[robinofloxley]

One of the problems with flat earth discussions is that there is no consensus behind any particular flat map, however the one thing I assume everyone does agree on is that whatever it looks like, it will be flat.

On any flat surface, there can only be one shortest route between any two points - a straight line - and therefore only ever one sensible direction of travel if you want to get to your destination via the shortest route.

On a sphere, the shortest route between two points is a great circle arc and in general there is only one great circle passing through both points and hence one shortest route. The exception occurs when the two points are antipodal (directly opposite each other). In this case many great circle routes are possible so there is no longer a single shortest route, there are many.

So for example, on a globe earth, the UK and New Zealand are almost antipodal, therefore you can head off in any direction you like from the UK, follow a great circle route for about 12500 miles and you will be more or less in New Zealand. On a globe travelling in opposite directions yet ending up in the same place makes perfect sense, on a flat earth it doesn't.

In reality, from the UK, I can fly west via Los Angeles to New Zealand or I can fly east via Tokyo and there are plenty of other routes to choose from such as via Hong Kong, Dubai, Manilla etc. These routes all have similar flight durations and cost similar amounts of money.

Anyone care to try and explain?

[Tom Bishop]

There is a chapter on Great Circle Sailing in Earth Not a Globe.

[stack]

There is a chapter on Great Circle Sailing in Earth Not a Globe.

The problem with Rowbotham's chapter is that his presumption is that pretty much all forms of transport are and have been navigating incorrectly for centuries. As in Great Circles versus Rhumb Lines. A flight from JFK to Heathrow follows a great circle, not a rhumb line. And I highly suspect shipping and airliners of the world would prefer the shortest route for myriad reasons, most notably cost versus profit.

Here's what a simplistic view of a typical flight path looks like on a great circle versus a rhumb line, seems counter-intuitive if the world was flat:

[Tom Bishop]

Paint the red route on the AE map.

[robinofloxley]

There is a chapter on Great Circle Sailing in Earth Not a Globe.

OK thanks for the link, I read through that and a lot of it seems more to do with the practicalities of 19th C sailing, so I'm not really clear whether Rowbotham is actually claiming that there is a shorter distance on a sphere (not talking about the Earth here, just a plain old sphere) than a great circle distance. If he is claiming that, then it begs the question - what is this shorter route?

Put plainly, on a sphere, is the shortest distance between any two points a great circle distance or not?

[stack]

Paint the red route on the AE map.

Which AE map?

[Tom Bishop]

Paint the red route on the AE map.

Which AE map?

I am gauging from your response that you have seen your error. We need not discuss the matter further.

[QED]

Paint the red route on the AE map.

Which AE map?

I am gauging from your response that you have seen your error. We need not discuss the matter further.

I think the matter is best served by discussing it further. In classic zetetic fashion, one should sail these routes, and bring the resulting data to this forum. Until you do so, you are simply appealing to an authority without personal evidence. That is not zetetic, but instead is exactly what FEers accuse REers of doing: believing some source without personal verification.

Hence, Rowbotham citations are no more valid than any other RE citation. True zetetic methods insist that one discards Rowbotham claims (or any other claims), until personally verified.

We do adhere to zetetic methods, yes? That is the whole point, yes?

[Tom Bishop]

Paint the red route on the AE map.

Which AE map?

I am gauging from your response that you have seen your error. We need not discuss the matter further.

I think the matter is best served by discussing it further.

Feel free to draw the route on the AE map and I will discuss it with you.

[stack]

Paint the red route on the AE map.

Which AE map?

I am gauging from your response that you have seen your error. We need not discuss the matter further.

A few things:

1) As you know, an AE map is a globe projection that can be centered on any point on earth.
2) You have previously stated, you don't subscribe to the FE mono-pole "model", but to the bi-polar "model". Hence my question. But I presume you mean one centered on the north pole. As in the FE mono-pole model.
3) Since all AE maps are globe projections, there is no such thing as an FE map as you have stated yourself many times. So using an FE earth map is moot, b/c one doesn't exist.

Moving on.

Comparing rhumb line to great circle navigation, using JFK to Heathrow, rhumb line is 219km more than a great circle.

Comparing rhumb line to great circle navigation, using Cape Town to Sydney, rhumb line is 1240km more than a great circle.

[QED]

Paint the red route on the AE map.

Which AE map?

I am gauging from your response that you have seen your error. We need not discuss the matter further.

I think the matter is best served by discussing it further.

Feel free to draw the route on the AE map and I will discuss it with you.

Do you ever NOT pass the buck? Or is your position always one of retreat? That is not a response of strength.

Anyway, see above. The lines have been drawn (as it were).

So go ahead and discuss it. If you have the nerve.

[Tom Bishop]

Maybe you should submit your queries to a repeating-Mercator Flat Earth group. We use none of those models.

[juner]

Do you ever NOT pass the buck? Or is your position always one of retreat? That is not a response of strength.

Anyway, see above. The lines have been drawn (as it were).

So go ahead and discuss it. If you have the nerve.

If you have nothing to add to the thread beyond asking other people to do your homework for you, then I will kindly ask that you refrain from posting in the upper fora. You are teetering on the edge of a permanent ban; you may want to stop being your normal charming self for a while.

[stack]

Maybe you should submit your queries to a repeating-Mercator Flat Earth group. We use none of those models.

I guess a couple of more things:

- I wasn't aware this was the Azimuthal Equidistant Flat Earth Society.
- You yourself have stated many times that there is no Flat Earth map.
- And FE really doesn't have a "model". The two main contenders are the mono-pole and the bipolar. But as you stated the other day, in FET, the layout of the continents is unknown.

