Wikipedia:Reference desk/Archives/Mathematics/2019 September 19

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September 19

Subtended angle

Is a subtended angle:- the angle opposite the curved edge, in a pseudo-triangle with two straight edges, but one curved edge...? Or, how can it be described as simply as that, i.e., without using the word "subtended" as the main word describing the "subtended" entity? Let's explain this with geometric shapes and no reference to the word subtended so that anybody can understand it. The example of the cricle with a vertex is not cutting it. Yes I am totally aware what a circle with a central vertex is. I am not really sure what subtended is. This is somewhat important as it is part of the description of what a parsec is... ~ R.T.G 07:17, 19 September 2019 (UTC)

OK, I'll have a shot at it.

Consider your eyeball as a point from which you are looking at an object (I know you have two eyes, but (a) I'm idealising, (b) at the scale of astronomy this makes no discernable difference, and (c) in many optical instruments you only use one eye at a time.)

How do you measure the apparent size of the object you're looking at? One way is to imagine a triangle with one point at your eye, and the other two at the opposite-most sides of the object. (In the diagram you've been looking at, where your eye is at Point A or Point B, the "curved edge" is an idealized substitute for an object – it's not an essential part of the concept of a subtended angle, and in my opinion, the article is misleading, because it's describing only a particular application of the concept.)

We say that the angle (or "arc", where it's projected on to the ideal Celestial sphere) of the triangle at your eye – which we can measure precisely with the right instruments – is the angle "subtended" by the object, and we can use that angle to calculate other things. For example, if we know the object's distance, we can now calculate its width; conversely, if we know its width, we can calculate its distance, using simple trigonometry.

In the case of the Parsec, the two "sides" of the "object" are actually the two different positions of a star observed from opposite sides of the Earth's orbit six months apart. (We have to assume the background stars against which we measure these positions are so far away that the Earth's movement doesn't measurably effect their positions.) These positions are different because of Parallax.

We now define 1 parsec as the distance a star (or other object) would have if the angle between the two observed positions – the angle they "subtend" – caused by the parallax is exactly one second of arc, also called 1 arcsecond or arcsec (which is one sixtieth of one minute, which is one sixtieth of one degree). It happens to work out at 3.26156 light years, and astronomers find this a convenient unit to use because (a) being somewhat bigger than a light year it sometimes makes the numbers more manageable and (b) it's how many distances are initially measured so the conversion to light years is an unnecessary calculation which might introduce errors.

I hope that helps. If not, I'll let others continue the clarification, as their approach might work better for you. {The poster formerly known as 87.81.230.195} 90.202.210.107 (talk) 12:19, 19 September 2019 (UTC)

Should this even be an article? Seem like the wiktionary entry wikt:subtend pretty much covers it. --RDBury (talk) 16:41, 19 September 2019 (UTC)

Well I am an inclusionist. The article may not yet be perfect, but it has scope to be more informative and in a different format, even as a stub. It is understandable however, choosing which stubby specialised info articles to permit is a huge grey area on the encyclopaedia sometimes. Can't delete em all, can't include em all...

From both of your explanations and suggestions I believe I am understanding, but I am asking, would there be a terminology similar to semi major/semi minor axis to fit the central pivot of the parsec diagrams? This would make a much easier job of tweaking the info for easy digest... though perhaps simply spreading out the three blue links around the word "subtended" in the lead section would work wonders... And thankyou for spelling it out, 90.202.210.107. ~ R.T.G 20:35, 20 September 2019 (UTC)

Or... semi major and minor axis is the term really isn't it? ~ R.T.G 07:25, 21 September 2019 (UTC)

I can't see how semi-major and minor axes are relevant to the situation. If the Earth's orbit was markedly more eccentric than it is (so that its major and minor axes were much more different than they are) one would have to take into account the actual dates (= orbital positions) when one made the observations, because different 'diameters' of the orbit would be significantly different in length. However, in actuality the divergence of Earth's orbit from circular (its eccentricity, about 0.0167) is sufficiently small that any error it introduces is generally less than the size of errors stemming from imprecisions in measurements and distortions by the Earth's atmosphere. (As instruments and measurements have become more precise, particularly when made by satellites, such an allowance would become necessary, but we've got round this by defining the value of the parsec as an exact figure.)

