Talk:Parabola

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focal length

From the equations:

y=x^2/4f ... (1)

and

y=ax^2 ... (2)

and

y=x^2 / (4 r sin theta ) .... (3)

surely it is obvious that f=r sin theta, using equations 1 and 3. Equation 2 is just confusing.

DOwenWilliams (talk) 22:55, 15 July 2015 (UTC)[reply]

Thanks for clarifying (and also the 2 "thanks"), DOwenWilliams. My point is the phrasing. The reason I think

...if a parabola has an equation of the form

y=ax^{2}

where

a

is a positive constant, then its focal length is

f={\frac {1}{4a}}

.

is clearer than

...if a parabola has an equation of the form

y=ax^{2}

where

a

is a positive constant, then

a={\frac {1}{4f}}

where

f

is its focal length.

is that the subsection is entitled "Focal length". The former gives the focal length in terms of an arbritary constant a, whereas the latter gives a in terms of f.

As an analogy, suppose there was a section "x-intercept" in the article "Equation of a line". It seems clearer to me to have the statement

If a line has an equation of the form y = mx + c , then the x-intercept, x₀ is -c / m .

rather than

If a line has an equation of the form y = mx + c , then c = -mx₀ , where x₀ is the x-intercept.

Just my 2p :-) cmɢʟee⎆τaʟκ 19:35, 20 July 2015 (UTC)[reply]

I've just changed it to get rid of a altogether. We're interested only in f. a is unnecessary and confusing. DOwenWilliams (talk) 20:12, 20 July 2015 (UTC)[reply]

Thanks, I think that's the clearest of them all. Well done! cmɢʟee⎆τaʟκ 11:28, 21 July 2015 (UTC)[reply]

Similarity

The assertion that all parabolas are similar to each other (or that they have the same shape) is incorrect, and I'm planning on fixing that soon. But I just wanted to say something here first.

In fact, the whole lead is of fairly poor quality and could use a complete rewrite. I'll possibly work on that, too, but more slowly. — Preceding unsigned comment added by 24.56.116.202 (talk) 15:07, 16 August 2015 (UTC)[reply]

Of course it is correct: up to rigid motions, parabolas are a one-parameter family (take, say, the distance between the focus and directrix), and changing this parameter acts as a rescaling (I.e., similarity). I have no comment about the lead section more generally. --JBL (talk) 15:58, 16 August 2015 (UTC)[reply]

Changing the parameter scales in one direction only, while geometric similarity refers to uniform scaling. If all parabolas were similar, then all ellipses would be similar as well, but I don't think anyone would claim that. This article is even contradicting the page on similarity.24.56.116.202 (talk) 16:10, 16 August 2015 (UTC)[reply]

No, ellipses are a two-parameter family that can be scaled independently in two different directions to yield other (non-similar) ellipses. Parabolas are a one-parameter family, and the transformations "scale along one axis," "scale along the other axis", and similarity all have the same effect. This is extremely easy to verify in coordinates: dilating the parabola y=ax^2 by R gives y/R = a(x/R)^2, which has exactly the same effect as the result of scaling the parameter a by 1/R, or scaling along the y-axis by 1/R, or scaling along the x-axis by the square root of R. --JBL (talk) 16:30, 16 August 2015 (UTC)[reply]

I realize all that. But a figure which is obtained from another by scaling (in one direction only) is not similar to the original (generally). Check the definition of similarity that Wikipedia has. If it were, all ellipses and all rectangles would be similar to each other, respectively. The fact that parabolas depend on only one parameter is irrelevant here -- it's a purely geometric notion that parabolas don't satisfy. -24.56.116.202 (talk) 16:48, 16 August 2015 (UTC)[reply]

Please read what I wrote, beginning with "the transformations." Scaling a parabola along one axis gives literally exactly the same result as dilating (possibly by a different ratio). This is a special property of parabolas -- the fact that other shapes lack this property is totally irrelevant. --JBL (talk) 19:38, 16 August 2015 (UTC)[reply]

Every ellipse with a given eccentricity is similar to every other ellipse with the same eccentricity. An ellipse with e=0.5 is similar to every other ellipse with e=0.5. Every circle (an ellipse with e=0) is similar to every other circle. Every parabola (an ellipse with e=1) is similar to every other parabola. It's quite a general property, not peculiar to parabolas. DOwenWilliams (talk) 22:26, 16 August 2015 (UTC)[reply]

The thing that you refer to with the words "it's quite a general property" is different than the property that I was discussing with the IP editor in the comment to which you appear to be responding. (What you say about similarity and eccentricity is correct, of course, and maybe it will help the IP out.) --JBL (talk) 22:47, 16 August 2015 (UTC)[reply]

