# Talk:Rope around the Earth

I'm putting this up here just to show that it was written after the final review. I feel it's important that someone addresses the error in the picture that I posted about earlier (on June 24th).

- Nordhr 16:32 29 July 2011

- Nordhr 16:32 29 July 2011

## Contents

## Final Review

been Htasoff 19:17, 17 June 2011 (UTC)This page has been edited and finished for a while. The checklist was gone through a few weeks ago, so the comments on the checklist sections were not included, because I had not told to do so when I was reviewing it. Abram gave it the okay, but I wanted to formally post it for final review.

just a few small comments left! AnnaP 13:25, 19 July 2011 (UTC)

- Can you make this sentence "It should be noted that the Earth is not a perfect sphere. Not only does it bulge out around the equator, but mountains and valleys give it a rough surface. This roughness affects the amount of slack needed, as might be expected." link down to your explanation of concave/convex.Check
- In your explanation of the convex geometry, can you explain that the only problem arises when things are
*locally*concave?I'm uncertain what you mean. The problem would still exist if it were globally concave.

- True, but
*it's not*globally concave, just locally. So, while the earth is globally concave, you can a little places where this doesn't work because of local concave-ness, such as caves. Clarified the scale.

- True, but

- This sentence "We will prove that this is the case for any convex shape. " is not exactly true. Your proof applies to any regular polygon. I'm sure the proof could be extended to convex polygons, but as is, it's more limited. Explained how, since that interior angles of all n-gons add to the same number, this finding applies to regular and non-regular shapes.
- This sentence:"And θ has a value that we can use in the equation as well: " is a bit confusing. Can you change it to say something about using the general equation for a regular polygon?Check
- Right after equation 14, you have a & theta that didn't turn itself into a proper theta. Check

AnnaP I'm giving my notes in red. I have written some in each of the headings below, and additional comments will appear at the bottom of your response to the checklist.

- Here are the comments that still need addressing. I've pulled them out so that you don't have to sort through everything AnnaP 7/7

:*There is one, very big accuracy issue to be fixed. Critical to this entire page is the assumption that the Earth is a perfect sphere **which it is not.**

- You need to make this assumption
**crystal**clear to the reader, right up front, and remind the reader of it more than once.Check- Mentioning the equatorial bulge isn't quite enough--you should mention mountains. The Andes in Ecuador are quite beautiful, and very big!

- the Earth is actually quite smooth, roughly as smooth as a billiard ball, as this anonymous person demonstrates via calculations [1].

- Let me explain why I disagree so strongly with this--your equations depend on extreme accuracy. The thing is that your result of needed to add just 6.28 feet is totally dwarfed by the fact that the equator rises over 15,000 feet in Ecuador (it also rises to something like 6000ft in Kenya). It would take quite a bit more rope to make it even stretch up and over that those mountains. So, this lack of roundness does effect the problem, and I do believe that you should mention this.
- Abram, 7/12 Just to weigh in on this, Anna is right (as she explained to me) that the fact that mountains are pointy means that they do require significant extra rope length (plateaus wouldn't be a problem -- you can work out the calculations for yourself if you would like).

- You need to make this assumption

- For a circle of any given radius, the mount of slack needed to raise a rope around the circumference one foot is 2π feet. So if you imagine the circumference of the Earth as a piece-wise composite of circles of different radii, then, as long as the rope is initially tight over all the bumps and crags, 2π feet is still all that's needed. See the image below.
- At each topographic level, the shape is a circle. So the amount of slack needed to be added is 2π/ (the arc-length of the section). This will yield 2π slack needed for the entire object, as long as the rope is taught around the bumps. Now I realise that the verticle sections will be unaffected by tis increase in slack, however they don't have to. The one foot clearance can be created by moving the slack along the length of the rope, as in the second picture. Additionally, the topographical lines can be made infinitely close, creating a less jagged topographical cross-section. No matter how eneven the circumference of the world is, the 2π feet of slack should provide the clearance of 1 foot all around.
- The shape does not even have to be circular or differentiable, only convex (it gets weird if it's not convex.). The third image shows a rope around a square of side length 1. The rope is 1 unit away from the square at all points.
- the straight section of each side of the larger figure is 1 unit long. Each "corner" of the larger figure is a quarter circle with radius = 1. thus the perimeter of the larger figure as a whole is 1+1+1+1 + 2π(1)/4 + 2π(1)/4 + 2π(1)/4 + 2π(1)/4 = 4 + 2π, while that of the small square is 1+1+1+1 = 4. (4 + 2π) - 4 = 2π units of slack. QED

- This is awesome enough to add to the page.
- In summary: It's complicated. I will address this because the globe has concave topography, however I will add a section about this.

