#### ChrisTP

• 926
##### Re: Are plane tickets real?
« Reply #200 on: May 01, 2020, 04:57:47 PM »
Bingo. And not just the Bing API, but everything ever run on a computer has the same limitations. That's why computers typically deal differently with two types of number. The set of integers, in maths, referred to as are all the whole numbers, anything which can be written without a fractional part (e.g. -19, 0, 120). The set of real numbers, includes all the whole numbers (i.e. members of ), plus all fractions, plus all the irrational numbers such as π. In maths, is a superset of . In a computer they are treated separately.

This is then yet even more evidence that the bing API is not based on the haversine formula! The real true haversine formula is not constrained by limited memory where the bing API one is.
OK, say you are correct in that it's not 100% accurate because of computation limitations.. What now? It's still accurate enough to show that it's using a globe projection so what exactly is your point? It just seems like you're doubling down on something pointless.
Tom is wrong most of the time. Hardly big news, don't you think?

#### robinofloxley

• 203
##### Re: Are plane tickets real?
« Reply #201 on: May 01, 2020, 04:59:04 PM »
Bingo. And not just the Bing API, but everything ever run on a computer has the same limitations. That's why computers typically deal differently with two types of number. The set of integers, in maths, referred to as are all the whole numbers, anything which can be written without a fractional part (e.g. -19, 0, 120). The set of real numbers, includes all the whole numbers (i.e. members of ), plus all fractions, plus all the irrational numbers such as π. In maths, is a superset of . In a computer they are treated separately.

This is then yet even more evidence that the bing API is not based on the haversine formula! The real true haversine formula is not constrained by limited memory where the bing API one is.

Just run me through the logic here. If I say "here's a computer program which solves pythagoras's formula", then you're going tell me it doesn't, because it runs on a computer and the real pythagoras's formula is the perfect mathematical one. That seems to be the implication of what you're saying. And substitute for pythagoras literally every known mathematical formula.

I think you are missing the real point here. We have two implementations of something. I'm claiming the something I've implemented is called haversine. Microsoft wrote the other one and they claim theirs also implements something called haversine.

I've demonstrated that Microsoft's version rounds its numbers to the nearest 1cm, but actually found a way to stop their version from doing any rounding at all. Both versions are running on computers that will inevitably give imprecise results because they are dealing with real numbers. Both versions inevitably deal with pi because they calculate arc lengths on the surface of a sphere with a certain radius. The number pi is irrational and its true value is not precisely known, because it would contain an infinite number of digits. You could store a decimal digit of pi on every atom in the entire universe and you still don't have enough digits.

Given all that, when you compare 200 million results from their code with 200 million results from my code, the results match, 100%. The only way this could possibly happen is that the two implementations are virtually line for line the same, do exactly the same steps in exactly the same order, get exactly the same imprecise results from each of the internal calculations.

This is like getting two exam essays to mark from two different students. Not only do you find complete agreement on every point, but the same spelling and grammar mistakes are made throughout. It'd be pretty suspicious wouldn't it.

I don't have access to the Bing code, but the published versions of the haversine formula are almost certainly in my opinion where Microsoft looked when they were writing this stuff. It's simple enough code and there are a few minor variations and I tried a few of these, but in the end found one which, with a little extra experimentation, just matches exactly their results. You could just swap out their code and swap in my code and it would be impossible for anyone to find any difference.
« Last Edit: May 01, 2020, 07:03:22 PM by robinofloxley »

#### Nosmo

• 7
##### Re: Are plane tickets real?
« Reply #202 on: May 01, 2020, 11:35:21 PM »

in binary x + x does not always equal 2x.

An accurate statement is that in binary
x + x = 10x
Where for example
0.01 + 0.01 = 0.1
0.1 + 0.1 = 1
1 + 1 = 10
10 + 10 = 100

It seems iamcpc is under the impression that one tenth plus one tenth in binary does not equal two tenths.
This is incorrect and can be shown
(1/1010) + (1/1010) = (10/1010)

The problem that iamcpc is highlighting is that when expressing fractional numbers in a decimal fraction style format not all values can be exactly represented.
For the Base 2 number system one tenth is one of these fractions.
The problem here is not that oneX + oneX <> twoX.
It is that X is not exactly equal to one tenth.

This is not a unique problem for Base 2.
Base 10 has the same issue with a range of fractions, 1/3 for example which has been pointed out.