The pageview statistics for this article closely mirror activity in schools, with minima during school vacations and weekends. This presumably shows that many of the readers are school students. Editors should be aware of this. Don't make the article more sophisticated than its readers can appreciate. DOwenWilliams (talk) 19:28, 16 August 2015 (UTC)[reply]

I've added a short paragraph to the article addressing the main issue discussed above, that a parabola can be "stretched" or compressed in any direction and still maintain its parabolic shape. This is a property of which many people are unaware, and find surprising. Generally, citations are deemed necessary in Wikipedia only to support statements that might reasonably be doubted. Mathematical statements that are derived from first principles are beyond doubt unless the derivation is shown to be fallacious. There are many examples in this article and others of statements that are derived, without citations. This one I added today is just another. It is derived by JBL in the paragraphs above. I put an essentially identical derivation into the article. DOwenWilliams (talk) 03:21, 20 August 2015 (UTC)[reply]

It is simply untrue that I derive the property you mention above: there are more than two directions in a plane! The fact that the result is true is almost totally irrelevant to the question of whether it should be in the article: there are infinitely many true statements about parabolas, many of which have the properties you mention, and mostly they do not belong in an encyclopedia, let alone with a whole section. Adding random bits of trivia like that with no obvious filter tends to obscure the actually important information in the article. If you can find a source for the section you've added, that would be a piece of evidence in favor of there being enough weight to mention it; if not, I request again that you remove it. --JBL (talk) 15:09, 24 August 2015 (UTC)[reply]

By the way, the problem with the other sentence (that you removed earlier) is that it strongly implied that if you rescale an ellipse or hyperbola, you get something other than an ellipse or hyperbola (respectively). --JBL (talk) 15:27, 24 August 2015 (UTC)[reply]

Yes. I can see that you might make that inference. If I had left that sentence in, I would probably have fixed it. But that's irrelevant now.

Why is this particular little section more worthy of deletion than others, such as the one about perpendicular tangents intersecting on the directrix?

I wrote a long reply to you a few minutes ago, then got an edit conflict when I tried to save it. You had added a sentence. I don't have time to re-write it. This is a storm in a thimble. Enough.

DOwenWilliams (talk) 16:07, 24 August 2015 (UTC)[reply]

That is unfortunate about the EC. (FWIW, In my experience there is usually a copy of the text I've written somewhere on the EC page, which I can copy and then paste to make a new comment.) The answer to your question is that, for convenience and because I have limited amounts of time to devote to WP, I typically edit by watch list, so that I evaluate new changes to articles much more than I read articles from scratch. This particular change relates rather directly to me, as well, hence my interest. But this has very little to do with the question of whether this subsection should be in this article. My instinct is still to remove it entirely, but with a citation in place of the partial derivation I could easily see it as a sentence fitting in somewhere. I also do not see any signs of a storm here -- just two editors discussing an editorial question. If you like, I imagine we could request additional opinions from, say, WikiProject Math. --JBL (talk) 17:02, 24 August 2015 (UTC)[reply]

Oh, okay. You've talked me into going back to before this discussion. DOwenWilliams (talk) 06:44, 25 August 2015 (UTC)[reply]

Usage of "parabolic"

"Strictly, the adjective parabolic should be applied only to things that are shaped as a parabola, which is a two-dimensional shape. However, as shown in the last paragraph, the same adjective is commonly used for three-dimensional objects, such as parabolic reflectors, which are really paraboloids." This statement is obviously nonsense. Parabolic reflectors are, in fact, parabolic. What would they be called otherwise? "Paraboloidic"? This paragraph should be stricken from the article. — Preceding unsigned comment added by 50.255.2.98 (talk) 00:17, 11 January 2017 (UTC)[reply]

I agree and have removed it. (You could have done so, too!) --JBL (talk) 01:16, 11 January 2017 (UTC)[reply]

A Geometrical Constructions to Find the Area of a Parabolic Sector

I don’t know how to fight my corner with Wikipedia

It seems to me this is a lovely property of parabolas that has not previously been described in this entry on Parabolas. It has as much right to appear here as many of the other entries

Is there anybody else who can consider this entry or am is it to be damned on the whim of a single individual?

Obviously, I’m not familiar with all the subtleties but I do know that there are many contributions that relate to my field of expertise, Isaac Newton, that are very poor. Perhaps the people who have the power to remove items should have a look at those.

If there is anything wrong with this entry I will be happy to change it.

The construction

S is the Focus and V is the Principal Vertex of the parabola VG. Draw VX perpendicular to SV.

Take any point B on VG and drop a perpendicular BQ from B to VX. Draw perpendicular ST intersecting BQ, extended if necessary, at T. At B draw the perpendicular BJ, intersecting VX at J.