- For a circle of any given radius, the mount of slack needed to raise a rope around the circumference one foot is 2π feet. So if you imagine the circumference of the Earth as a piece-wise composite of circles of different radii, then, as long as the rope is initially tight over all the bumps and crags, 2π feet is still all that's needed. See the image below.

- D'oh, I forgot to round corners as parts of a circle. That makes a lot of sense. BUT something funny is going to happen at the base of a triangle--I'll create a picture to show what I mean . AnnaP 7/12

Yah, I noticed that too. It cant be exactly one foot away from the concave corner, but I'm pretty sure that once this is set as a limit, that will no longer be a problem. Even rounding the corners, vs translating the slack, will become a non issue in a limiting case. The primary problem is that we are now in fractal dimensions. The only thing that does not work with this model are the overhangs under cliffs and in caves.

- I think part of the problem is whether we define "one foot above the ground" at a point to mean "one foot up along the normal" or "one foot up along the radius". If you model mountains as triangles instead of a Reimann-sum looking thing, it seems more natural to define "one foot above the ground" in terms of the normal, which means that you then cannot use the argument about circles of different radii. See picture below. -Kate 20:47, 12 July 2011 (UTC)

- <font=darkred>
**What radius are you using for the Earth's radius? The polar and equatorial radii are significantly different from each other and from the mean radius. You mention wrapping the rope around the equator, but don't specify if you're using the equatorial numbers**

- <font=darkred>This comment was not addressed

- I'm using an decently accurate amalgam of mean radii. Unfortunately, this is a bit af a short-comming, however the radius I use is not inaccurate, and I don't think it is worth redoing every calculation on the page with an exact mean equatorial radius.

- While I agree it isn't worth redoing your calculations, I do think that you should explain where your number is coming from. You can site a particular source for the number that you do use. Again, I think that this is important since your page makes a big deal out of very small numbers.

- Abram, 7/12: I also think that citing your number is a good idea. Anyway, it takes very little time, so why not?
- I derived the value I used, but I don't exactly remember how I went about doing it. I began with a value given in metric, and remember at one point I converted from metric to imperial, which made it so that my value does not precisely match other values available. The value I use is accurate, but I can't exactly retrace my steps.

Messages to the Future

- Did not feel that there was anything in particular I needed to say.

References and footnotes

- All images are attributed except for those I made entirely myself.
- Make sure to note this on the pages for the files themselves. When I click on the image it should be noted somehow Check
- I cited both the book and the website that I used for the math on this page.

Good writing

The following items are just meant to be reminders. If one of these items needs clarification, or seems like a great idea that you don't know how to implement, see What Makes a Good Math Images Page?.

Context (aka Generating interest aka Who cares?)

- tried to turn the counter intuitive result into a logical one by explaining it via ratios, emphasizing how intuitive such an unintuitive result can be when considered from the proper perspective.
- I think that your first paragraph at the very beginning of the page provides a nice context Thanks

Quality of prose and page structuring

- The beginning paragraph(s) of the page clearly define the topic or purpose of the page as a whole, and may outline the page or preview conclusions that will be reached later in the page.
- All check, no further comment.

Your overall structure works well, but here are some other fixes

- The very first sentence of the page should be rephrased to avoid the phrase set aside by commas.I'm not too concerned about it, but I took care of it.
- work on the entire "In the puzzle..." paragraph. You might want to define your variables in another bulleted list as you did above.The only variable not in the list is
*l*, which doesn't occur in the diagram. I define it when I introduce it, so am uncertain what needs to be changed. - The sentence starting "Just as bizarre..." needs to be rephrased, potentially broken into two sentencesCheck
- The "Here is a way to illustrate..." paragraph is not entirely necessary. It is also confusing in its current form. If you want to include these points, find another way to say it. Steve and I have been talking about the accuracy of the approximations, and this was the best way I could explain the varying degrees of accuracy.
- Okay
- Make one more pass on the entire piece and try to make your sentences simpler. In general, try to avoid having more than 2 commas in a given sentence. You tend towards what I call "Sentences of doom" which are long and confusing for the reader. Short and sweet is best.