It seems that iamcpc does not understand the use of precision in the real world.
It is also strange that when using computers such an exacting level of precision is being asked for, when one of the instruments that iamcpc has used to verify Bin’s distances is such inaccurate and imprecise device.

I have used a use a rolling measuring tape, GPS, I've driven a route that I've walked along the street and used my Car odometer, I've used wearable fitness trackers, and for a while I wore a pedometer which also tracked distance.

A Car odometer, a device that measures to the one tenth of a kilometre, (one tenth of a mile in the UK or USA), so not very precise. They also generally measure distance based on the rotation of a wheel, whose diameter and circumference vary over time (as tyres wear), so also not exactly accurate.
This seems to be quite a different standard to the one Robinofloxley is being held too.

I must say Robin that I have enjoyed your explanation and work in this thread, it is an example of the unexpected educational threads that pop up from time to time.

#### robinofloxley

• 203
##### Re: Are plane tickets real?
« Reply #203 on: May 02, 2020, 09:23:26 AM »
A Car odometer, a device that measures to the one tenth of a kilometre, (one tenth of a mile in the UK or USA), so not very precise. They also generally measure distance based on the rotation of a wheel, whose diameter and circumference vary over time (as tyres wear), so also not exactly accurate.
This seems to be quite a different standard to the one Robinofloxley is being held too.

I must say Robin that I have enjoyed your explanation and work in this thread, it is an example of the unexpected educational threads that pop up from time to time.

Thank you for saying that, I'm glad you've enjoyed it. I've learned quite a lot myself putting it all together and have enjoyed doing it.

I do think iamcpc has a somewhat varied standard when it comes to evaluating evidence. He believes Bing distances are based on measured distances in the real world and take into account elevation. No evidence has ever been given for this belief, so unless he's holding something back, that's a strong belief based on zero evidence, however when it comes to this whole haversine business, there's a mountain of evidence now and he's still a very long way from being convinced.

#### iamcpc

• 832
##### Re: Are plane tickets real?
« Reply #204 on: May 04, 2020, 05:44:40 PM »

A Car odometer, a device that measures to the one tenth of a kilometre, (one tenth of a mile in the UK or USA), so not very precise. They also generally measure distance based on the rotation of a wheel, whose diameter and circumference vary over time (as tyres wear), so also not exactly accurate.
This seems to be quite a different standard to the one Robinofloxley is being held too.

I must say Robin that I have enjoyed your explanation and work in this thread, it is an example of the unexpected educational threads that pop up from time to time.

Again we are talking about math and a mathematical formula. Go take a first grade math test. On the question 1+ 1 = I want you to put 1.999999995 and see if the answer is correct or not. Now try again with the answer 2.0000000000005. This is the results of a mathematical calculation, so as a result both of the answers listed before are WRONG even though they are very very close to being right.

#### ChrisTP

• 926
##### Re: Are plane tickets real?
« Reply #205 on: May 04, 2020, 06:09:55 PM »

A Car odometer, a device that measures to the one tenth of a kilometre, (one tenth of a mile in the UK or USA), so not very precise. They also generally measure distance based on the rotation of a wheel, whose diameter and circumference vary over time (as tyres wear), so also not exactly accurate.
This seems to be quite a different standard to the one Robinofloxley is being held too.

I must say Robin that I have enjoyed your explanation and work in this thread, it is an example of the unexpected educational threads that pop up from time to time.

Again we are talking about math and a mathematical formula. Go take a first grade math test. On the question 1+ 1 = I want you to put 1.999999995 and see if the answer is correct or not. Now try again with the answer 2.0000000000005. This is the results of a mathematical calculation, so as a result both of the answers listed before are WRONG even though they are very very close to being right.
Beginning to think you don't have a point to this or don't understand what people are saying. I asked above what you're trying to get across but you ignored that. We all know the limitations of computers when dealing with large numbers.

You can't give the absolutely correct value for pi, so does that mean the value you do give will always be incorrect for any purpose?
« Last Edit: May 04, 2020, 06:12:47 PM by ChrisTP »
Tom is wrong most of the time. Hardly big news, don't you think?

#### iamcpc

• 832
##### Re: Are plane tickets real?
« Reply #206 on: May 04, 2020, 10:21:49 PM »

A Car odometer, a device that measures to the one tenth of a kilometre, (one tenth of a mile in the UK or USA), so not very precise. They also generally measure distance based on the rotation of a wheel, whose diameter and circumference vary over time (as tyres wear), so also not exactly accurate.
This seems to be quite a different standard to the one Robinofloxley is being held too.