Incidentally, my description above of defining the parsec misleadingly omitted a detail. While it's true that we make parallax measurements 6 months apart, so using the diameter of earth's orbit to separate the two observations, a parsec is defined using half that diameter, i.e. the radius of the earth's orbit, defined as 1 Astronomical unit or 1 AU. {The poster formerly known as 87.81.230.195} 90.202.210.107 (talk) 05:29, 22 September 2019 (UTC)

In relation to the central pivot point referred to as subtended, not the Earths orbit. I'm trying to see if I can draw the same picture in words as the diagrams are drawing in the pictures. The current wordings are dumping the terminology of geometry out, but sort of skipping the actual geometry itself in favour of highlighting the terms. Both are possible, and necessary... ~ R.T.G 08:25, 22 September 2019 (UTC)

The "central pivot point" (which is the star, whose position is presumed fixed) is not subtended, it's the angle at that point/star (between the directions to the two different points on the Earth's orbit from which the two observations are made) that is subtended. "Subtending" is something that only applies to an angle, or to the apparent diameter of something considered as an arc. Major and minor axes are attributes of an elliptical orbit (or any ellipse, and are equal in value in the extreme case of a circular orbit or any circle), so they cannot be applied to a fixed point. {The poster formerly known as 87.81.230.195} 90.202.210.107 (talk) 22:16, 22 September 2019 (UTC)

(They are so fixed that, at any moment you measure them, they are a constant in space and time of course!) Apparently, the major and minor axis also apply to hyperbola. The shape from which the parsec is taken, including the far back part, is a segment of a hyperbola, the base of which is marked by the elliptical orbit of the Earth. They differentiate as major and minor because one is longer than the other and semi- is added to both to note the halfway point where they meet, while the meeting point itself is the subtended angle. And I am believing now the way to improve the description is to describe these individual parts all rather than simply throwing out the "subtended" descriptor on its own with the starting points, i.e. the AU and the arcsecond... (it's not so significant a change as the verbage used to discuss it, but the improvement to the dissemination of parsec in particular may be to no end here if done carefully...) ~ R.T.G 07:28, 23 September 2019 (UTC)

The semi-major axis of a hyperbola, which is an open curve, is a completely different entity (being outside it when it is graphed) from the major or minor axes (essentially inside diameters) of an ellipse or circle, which are closed curves.

There is no hyperbola involved in any way with the parsec that I am aware of. If you are referring to the arc of "distant stars" in the first diagram of the Parsec article, that is actually a diagrammatic representation of a straight line (or arc) on the surface of the (theoretically infinitely distant) Celestial sphere, which is only depicted as curved in the diagram because of space constraints and drawing conventions.

Perhaps it's too long since I retired from active Astronomy (which I read at University) but I have no grasp of what misconceptions you have or how to correct them, so I think it's best if I bow out of the discussion. I think it would be more profitable for you to find a teacher of mathematics or astronomy local to you so that you can discuss these questions face-to-face while drawing or pointing to diagrams. {The poster formerly known as 87.81.230.195} 90.202.210.107 (talk) 17:27, 24 September 2019 (UTC)

@90.202.210.107:According to Wikipedia, the hyperbola is the intersection of a 2 dimensional segment of a double cone. That's what the parsec article seems to be showing as the metric. Hyperbola show two diagrams, one of a double cone with a section cut through, and the other of a flattened symmetrical figure featuring crossed lines which begin and end along opposing curves, pretty much matching the parsec diagram closer to anything else I found. This, plus searching through university webpages based on the advice above (and a hint from a fansite at the Fandom website dedicated to... I can't remember, mathematic formula or geometry maybe, but this page was on the "subtended angle").

So I have looked closer (I'm giving up now too for now but nevertheless) and the article is saying the hyperbola is any sort of section through such a double cone, any sort of section but the one which cuts through the apex itself, i.e., the point representing the extent of the parsec. There is even an article in the wiki called conic sections of 67MB with detailed and historic diagrams of every sort of conical section except the one which goes symmetrically through the center...

From these I have found literally dozens of articles just on Wikipedia detailing various forms of section through 2D and 3D curved and triangular formations based in large part on the cone, except the simplest straight down the middle one we are talking about here, so even if I can't find the diamond, I know I have reached the mountain, I have drunk from the correct eternal spring. If the answer cannot be had from up here, it must have been buried. I'm going back down too, thanks truly for humouring my curiosity. I know significantly more than I did before, even if I still do not know it all now and will forget it again tomorrow, cheers o/ ~ R.T.G 19:03, 24 September 2019 (UTC)

Funnily enough, a question below mentions calculations involving ellipse, and points out something like "it's not an exact science". And neither is the calculation of the parsec an exact science... so the section we are calculating can never go exactly through the apex as we don't have an exact science on that, just really really closely, so a hyperbole, it is, and major and minor axis, they are. I'm not sure if you would agree, but I feel now all that is required is (for want of the more exact terms and definitions)... to use that information to describe the situation any more simply that it is now. It is at least possible, I'm sure of that much... x)~ R.T.G 09:31, 25 September 2019 (UTC)

Fundamental Theorem of Line Integrals

I'm trying to answer a calculus question about the Fundamental Theorem of Line Integrals. The question is: "Show the following integral is independent of path and use this to calculate the integral from (0,0) to (1,2)." The integral in question is:

