proposal. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the proposal was no consensus to move the page, per the discussion below. Dekimasuよ! 02:30, 31 July 2007 (UTC)

---

Plane (mathematics) → Plane — Most basic usage for plane, on which all other uses for plane are based (even airplane) ~ JohnnyMrNinja 01:06, 26 July 2007 (UTC)

Oppose "most basic" does not equate to "most common" (or "least ambiguous"). Ewlyahoocom 02:06, 26 July 2007 (UTC)

• Oppose. Even if once based on the geometric term (and I'm not even convinced of that) there are now several other significant meanings. Andrewa 14:41, 26 July 2007 (UTC)
• Oppose. As above. --Juiced lemon 16:06, 26 July 2007 (UTC)
• Oppose. When I hear the word "plane", I think of an airplane. Georgia guy 16:52, 26 July 2007 (UTC)
• Oppose, and a wood plane is not based on the mathematical plane. 132.205.44.5 21:50, 26 July 2007 (UTC)
• Oppose - There is not one primary usage for "plane". Clarification with (mathematics) works fine. Raime 13:23, 27 July 2007 (UTC)
• Oppose - as above, better to leave the dab page where it is. -- Beardo 17:42, 29 July 2007 (UTC)

Discussion

These arguments (among others) also appear to apply to the requested move of Square (geometry) to Square, again displacing a disambiguation page which was recently moved to Square (disambiguation). Andrewa 04:00, 27 July 2007 (UTC)

The above discussion is preserved as an archive of the proposal. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Intersection

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)

Notation Question

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)

Note that D is not matrix but determinant and hence a scalar. — Preceding unsigned comment added by 180.149.51.66 (talk) 11:49, 3 February 2012 (UTC)

"In jargon"

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)

What?

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)

Colinear lines

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)

Sorry but i am not able to understand what does overlapping lines mean? — Preceding unsigned comment added by 180.149.51.67 (talk) 11:44, 3 February 2012 (UTC)

Wow - A rewrite is required

The article contains the text:

"In this setting planes differ from lines Differing from lines, however, planes cannot be skew."

This is simply dreadful.

I haven't got past the "Euclidean geometry" section!

Plane defined by 3 points, method 2 -- what about planes crossing (0,0,0)

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)

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.--JohnBlackburne^words_deeds 13:53, 8 April 2010 (UTC)

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)

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)

Signed point-plane distance

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)

Distance between two parallel planes

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)

Space in first line

Several editors have changed the wikilink in the first line from [[Space (mathematics)|space]] to [[Euclidean space|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)

Half-plan

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)

Half-plane? —Tamfang (talk) 17:56, 30 October 2012 (UTC)

So okey, this notion is pooling with the half-space, my mistake.--82.224.165.208 (talk) 00:16, 31 October 2012 (UTC)

visual gravity explanation

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)

Definition/forms

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)

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)

Thanks! Vegard (talk) 10:15, 26 January 2014 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Euclidean planes in three-dimensional space/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

If you go to the "discussion" section of this article you will see that added about 4 sections highlighting key flaws in this article. In my opinion the grammar is dreadful. If this article has passed an editorial board's review then it doesn't say much about the review process.

Last edited at 12:41, 10 June 2009 (UTC). Substituted at 03:06, 30 April 2016 (UTC)

Extending wall infinitely far

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)

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)

Implicit assumption of only three dimensions

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)

Split of Plane (mathematics)

@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, Vector space § Subspaces and quotient spaces (redirects from Vector plane), Displacement (geometry), Euclidean vector, Bivector, Cartesian coordinate system § Two dimensions, Polar coordinate system, Real coordinate space § n = 2, 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)

Aside: Can we please avoid using the {{excerpt}} template in mathematics articles? The result is invariably inferior in content and style to just writing an appropriate section that fits into the host article, and prevents other editors from making improvements (because most improvements to the summary in the host article are inappropriate to the lead of the source article, and vice versa). Even just copy/pasting the existing lead section is an overall much better method because it facilitates future improvements. –jacobolus (t) 05:44, 20 February 2023 (UTC)

@Jacobolus I'd suggest proposing that template ban in WP Talk:SS. fgnievinski (talk) 02:05, 21 February 2023 (UTC)

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)

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)

@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)

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)

@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)

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)

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)

I've responded further down below. fgnievinski (talk) 05:50, 5 March 2023 (UTC)

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)

@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)

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)

Thank you for your suggestion! fgnievinski (talk) 05:40, 5 March 2023 (UTC)

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)

I've added instances of occurrence in nature, to make it more accessible for non mathematicians. fgnievinski (talk) 07:29, 5 March 2023 (UTC)