I must say Robin that I have enjoyed your explanation and work in this thread, it is an example of the unexpected educational threads that pop up from time to time.

Again we are talking about math and a mathematical formula. Go take a first grade math test. On the question 1+ 1 = I want you to put 1.999999995 and see if the answer is correct or not. Now try again with the answer 2.0000000000005. This is the results of a mathematical calculation, so as a result both of the answers listed before are WRONG even though they are very very close to being right.
Beginning to think you don't have a point to this or don't understand what people are saying. I asked above what you're trying to get across but you ignored that. We all know the limitations of computers when dealing with large numbers.

You can't give the absolutely correct value for pi, so does that mean the value you do give will always be incorrect for any purpose?

My point is that it's not robinofloxely who is being held to this standard of accuracy. It's the testing of the mathmatical formula which is being held to a high degree of precision which I have explained several time

#### ChrisTP

• 926
##### Re: Are plane tickets real?
« Reply #207 on: May 04, 2020, 10:43:09 PM »
Great! and so what is your point? it functions as intended and isn't off by any noticeable error. Now look at your idea of the flat earth map and notice the poles are stretched from 0 which it should be, to 24,901 miles. That's a massive error and doesn't appear to be happening at the equators. Why are you so extremely against such small roundings of numbers vs 24,901 miles or error? This is the point I'm trying to make. you could complain that computers are off by a even a mile but it's still sure as hell showing to be more of a spheroid shape than a flat one. you're being held to the same standards but your idea of what the map should be has a huge error which has been proven. You could get caught up in the debate about whether machines can calculate big numbers or you could stop doubling down on a silly point.
Tom is wrong most of the time. Hardly big news, don't you think?

#### robinofloxley

• 203
##### Re: Are plane tickets real?
« Reply #208 on: May 05, 2020, 09:30:40 AM »
My point is that it's not robinofloxely who is being held to this standard of accuracy. It's the testing of the mathmatical formula which is being held to a high degree of precision which I have explained several time

I guess I'm with ChrisTP and Nosmo on this, I'm really struggling to understand your point. I know you've tried to explain to us several times, but I'm just not really getting it.

Computers and maths are two different things. Maths doesn't have to live in the real world so things like π and √2 and ∞ are nothing special and the fact that their values can never be written down as actual numbers is an irrelevance. My favourite mathematical formula is:
eiπ = -1
Now you can't just plug in values for e, i and π and work this out, but it is nevertheless true.

Computers are just machines for performing calculations, nothing more. You can't just hand a computer an equation like that and expect it to solve it.

What you can do is take a mathematical formula and use it as a basis for some computer code to calculate a result from a set of values. The code is not maths, the syntax is not the same for a start. What we're comparing in this thread topic is not maths and computers, but two separately written bits of computer code which do something. The something they do is claimed to be the same thing, whatever name you like to give that thing is up to you. We're all referring to it as haversine, but if you object to that, fine let's call it Nigel instead. So now we have Bing and ourselves implementing Nigel. Do they really? Well we can just feed in endless different values to each one and every time, the two Nigels agree with each other 100% (and don't forget, the most recent code does away with all rounding, so that's no longer an issue). So Nigel1 and Nigel2 are in complete agreement with each other, always. They are identical twins.

#### iamcpc

• 832
##### Re: Are plane tickets real?
« Reply #209 on: May 07, 2020, 03:43:54 PM »
Great! and so what is your point? it functions as intended and isn't off by any noticeable error.

I guess we can just agree to disagree on what level of variation is considered noticeable.  if I said that 1 + 1 = 1.9999999995 I would consider the amount that equation is off to be noticeable. You don't. There is nothing wrong with us having a difference of opinion here.

Now look at your idea of the flat earth map and notice the poles are stretched from 0 which it should be, to 24,901 miles. That's a massive error and doesn't appear to be happening at the equators.

Bing maps has an interactive scale.

Why are you so extremely against such small roundings of numbers vs 24,901 miles or error?
I'm not at all against rounding.

you could complain that computers are off by a even a mile but it's still sure as hell showing to be more of a spheroid shape than a flat one.

Spheroid? I thought the earth was a Sphere, or was it an oblate spheroid? I'm confused.

your idea of what the map should be has a huge error which has been proven.