For the parabola, the segment VBV, the area enclosed by the chord VB and the arc VB, is equal to ∆VBQ / 3, also $BQ={\frac {VQ^{2}}{4SV}}$

The Area of the Parabolic Sector SVB = ∆SVB + ∆VBQ / 3 $={\frac {SV.VQ}{2}}+{\frac {VQ.BQ}{6}}$

Since triangles TSB and QBJ are similar:

$VJ=VQ-JQ=VQ-{\frac {BQ.TB}{ST}}=VQ-{\frac {BQ.(SV-BQ)}{VQ}}={\frac {3VQ}{4}}+{\frac {VQ.BQ}{4SV}}$

Therefore, the Area of the Parabolic Sector $SVB={\frac {2SV.VJ}{3}}$ , and can be found from the length of VJ, as found above.

It should be noted that a circle through S, V and B also passes through J.

Conversely, if a point, B on the parabola VG is to be found so that the Area of the Sector SVB is equal to a specified value, determine the point J on VX, and construct a circle through S, V and J. Since SJ is the diameter, the centre of the circle is at its mid-point, and it lies on the perpendicular bisector of SV, a distance of one half VJ from SV. The point required, B is where this circle intersects the parabola.

If a body traces the path of the parabola due to an inverse square force directed towards S, the area SVB increases at a constant rate as point B moves forward. It follows that J moves at constant speed along VX as B moves along the parabola.

If the speed of the body at the vertex, where it is moving perpendicularly to SV is v, then the speed of J is equal to 3v/4.

The construction can be extended simply to include the case where neither radius coincides with the axis, SV as follows. Let A be a fixed point on VG between V and B, and point H be the intersection on VX with the perpendicular to SA at A. From the above, the Area of the Parabolic Sector $SAB={\frac {2SV.(VJ-VH)}{3}}={\frac {2SV.HJ}{3}}$

Conversely, if it is required to find the point B for a particular area SAB, find point J from HJ and point B as before. By Book 1 Proposition 16, Corollary 6 of the Principia, the speed of a body moving along a parabola with a force directed towards the focus is inversely as the square root of the radius. If the speed at A is v, then at the vertex, V it is ${\sqrt {\frac {SA}{SV}}}v$ , and point J moves at a constant speed of ${\frac {3v}{4}}{\sqrt {\frac {SA}{SV}}}$

Note: the above construction was devised by Isaac Newton and can be found in Book 1 of the Principia as Proposition 30.

— Preceding unsigned comment added by Mikerollem (talk • contribs) 10:56, 14 September 2017 (UTC)[reply]

Hi Mikerollem. When there's a dispute about whether to include material, the best way to approach the problem is to start a discussion on the article's talk page. (Here). On the talk page, we can discuss the material, revise it if necessary, and then reinsert it in the article if appropriate.

I wasn't the one who removed your original version of this, but I agreed with that decision. By policy, Wikipedia is an encyclopedia, not a textbook: our purpose is to present facts, not to teach subject matter. Long derivations and mathematical proofs are usually not appropriate. Lovely properties of geometry are good to include, though. This version seems much more focused than the original, so I'm going to put it back into the article without the editorial comments. Other editors may have other input, though...--Srleffler (talk) 18:56, 14 September 2017 (UTC)[reply]

Is there any secondary source for this? Articles like this one, on relatively accessible objects with long histories, tend to become larded down with trivia, including sometimes long proofs of unimportant facts. The requirement to be discussed in secondary sources provides a baseline standard for judging whether content is significant enough to include (as well as preventing original research). --JBL (talk) 19:24, 14 September 2017 (UTC)[reply]

Motion of projectile under gravity in vacuum is NOT a parabola

My physics master Dennis 'Dicky' Dyson in 1959 was a stickler for accuracy and for thinking things through from first principles irrespective of commonly held views. For example, he held scientists make new discoveries by 'Trial and success', not 'Trial and error'.

He taught me that a projectile moving freely under earth's gravity (neglecting air resistance) is in orbit round the earth - therefore its path (according to Newton) is that of a satellite, i.e. an ellipse with the centre of the earth at one focus. The lines of gravitational force are NOT parallel (which would produce a parabola) but converge towards the centre of the earth. That this effect is infinitesimal and unmeasurable over common distances, is immaterial - theoretically the path is a portion of an ellipse, even though it may very, very, very, closely approach a parabola; so closely you can't see the difference. But truth is truth!

86.187.168.50 (talk) 23:34, 12 January 2018 (UTC)[reply]

"... (neglecting air resistance) ... (according to Newton) ... truth is truth" No further comment needed. --JBL (talk) 23:48, 12 January 2018 (UTC)[reply]

What is this meant to mean? 86.187.174.113 (talk) 09:47, 18 October 2018 (UTC)[reply]

It means that you consider this page as a forum (see WP:NOTAFORUM) for publishing your own thought (see WP:NOR). As this page is aimed for discussing improvements of the article, and that is not what your post is about, it does not belong to this talk page, and thus does not deserve to be discussed here.