Integration of Images and Text

- All check, no further comment.
- Image 1 is not to scale. Make sure that you note that!I figured that was obvious, but made sure to note it explicitly anyway.
- Use an image to explain your arclength equation Although I think it is a worthy topic, I also think that the image I have to explain it, and the explanation itself, will digress from the rest of the section. Perhaps it could go on another helper page?

- I vote for a helper page. Just leave the link red Check

Connections to other mathematical topics

- Because this is a specific puzzle, and has relatively simple math, not many connections are natural. It seems to be just a neat little find. I did connect the height section to an approximations page (not yet up), which may well link to many things like Taylor series, summation, etc. eventually.
- You're in good shape here Check

Examples, Calculations, Applications, Proofs

- The proof for the puzzle is in the MME, followed by an discussion to a part of the original puzzle which I added. I discuss this second, much more complicated part as best I can, and hide it, as it is not necessary for the original puzzle.

* I would actually go through the calculation of how much distance you'd need to add if you wanted to lift the rope a different distance from the surface of the Earth--say 35,000ft, or the cruising elevation of a commercial get. Check

Mathematical Accuracy and precision of language

- The height section becomes more complicated, and had to be done with approximations. This topic was introduced, and a helper page proposed. Everything else is meticulously explained and defined.
- There is one, very big accuracy issue to be fixed. Critical to this entire page is the assumption that the Earth is a perfect sphere
**which it is not.**

You need to make this assumption **crystal** clear to the reader, right up front, and remind the reader of it more than once.Check

- Mentioning the equatorial bulge isn't quite enough--you should mention mountains. The Andes in Ecuador are quite beautiful, and very big!
- the Earth is actually quite smooth, roughly as smooth as a billiard ball, as this anonymous person demonstrates via calculations [2].

Onto the smaller points

- Make sure you clearly define what you mean by arc.See comment on arclength.
- Similarly, looking at image 3 and reading your text, it is unclear if refers to the entire length where the rope is not touching the Earth or if it refers only to the length opposite . This becomes clear later on, but it should be more clear up front.
- Using l instead of is confusing in equations 4-6. If you want to do this, be sure to have an entire small paragraph explaining your use of l immediately before your equation.

I do devote the first full paragraph of the section to why I use *l* I feel as though any more would be redundant, and I got comments telling me that parts of the page were redundant.

- <font=darkred>
**Can you make the "l" in that paragraph bigger, or bold to make it stand out more? That might be an easy fix to highlight that explanation.**I can, but Steve has STRESSED consistancy in my variables almost every time I met with him about this page. - <font=darkred>
**I strongly advice against using rounded numbers***anywhere*on the page, given that your work depends on very accurate numbers. - <font=darkred>This comment was not addressedI agree for the calculations, but disagree for the explanations. The math obviously needs exact values, but when describing the scope, ratios, and overall gist of the puzzle, I firmly believe that smoothing the numbers allows for a more straightforward explanation.
- <font=darkred>
**What radius are you using for the Earth's radius? The polar and equatorial radii are significantly different from each other and from the mean radius. You mention wrapping the rope around the equator, but don't specify if you're using the equatorial numbers** - <font=darkred>This comment was not addressed I'm using an decently accurate amalgam of mean radii. Unfortunately, this is a bit af a short-comming, however the radius I use is not inaccurate, and I don't think it is worth redoing every calculation on the page with an exact mean equatorial radius.

<font=darkred>***your definition of "argument" in the mouse over isn't going to help anyone who doesn't already know what argument means. Try to rework that.**

- <font=darkred>This comment was not addressed Check
- rephrase the "root finder" mouse over to be a bit more clear or simply explain that you are finding the roots without using such specific phrasing Check

Layout

- All check, no further comment

Misc

- Can you make your equation numbering consistent through the entire page? You switch methods part of the way through, which throws me off.

If you want to separate out the equation numbering by section, you can always make it "Equation 1-1" or some similar method.Check

## General Comments not on one section

I was looking at the main picture on this page, and it says that the rope is 1 ft longer than the circumference of the Earth. Should it be 6.28 feet longer and the '?' is equal to one foot?