That's funny because like 99% of the population would agree that Bing maps, with it's interactive scale, represents a map of the earth.

Computers and maths are two different things.

Yep.  This has been strongly demonstrated here. So trying to compare non computer math to computer math is a little challenging.
« Last Edit: May 07, 2020, 03:52:11 PM by iamcpc »

#### ChrisTP

• 926
##### Re: Are plane tickets real?
« Reply #210 on: May 07, 2020, 04:13:29 PM »
Quote
That's funny because like 99% of the population would agree that Bing maps, with it's interactive scale, represents a map of the earth.
99% of the population would also agree that Bing maps work because it's representing a map of a globe earth. you think it's representing a flat earth, that's great and all but you're wrong. Stop with the "interactive scale" stuff, repeating it doesn't make Bing maps any less of a sphere projection.
Tom is wrong most of the time. Hardly big news, don't you think?

#### iamcpc

• 832
##### Re: Are plane tickets real?
« Reply #211 on: May 07, 2020, 04:38:09 PM »
you think it's representing a flat earth, that's great and all but you're wrong.

This is where you are wrong. I think that Bing maps represents a map of the earth regardless of what shape the earth is.  So please leave the shape of the earth out of it.

It's very common among the FE community to say that there is no map of the earth, regardless of what shape the earth is, or some variant of that concept or idea such as:

-We don't know what a map of the Earth looks like
-There is no map of the Earth
-We don't know the size and positions of the continents on this planet
-Long distances are unknown or unmeasured

Do you notice how NONE of those statements or ideas say that the earth is flat?  So please leave the shape of the earth out of it.

To further illustrate it:

-If the earth is a cube I can still use Bing maps, or a variant of it, to accurately navigate this planet.
-If the earth is a sphere I can still use Bing maps, or a variant of it, to accurately navigate this planet.
-If the earth is a spheroid I can still use Bing maps, or a variant of it, to accurately navigate this planet.
-If the earth is half sphere dome I can still use Bing maps, or a variant of it, to accurately navigate this planet.
-If the earth is a half spheroid dome I can still use Bing maps, or a variant of it, to accurately navigate this planet.
-If the earth is a half oblate spheroid dome I can still use Bing maps, or a variant of it, to accurately navigate this planet.
-If the earth is an oblate spheroid I can still use Bing maps, or a variant of it, to accurately navigate this planet.
-If the earth is a flat circular dinner plate I can still use Bing maps, or a variant of it, to accurately navigate this planet.

We could repeat this with an infinite number of shapes/sizes. So again please leave the shape of the earth out of it.

If the earth is [Insert shape here] I can still use Bing maps, or a variant of it, to accurately navigate this planet.

I very strongly disagree with this concept or idea that we don't have a map. I'm not talking about the shape of the earth, I'm only talking about a map, so again please leave the shape of the earth out of it. I believe that after hundreds, if not thousands, of years of advancement in things like navigation, shipping, GPS, surveying, cartography, etc that we are able to make a map of the earth. I'm not talking about the shape of the earth, I'm only talking about a map, so again please leave the shape of the earth out of it.

No. I do not think that Bing maps represents a flat earth. I think that Bing maps, with an interactive scale, represents a map of the earth. I'm not talking about the shape of the earth, I'm only talking about a map so this is where you are confused on my position.

Stop with the "interactive scale" stuff

Then stop with the "Bing maps is not accurate because the distances are wrong the further you get away from the equator stuff" because that is the canned response to that statement.

#### ChrisTP

• 926
##### Re: Are plane tickets real?
« Reply #212 on: May 07, 2020, 05:14:25 PM »
Well if I wrongly assumed you to be a flat earther I am sorry. I will say though that there is no mystery behind Bing maps, humans made it to work under the assumption that the shape is mapped out based on a spheroid. It simply wouldn't all work exactly the same with any other shape (unless it was projected from a spheroid). I'm not sure how else to put this "If the earth is [Insert shape here] I can still use Bing maps, or a variant of it, to accurately navigate this planet." No, because unless the visual representation is a projection from a spheroid the base shape of the thing being mapped out has to be a spheroid regardless. You can project that onto a cube, a plane, whatever... The calculations are still based on a globe shape.
Tom is wrong most of the time. Hardly big news, don't you think?