It means also that you discuss the page without having read it, in particular the footnote (i) that says, in a more accurate way, the same thing. D.Lazard (talk) 10:12, 18 October 2018 (UTC)[reply]

I beg to differ, I was not using this page as 'a forum for publishing my own thought' but stating an indisputable fact (do you dispute it?) to improve the article. The section to which footnote I relates is headed 'In the Physical World' - but the first paragraph relates to the *theoretical* path of a projectile under gravity, and it is wrong, but a very commonly held belief. At the very least the first paragraph should be modified to say that theoretically the path is an ellipse with the earth at one focus, but at common scales this is indistinguishable from a parabola, and/or Footnote I should be moved to the first paragraph. 86.187.175.96 (talk) 20:44, 9 January 2019 (UTC)[reply]

Split

At a size of over 74K this article has become too unwieldy and should be split. It has been suggested that the main readership of this page consists of students whose interests generally don't include some of the more esoteric topics that are included in the page. I would think that a natural split would break out those topics of interest to this population and have a second page devoted to more advanced (that is, requiring more background or being less directly relevant) topics. Any suggestions or other ideas? --Bill Cherowitzo (talk) 19:25, 24 January 2018 (UTC)[reply]

I would suggest to split into Parabola (classic geometry) and Parabola (coordinates geometry). Probably, for modern students, the former would appear as advanced, and the latter more elementary. A century ago this would probably have been the opposite. D.Lazard (talk) 21:38, 24 January 2018 (UTC)[reply]

I think most students are interested in parabolas as graphs of quadratic functions. French, German and Spain WIKIs have convenient pages on this subject. If the article Quadratic function would be restricted to the parabola case, it could be convenient for English students. So, in the lead of parabola there should be at the top an obvious link to Quadratic function. But nevertheless, the parabola article should be cleaned up.--Ag2gaeh (talk) 08:58, 25 January 2018 (UTC)[reply]

I have moved toward the beginning the description as the graph of a function. Nevertheless, the remainder of the lead remains too technical and too detailed for most readers. D.Lazard (talk) 10:46, 10 January 2019 (UTC)[reply]

Basic equations

Note: This is about Special:Diff/905194553 ~ ToBeFree (talk) 21:00, 8 July 2019 (UTC)[reply]

All the theory is fine, but for people looking for equations they can use in their daily lives it would be nice with a listing in the beginning for finding the vertex, foci, directrix and intersect. I realize that the addition I made was incomplete, but some kind of user guide before the reference guide would make the page useful for people like me. The reference that I added had exactly what I needed. It would be nice to have both in the wiki page.

By the way how do I comment on a change. I would rather have made this post as a comment.

Freeduck (talk) 20:04, 8 July 2019 (UTC)[reply]

The data that you are asking for are given in the section § As a graph of a function. D.Lazard (talk) 20:16, 8 July 2019 (UTC)[reply]

Move Down the General Case Graph

In the section "Axis of symmetry parallel to the y axis" to the right of the text "More generally, if the vertex is ..." There is a plot called, "Parabola: general case". Given the matching text and the fact they appear together on the same horizontal position it is natural to assume the paragraph goes with the plot -- but it does not. That text is actually describing the previous plot. This is confusing.

Lower down, even past the subsection, "remarks¨, there is another full section called "General Case¨. That is where the "Parabola: general" case plot belongs. I hope someone will move the plot down. 86.233.234.10 (talk) 15:47, 2 July 2021 (UTC)[reply]

Fixed. D.Lazard (talk) 16:40, 2 July 2021 (UTC)[reply]

Redundancy in lead

I'm not an expert in math, but isn't the sentence "The graph of a quadratic function y=ax²+bx+c is a parabola if x!=0 and, conversely, a parabola is the graph of a quadratic function if its axis is parallel to the y-axis." saying the same thing twice, or is there a particular usefulness to list the converse? --laagone ^talk 13:37, 12 March 2024 (UTC)[reply]

The first assertion says that some parabolas are graphs of quadratic equations, and the second assertion says that every parabola can be obtained this way. That is, the indefinite article after "conversely" must be understood as "every" (this is standard mathematical jargon). I have edited the article for clarifying this. D.Lazard (talk) 14:11, 12 March 2024 (UTC)[reply]

Inequalities obtained through the coefficients of the equations of parabolae

When a and c are positive quantities but b is a negative quantity such that b² > 4ac, inequalities include:

0 < a < ⁠b²/4c⁠ where b < 0 but c > 0.
b < –2√ac where a > 0 and c > 0.
0 < c < ⁠b²/4a⁠ where a > 0 but b < 0.

2603:7000:B500:5D4:FCF3:6580:ECA7:89DF (talk) 23:08, 6 August 2024 (UTC)[reply]