$\int \limits _{C}y^{2}dx+2xydy$

I calculate the curl of this function as 0, so it is independent of path. Evaluating the integral using the Fundamental Theorem at (0,0) and (1,2), I calculated the answer as 8. However, when I parameterize the curve C as <t,2t> and calculate a standard line integral, I get 16/3. I'm thought these answers should be equal (assuming I'm calculating everything correctly). OldTimeNESter (talk) 18:07, 19 September 2019 (UTC)

The integral is actually equal to 4. I obtained this results both ways. You should check your calculations. Ruslik_Zero 19:04, 19 September 2019 (UTC)

Thanks for taking the time to look at it. I'm not sure what I'm doing wrong. Here is my evaluation of the integral using the Fundamental Theorem:

$\int \limits _{C}y^{2}dx+2xydy=xy^{2}+xy^{2}=2xy^{2}\vert _{(0,0)}^{(1,2)}$

That should equal 8. Am I evaluating the integral correctly? OldTimeNESter (talk) 16:10, 20 September 2019 (UTC)

No, because the gradient of the function you're evaluating between the endpoints isn't what's being integrated. Note that the statement you're making: $\int _{C}y^{2}\,dx+2xy\,dy=xy^{2}+xy^{2}$ doesn't actually make any sense – the LHS is an integral, which evaluates to a number, and the RHS is a function. You need to find a potential function for the vector-valued function you're trying to integrate. That is, a (scalar-valued) function $F(x,y)$ such that $\nabla F(x,y)=\langle y^{2},2xy\rangle .$ –Deacon Vorbis (carbon • videos) 16:57, 20 September 2019 (UTC)

The solution is:

$\int \limits _{C}y^{2}dx+2xydy=\int \limits _{C}{\frac {\partial (xy^{2})}{\partial x}}dx+{\frac {\partial (xy^{2})}{\partial y}}dy=\int \limits _{C}\nabla (xy^{2})d\mathbf {l} =xy^{2}\vert _{(0,0)}^{(1,2)}=4$,

$\int \limits _{C}y^{2}dx+2xydy=\int \limits _{0}^{1}((2t)^{2}dt+2t(2t)d(2t))=\int \limits _{0}^{1}12t^{2}dt=4t^{3}\vert _{0}^{1}=4$.

Ruslik_Zero 18:54, 20 September 2019 (UTC)

Composite numbers whose divisors are all perfect squares plus one

Is 10 the only composite number all of whose divisors (including itself) are of the form n²+1? The OEIS sequence A054964 claims so, but a proof is needed. GeoffreyT2000 (talk) 18:10, 19 September 2019 (UTC)

I think the first step is to prove if (a²+1)(b²+1)=c²+1 where a,b>0, a+i and b+i are Gaussian primes, then either a=1, b=2 or a=2, b=1. (Note that this is false for a+i and b+i are not assumed prime since (2²+1)(3²+1)=7²+1.) Wlog a≤b, and we have a²+1 and b²+1 (normal) primes. (a²+1)(b²+1)=c²+1 so a+i|(c+i)(c-i), and since a+i is prime, a+i|c+j where j=±i. Say c+j=(a+i)(p+iq). (a²+1)(b²+1)=c²+1=(a²+1)(p²+q²) so b²+1=p²+q², (p+iq)=u(b±i) where u∈{±1, ±i}. I'm invoking the uniqueness part of Fermat's theorem on sums of two squares but it turns out that part doesn't seem to be covered in our article; it's common knowledge though. The upshot is either (a+i)(b+i)=u(c±i) or (a+i)(b-i)=u(c±i) where u∈{±1, ±i} If a=b then c²+1 is a square, which implies c=0 and this is impossible. Therefore a<b. Arg(a+i) and Arg(b+i) are both between 0 and π/4 so (a+i)(b+i) is in the first quadrant. Also Arg(b+i)<Arg(a+i) so (a+i)(b-i) is in the first quadrant. The reduces the cases to 1) (a+i)(b+i)=c+i, 2) (a+i)(b+i)=1+ic, 3) (a+i)(b-i)=c+i, 4) (a+i)(b-i)=1+ic. In case 1, a+b=1 which is impossible if 0<a<b. In case 2, ab-1=1, ab=2, a=1, b=2. In case 4, ab+1=1, ab=0, which is impossible. In case 3, b-a=1, one of a or b must be odd, which implies 1+i divides a+i or b+i, and since these are prime either a or b must be 1. But b=a+1 so a=1, b=2 the same conclusion as case 2. All cases lead to the same solution (or a dead end), so this completes the proof.

To prove the original statement, suppose r is a composite number all of whose divisors are of the form n²+1. Factor r into primes, which must all be of the given form, so r=(a²+1)(b²+1)...(k²+1) where a²+1, b²+1, ... k²+1 are primes. The product of any two of these has the same form, so by the above the only possible factors are 2 and 5 and neither of these can be repeated. This leaves 2⋅5 = 10 as the only possible value of r. --RDBury (talk) 00:58, 20 September 2019 (UTC)