#### ChrisTP

• 926
##### Re: Are plane tickets real?
« Reply #213 on: May 07, 2020, 05:16:03 PM »
Quote
Then stop with the "Bing maps is not accurate because the distances are wrong the further you get away from the equator stuff" because that is the canned response to that statement.
Man, I wouldn't have to keep repeating it if you understood... The flat map shown is a projection from a globe and just a representation of the underlying base shape, which we know is spheroid and nothing else.
Tom is wrong most of the time. Hardly big news, don't you think?

#### Nosmo

• 7
##### Re: Are plane tickets real?
« Reply #214 on: May 08, 2020, 11:21:40 PM »

If I took a math test and I put 1 + 1 = 1.9999999999999999999999999999995

I'm pretty sure everyone here would agree with you on this.
You have clearly introduced an error.

Again we are talking about math and a mathematical formula. Go take a first grade math test. On the question 1+ 1 = I want you to put 1.999999995 and see if the answer is correct or not. Now try again with the answer 2.0000000000005. This is the results of a mathematical calculation, so as a result both of the answers listed before are WRONG even though they are very very close to being right.

I guess we can just agree to disagree on what level of variation is considered noticeable.  if I said that 1 + 1 = 1.9999999995 I would consider the amount that equation is off to be noticeable.

Lets couch this in a slightly different way, in a practical real world application.
If I asked you to machine some bar stock and produce three 2cm lenghts, and the three pieces were actually 1.999999995, 2 and 2.0000000000005 cm respectivley. Could you really notice the difference?
You couldn't measure the difference with a vernier guage or a micrometer, you would need very sophisticated equipment to detect the difference.
In what real world application can you see this difference mattering?

If I ask you to measure out a distance to 2000km, and you measure it to the above accuracy you would be withn 5mm.
In what real world application can you see this difference mattering?

#### iamcpc

• 832
##### Re: Are plane tickets real?
« Reply #215 on: May 13, 2020, 08:01:31 PM »
Lets couch this in a slightly different way, in a practical real world application.
If I asked you to machine some bar stock and produce three 2cm lenghts, and the three pieces were actually 1.999999995, 2 and 2.0000000000005 cm respectivley. Could you really notice the difference?
You couldn't measure the difference with a vernier guage or a micrometer, you would need very sophisticated equipment to detect the difference.
In what real world application can you see this difference mattering?

First off math is math. The answer is right or it's wrong. If you are trying to solve for x in the formula x + 1 = 2 then, out of all of the infinite possible answers, there is only one that is correct. In measuring the distance to grandma's house i would say that .0000005 CM does not really matter. But we're not talking about the distance to grandma's house we are talking about reverse engineering the distance formula used on a website. They are two totally different things with two totally different degrees of accuracy.

Second off small distances do matter.
There is an entire branch of science called metrology dedicated to precise measurements measuring to the milliong/billionth of an inch actually is important and matters from things like nano technology, computer processors, or the precision engineering needed in a modern jet engine where, if one blade is long by .00000005 CM and another blade is short by .00000005 CM the entire engine would either not function or function much less efficiently.

Here's

#### stack

• 3427
##### Re: Are plane tickets real?
« Reply #216 on: May 13, 2020, 08:54:56 PM »
Lets couch this in a slightly different way, in a practical real world application.
If I asked you to machine some bar stock and produce three 2cm lenghts, and the three pieces were actually 1.999999995, 2 and 2.0000000000005 cm respectivley. Could you really notice the difference?
You couldn't measure the difference with a vernier guage or a micrometer, you would need very sophisticated equipment to detect the difference.
In what real world application can you see this difference mattering?

First off math is math. The answer is right or it's wrong. If you are trying to solve for x in the formula x + 1 = 2 then, out of all of the infinite possible answers, there is only one that is correct. In measuring the distance to grandma's house i would say that .0000005 CM does not really matter. But we're not talking about the distance to grandma's house we are talking about reverse engineering the distance formula used on a website. They are two totally different things with two totally different degrees of accuracy.

It's been made infinitely clear that Bing is using globe 'math' for its distance calculations. Nothing has even remotely been shown as evidence to the contrary. Even Microsoft says they use it. Case closed.

Second off small distances do matter.
There is an entire branch of science called metrology dedicated to precise measurements measuring to the milliong/billionth of an inch actually is important and matters from things like nano technology, computer processors, or the precision engineering needed in a modern jet engine where, if one blade is long by .00000005 CM and another blade is short by .00000005 CM the entire engine would either not function or function much less efficiently.

Here's

Cool video, but has nothing to do with the topic.