- Nordhr 14:44, 24 June 2011 (UTC)

Kate 21:00, 26 May 2011 (UTC):

- Your image links go to the captions, which I think can be confusing. Try moving the div tag so that it either just precedes the image or so that it goes around the image. CHECK

Smaurer1 20:37, 24 May 2011 (UTC)

The Why it is Interesting Section is somehow inside the More Mathematical Explanation Section, because it is hidden until the MME section is completely unhidden. This should not be. CHECK

## Opening Caption

The wording "This is a puzzle in that the answer is surprising" feels a bit awkward to me. What about wording like this:

- There is a famous puzzle about a rope tied taut around the equator. It asks how much the rope must be lengthened so that it will be able to hover one foot above the surface of the earth around the entire equator. While finding the answer requires only basic geometry, even professional mathematicians find the answer strangely counter-intuitive.CHECK

-*Abram 5/24*

I don't think you need a question mark after the first sentence. Rebecca 15:36, 25 May 2011 (UTC) CHECK

## Basic description

Kate 21:00, 26 May 2011 (UTC):

- I would leave some blank space at the end of your Basic Description so that the heading for the MME is all the way left instead of wrapping around your picture. CHECK

- I also still find the placement of the max. height image confusing - if it could even go on the right I think that would be much better, because as it is, you notice the image before you start reading the basic description, and then that can get confusing.
- This issue might also be solved by adding a caption to Image 1 I recognize your point, and tried to implement it; this is now the least awkward position I can find.

- Re-wording suggestion for the last paragraph:

*Just as bizarre is that if this newly lengthened rope were lifted at one point to pull the rope taut around the earth again, the clearance under that point of lifting would be quite large.*CHECK- Also, maybe clarify that "this specific case" refers to the specific case of the earthCHECK

-*Abram 5/24*

~~*I would add a picture of the rope being lifted up so that it is taut again, and show what you mean by maximum clearance for the basic description. This will clarify what you're saying and its always good to have more pictures. The picture could be similar to that one you use in a more mathematical explanation, but you don't need the labeling so you could maybe use and earth instead of just a blue ball. Just a thought. CHECK
Rebecca 15:37, 25 May 2011 (UTC)~~

- Rebecca 20:09, 1 June 2011 (UTC) A suggestion about how to fix the placement of Image 1- I would crop the image so there's much less black space above and below the earth. I would also make the image slightly smaller, and I would add an extra space before the paragraph beginning with "Just..". This will help move the picture down to the second half of this section, and I think it will be less confusing.

- Rebecca 22:11, 10 June 2011 (UTC) This looks much better. I would still consider adding an extra space before the paragraph with "Just" to help the layout.

## More mathematical explanation

~~Rebecca 12:34, 31 May 2011 (UTC) When you're explaining Image 4 in the mathematical section, I'm not sure if you need to keep linking back to the equations. I think you could just link back the first time and the paragraph will look less cluttered. CHECK~~

Kate 21:00, 26 May 2011 (UTC):

*The puzzle states that we've lengthened the rope and made the rope hover 1 foot of the surface of the earth.*

- I don't like that this sentence is past tense-y… I would like it better if it was "The puzzle states that we have to lengthen the rope…" CHECK

*Indeed, one can see that the additional 2π is a result of distributing the 2π to the 1 in Eq. II, which always yields 2π.*CHECK

- I think it would be clearer to say "which yields an increase of 2π no matter what the radius of the ball is."

- Typo:
*represents θ, the angle whose cosine is angle whose cosine is*CHECK - Taut/taught typo:
*. Because the rope is taught around most of the earth,*CHECK

When you point out that R_2 is 1 foot larger than R_1, maybe remind the reader that this is because the puzzle states that we've made the rope to hover 1 foot of the surface of the earth. This is an example of a general principle that when you are deriving an equation, statement, etc, it's good to always be explicit on where each statement/step is coming from (in this case, the hypotheses of the problem).CHECK
-*Abram, 5/24*

Two problems with the statement: "Indeed, one can see that the additional 2 π is a result of extending the radius of the rope circle by one foot, an extension that will by definition be the same no matter the initial radius of the object being enclosed":

- The small problem is that the extension is not independent of the initial radius "by definition"; if it were a definition, you wouldn't have had to prove it just now!CHECK
- The larger problem is that one might not see why the 2*pi extension is independent of the initial radius, and you don't do anything to explain why. Maybe add a couple sentences pointing out that because you didn't make the value of R_earth any number in particular, that your answer didn't depend on the specific radius of the earth in any way, so this 2*pi extension would be the same for any ball or planet or star of any size.CHECK