Given the above, why do you have an issue with one projection over the other? FE doesn't have a map, AE, Mercator or otherwise and all the maps we do have are Globe projections. The mercator projection is especially well suited for Rhumb Line plotting for short distances. As well the mercator projection is the most widely used map for the bulk of all sea, air, and land transport and has been for a couple 100 years. And seems to do quite a good job.

So I don't think you really have a point to make. Regardless of map projection or model, you just simply have a problem arguing against widely used great circle navigation versus Rowbotham's "all the world's navigation is wrong as everyone should actually be using rhumb lines" notion.

[AATW]

Paint the red route on the AE map.

Or. Construct a flat earth map using known distances. The global transport industry is reliable enough to have confidence in their data.
Two possibilities:
1) You succeed and become the first flat earther to successfully construct an accurate flat earth map
2) You can't and realise that the earth can't be flat.

Obviously the 3rd option is to claim that a global industry which reliably gets people and goods around the world don't know how far places are apart of how fast their vessels move. Although quite how they run their businesses without that information is a mystery.

[robinofloxley]

I'm still keen to get an answer to the original point which is that if I'm travelling between say UK and New Zealand, on a globe I believe I can head off in any direction I like, following a great circle route for 12500 miles and I will arrive at my desired destination. Does anyone disagree that if we are on a globe, that assertion would hold true?

The only assumption being made is that a great circle distance is the shortest distance between any two points on a sphere and that flights on a globe would presumably follow such routes where practicable (and with minor adjustments for weather) for efficiency. Is there agreement on this or not?

I've read Tom's ENAG link and I'm not clear at all whether Rowbottom is disputing that a great circle distance is always the shortest on a sphere or if he's simply saying sailors in the 19th C and earlier aren't sailing great circle routes so that means we're not on a globe.

[edby]

VBA code for anyone interested. Calculates distance between two points using different assumptions about the shape of the earth

See also this thread where the code was used to plot flight times against predicted distance.

---
'enter latitude and longitude for the two points, in degrees (not radians)
'Calculates the distance between the two points on the assumption that the earth is spherical
Function haversine(lat1 As Double, long1 As Double, lat2 As Double, long2 As Double)
Dim x As Double, r As Double
    r = 6371#
    'convert to radians
    lat1 = lat1 * WorksheetFunction.Pi() / 180
    long1 = long1 * WorksheetFunction.Pi() / 180
    lat2 = lat2 * WorksheetFunction.Pi() / 180
    long2 = long2 * WorksheetFunction.Pi() / 180
   
    haversine = 2 * r * WorksheetFunction.Asin(Sqr((1 - Cos(lat2 - lat1)) / 2# + Cos(lat1) * Cos(lat2) * (1 - Cos(long2 - long1)) / 2#))
End Function

'enter latitude and longitude for the two points, in degrees (not radians)
'Calculates the distance between the two points on the AE map
Function FEdistance(lat1 As Double, long1 As Double, lat2 As Double, long2 As Double)
Dim l1 As Double, l2 As Double, l3 As Double, a1 As Double, l4 As Double, h1 As Double
   
    l1 = (90 - WorksheetFunction.Max(lat1, lat2)) * 111.19
    a1 = (long2 - long1) * WorksheetFunction.Pi() / 180
    h1 = Sin(a1) * l1
    l2 = Cos(a1) * l1
    l3 = (90 - WorksheetFunction.Min(lat1, lat2)) * 111.19
   
    l4 = l3 - l2
    FEdistance = Sqr(l4 ^ 2 + h1 ^ 2)
End Function

[edby]

I'm still keen to get an answer to the original point which is that if I'm travelling between say UK and New Zealand, on a globe I believe I can head off in any direction I like

I don't think you can head in any direction you like. Only one great circle passes through any two points, and the haversine formula above gives you the distance along that great circle.

[AATW]

I'm still keen to get an answer to the original point which is that if I'm travelling between say UK and New Zealand, on a globe I believe I can head off in any direction I like

I don't think you can head in any direction you like. Only one great circle passes through any two points, and the haversine formula above gives you the distance along that great circle.

Hmm. But if those points are antipodal then you pretty much can, can't you? Imagine the two poles, if you're standing at the North Pole you can follow any line of Longitude you like and you'll end up at the South Pole.

Obviously when it comes to flights there are practical considerations like where there is a conveniently placed airport part of the way and when you take a flight with a stop over that stop over is unlikely to be exactly on the great circle between the two places.

[robinofloxley]

I'm still keen to get an answer to the original point which is that if I'm travelling between say UK and New Zealand, on a globe I believe I can head off in any direction I like

I don't think you can head in any direction you like. Only one great circle passes through any two points, and the haversine formula above gives you the distance along that great circle.

Hmm. But if those points are antipodal then you pretty much can, can't you? Imagine the two poles, if you're standing at the North Pole you can follow any line of Longitude you like and you'll end up at the South Pole.

Obviously when it comes to flights there are practical considerations like where there is a conveniently placed airport part of the way and when you take a flight with a stop over that stop over is unlikely to be exactly on the great circle between the two places.

Ah yes, and that's exactly what I'm trying to get at. UK to New Zealand flights do in fact offer a very wide choice of conveniently placed airports serving as stopovers in very different locations across the globe, from North America (e.g. Los Angeles) through the Middle East (e.g Dubai) and the far east (e.g. Tokyo).

I don't agree that...

when you take a flight with a stop over that stop over is unlikely to be exactly on the great circle between the two places.

On the contrary, because the two endpoints are antipodal then there must always exist one great circle route which passes through both the endpoints and the stopover. Only one great circle passes through the departure and stopover and only one through the stopover and destination, so they must be one and the same great circle.

Put it another way. Pick any two non-antipodal points A and B and there is one great circle route between them. Extend that route all the way round and you must necessarily pass through the antipodal points of both A and B.