#### BRrollin

• 265
##### Re: Are plane tickets real?
« Reply #217 on: May 13, 2020, 08:57:27 PM »
Lets couch this in a slightly different way, in a practical real world application.
If I asked you to machine some bar stock and produce three 2cm lenghts, and the three pieces were actually 1.999999995, 2 and 2.0000000000005 cm respectivley. Could you really notice the difference?
You couldn't measure the difference with a vernier guage or a micrometer, you would need very sophisticated equipment to detect the difference.
In what real world application can you see this difference mattering?

First off math is math. The answer is right or it's wrong. If you are trying to solve for x in the formula x + 1 = 2 then, out of all of the infinite possible answers, there is only one that is correct. In measuring the distance to grandma's house i would say that .0000005 CM does not really matter. But we're not talking about the distance to grandma's house we are talking about reverse engineering the distance formula used on a website. They are two totally different things with two totally different degrees of accuracy.

Second off small distances do matter.
There is an entire branch of science called metrology dedicated to precise measurements measuring to the milliong/billionth of an inch actually is important and matters from things like nano technology, computer processors, or the precision engineering needed in a modern jet engine where, if one blade is long by .00000005 CM and another blade is short by .00000005 CM the entire engine would either not function or function much less efficiently.

Here's

I see what you’re saying, and I get your point, but it’s just not technically true.

x+1=2

In real space decimal, the solution is x=1.

But in complex space, you can have x=1,-i^2, and in fact, -i^(2+4n) where n is a natural number. So there are infinitely many solutions.

Also, if we move to a different number basis, then you obtain different solutions adherent to that basis.

The point I’m making is that even simple equations have details that are implicitly made. Which is fine. But if those details are not understood, and one of them changes, then someone could be entirely off base and not understand why.
“This just shows that you don't even understand the basic principle of UA...A projectile that goes up and then down again to an observer on Earth is not accelerating, it is the observer on Earth who accelerates.”

- Parsifal

“I hang out with sane people.”

- totallackey

#### DuncanDoenitz

• 329
##### Re: Are plane tickets real?
« Reply #218 on: May 14, 2020, 08:04:35 AM »
Quote from IAMCPC on 13 May 2020:

"Second off small distances do matter.
There is an entire branch of science called metrology dedicated to precise measurements measuring to the milliong/billionth of an inch actually is important and matters from things like nano technology, computer processors, or the precision engineering needed in a modern jet engine where, if one blade is long by .00000005 CM and another blade is short by .00000005 CM the entire engine would either not function or function much less efficiently".

I don't know how to make a computer chip, but I am a Licensed Jet Engine Engineer.  In engine maintenance I've got no way of measuring a turbine blade to 5/millionth of a millimeter and, you know what, it doesn't matter.  The engine is going to work, and to an acceptable level of efficiency. If someone made a blade to that level of accuracy, it would be a few molecules smaller by the time he'd cleaned it and put it in a shipping package.  I'm looking for around 0.02mm (0.002cm). You're talking out of your jetpipe.

On a similar note, I've got an apple in a fruit-bowl; one apple.  I add a second apple.  How many apples in the bowl?  Well, while I was picking the second apple, bacteria started reacting on the first apple and decay set in so, I've probably got 1.9999999995 apples.

See my point?  Math is math.  Reality is reality.

#### robinofloxley

• 203
##### Re: Are plane tickets real?
« Reply #219 on: May 14, 2020, 02:58:19 PM »
Just to get back to haversine for a moment. The mathematical haversine formula works by first calculating the central angle, θ, between two arbitrary points on a sphere. Then with the central angle determined, you work out the great circle distance between them. So a very simple example would be, what's the distance between 45°N, 10W and 46°N, 10W. Well the central angle, θ, is exactly 1°, so that bit is simple. All we need now is to work out the great circle distance. And that's easy too, it's πrθ/180 and that's an absolutely precise and correct mathematical solution to the problem. Unfortunately, it's useless. "Just drive north a distance of πrθ/180 and you'll be at your destination". To be of any use whatsoever, we need actual numbers for answers, so let's go ahead and assign a value for r of 6378.137km, but what do we use for pi? How many of the infinite decimal places do we need?

That's the problem with this, you can have an absolutely precise mathematical answer - πrθ/180 - or you can have an approximate, rounded actual value, e.g. in this case if we take pi to be 3.14159 then the answer is approximately 111.32km. You just can't have it both ways.