-*Abram, 5/24*

~~ GREAT GREAT first section of a more mathematical description. I found it very clear.Rebecca 15:39, 25 May 2011 (UTC) GOOD~~

### Maximum Height of Rope

See if you can be a little more careful or precise in your definition of x_1. If you don't know what I mean, let me know.CHECK
-*Abram, 5/24*

As we just discussed in person, the phrasing in the paragraph, "Now we will develop a formula..." signals to the reader that the amount of slack is something that has to be derived through a new formula, not something that we just learned to be 2*pi. Do something to make it clear that the latter is actually the case, and that this "generalization" refers to an entirely different (as yet unidentified) puzzle.
-*Abram, 5/24*

In formula A, the variables L_arc and r are undefined.CHECK
-*Abram, 5/24*

In the sentences beginning "Using the Pythagorean Theorem..." and "Also, since theta is the angle whose cosine", point out which triangle readers should be looking at.

- It could help to name the vertices with letters (yes, I know this would mean changing the drawing again). CHECK
- Also, at some point you should maybe justify why, or at least explicitly state, that the angle between the radius of the earth and the tangent line from that radius to the apex, is 90 degrees.

-*Abram, 5/24*

The sentence "Keeping in mind..." seems mighty redundant. It's really unusual that I'll say something has been over-explained, but in this case, maybe it has. CHECK
-*Abram, 5/24*

"Keeping in mind..." and "Where theta is the angle..." are not complete sentences, so the first word shouldn't be capitalized. The standard, I think, is to put a comma after the equation, and then make the words "keeping" and "where" lower-case.CHECK
-*Abram, 5/24*

The derivation of Eq. 3 could be expanded in a few ways:

- Add an introductory sentence stating the role of what you are about to do, e.g.
*Now that we have a formula for x_0, were are going to relate x_0 to x_1.*CHECK - Point out that because the rope is taught around most of the earth, all the slack comes from lengthening arc AC to segments AB and BC (see what I mean about adding names to these points)? CHECK

-*Abram, 5/24*

A few things about the paragraph "due to the presence of h within the cos^-1":

- It should say "...within the argument of cos^-1", with maybe a mouse-over explaining that "argument" means "input" (or you can just use "input") CHECK
- The problem isn't that h is within the cos^-1; it's that h is within the cos^-1 and it's also somewhere else in the equation. You might talk to Steve to get a somewhat precise statement that describes this phenomenon. CHECK I THINK
- Maybe add a mouse-over explaining what a root-finder isCHECK
- Many of these equal signs should be approximately equals signsCHECK I THINK

-*Abram, 5/24*

Just FYI, I haven't read the series section at all yet.
-*Abram, 5/24*

~~ No comma after "again be held taut" Rebecca 15:42, 25 May 2011 (UTC)CHECK~~

## Why It's Interesting

I've got ideas about this section, too, but I'll get to them later.
-*Abram, 5/24*

I would change you second and third sentences of "Why it's interesting" to something like "However, it is also interesting to explore why this confusing result proves true." I just think the way you have it is a little confusing. I realize what you are saying, but I can't find the proper connective to use. All the ones I've tried only slowed down the text. Also, I don't know what image to use, if you have an idea, let me know.

Can you think of a picture to add to this section as well? Rebecca 15:43, 25 May 2011 (UTC)

## Unsorted Comments ALL CHECK

Smaurer1 10:13, 20 May 2011 (UTC)

First, look up these two words; they don't mean the same thing and you only mean accuracy. Second, there are two meanings of accuracy, absolute and relative. I would say that the second order approximation is **much much** more accurate than the first order approximation. Perhaps we need a helper page on accuracy. I know that there is a difference, and am pretty sure I mean both. Both approximations are accurate, the first to 1 decimal place, the second to three. Both are precise, the both yielded 3 decimal places of precision, with the second being more accurate over the three places.

The intro should make clear the sense of puzzle. It is not a puzzle in that it is hard. It is a puzzle in that it is counterintuitive to most people. The second question, not even mentioned in the current intro, is even more counterintuitive. NEED TO DO

Becky raises an important point below about explanations that come *after* formulas. The rule is that this is not allowed unless it is made clear at the time of the formula or beforehand that the explanation is coming afterwards. At the very least the formula was end with a comma, not a period. But better to say "as we show below..." CHECK

It's good that Harrison made a table of his variables, but in fact his very first equation doesn't agree with his definitions. Then he goes on to use new variables (with rope1 and rope2) that he never defined. CHECK

The Why It's Interesting section is not really about why it's interesting. It is about how Harrison comes to peace with the nonintuitive answers; the non-intuitiveness is why it is interesting.

I am putting a few comments in the document itself and handing Harrison a paper copy with many more comments. We will go through it in person, but maybe not today as I am busy with honors examiners.

When the rope is taut around the globe, its length equals the circumference of the Earth.

Lengthening the rope so that it is 1 foot off the ground at all points simply means changing the radius of the circle it forms from:

- R
_{rope 1}= R_{earth}

to

- R
_{rope 2}= R_{earth}+1 ft. All this can now be simplified and the sub1-2 notation eliminated

Below are comments by Becky on May 19-20

Harrison, this is a great page already! I really like the idea and the execution. The topic is visual and I find it very interesting!

Great things about the page:

- Its great that your paragraphs are so short! The page is easy to read as a result.

- Great usage of the mouseover for pythagorean theorem.

- Overall, the message is clear.

- Good use of bullets to define the variables.

Some suggestions I have:

- "Lengthening the rope so that it is 1 foot off the ground at all points simply means changing the radius of the circle it forms from:..." I'm not sure that this is the best way to say this. Maybe try to reword it a bit. NOT DONE YET

~~Also, as a pointer, we learned last year that people get scared off by words like "simply" and phrases like "we can easily see that" because its intimidating if they haven't grasped the concept yet and they don't find it simple. I actually think your usage of "simply" isn't bad, but later you say "now it is clear that," which you should probably avoid. CHECK~~

- Some of your equations look like this:

- R
_{rope 1}= R_{earth}

- while others look like this:

~~I think the issue is that the first equation isn't encased in math brackets. You should be consistent and put these in math font. DONE PURPOSEFULLY, IF I'M TALKING ABOUT EQUATIONS OF THE ENTITIES WITHIN THEM, I WANT IT TO BE HTML SIZE CHARACTERS. ALSO, THIS REDUCES THE RISK OF MAKING IT LOOK LIKE THERE ARE AN OVERWHELMING NUMBER OF EQUATIONS ON THE PAGE.~~

~~When you move from equation 1 to equation 2, its not immediately clear what's going on because you don't explain until after. I know that I personally wont continue reading until I understand where an equation came from (so even if the explanation is the next sentence I might miss it). So it might be good to explain what's going on before you give the equation. I'd consider doing something like this instead:~~

- Since θ is the angle whose cosine is , we can replace θ with . Thus, Equation 1 is equivalent to:

~~. CHECK~~

- The paragraph that you begin with "As a result of the cos-1 (R/R + h)", you should try to be clear about why this means there is no explicit formula to find the height. CHECK, I THINK

- I think you have a good start to the why it's interesting section. It does seem counter-intuitive that the increase would be so small when the earth is so large. However, I'm not sure that you're saying this in the best way. Try to work with the section and maybe ask the other researchers for help if you need it. Also, you might want to discuss how it seems like the size of the ball should matter.

- I think a picture would be really good in this last section. One idea for a picture might be to help elaborate on the idea that the size of the ball doesn't matter. If you have a different idea- go with it!

I hope these suggestions help!

-Becky

Harrison 5/19/11

Rope around the Earth is ready to be reviewed.

### Comments by xd 01:10, 25 May 2011 (UTC)

Overall, I like the page. Short and sweet. I offer two suggestions.

Under Max Height of Rope, Possible Extension

1. Root finder program -> can you write such a numerical root finder program in Matlab or Python? Matlab even has its own program. You can even explain that and show a sample code. It will be a very nice to have such a program in there for the more involved readers. The math of the this page is not very dense so it may be very appropriate to do this. You can refer to Prof. Mewes' Matlab exercise root finder.

2. Series solution -> do intend to write a helper page or a separate page for the explanation? From Eq. 5 to Eq. 6, there is a huge gap and you are not very clear on whose Taylor Series approximation are you using i.e. the sqrt or the cosine inverse? You also need to explain the order of approximation.