[edby]

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.

Still don't follow. Suppose I travel down the meridian (0 longitude) from London A to some point on the equator B. Then suppose I travel along the equator (which is another great circle) to somewhere else C. It clearly doesn't follow that ABC is 'one and the same great circle'. Clearly there is a third great circle AC, but B does not lie on it.

Perhaps I misunderstand.

[QED]

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

Perhaps.

Perhaps maybe you should just try to answer a question honestly.

Mercator methods are un-zetetic, because they forfeit basic calculus to find their answers. Perhaps that is why you do not like them.

So rather than shine the lampshade-Sun onto models you don't believe in, would it not be more productive to emphasize the models you do champion?

I invite you.

[robinofloxley]

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.

Still don't follow. Suppose I travel down the meridian (0 longitude) from London A to some point on the equator B. Then suppose I travel along the equator (which is another great circle) to somewhere else C. It clearly doesn't follow that ABC is 'one and the same great circle'. Clearly there is a third great circle AC, but B does not lie on it.

Perhaps I misunderstand.

I think it's more likely that I'm not explaining this very well. I'll have another go.

In your example you have points A, B and C, but you don't have two antipodal points and that difference is crucial. So let's fix that and make A and C antipodal.

Travel down the meridian (0 longitude) as you did before to get to B (on the equator). Now you need to head for C, but C is constrained to be directly opposite A, which means it must also be on the 0 longitude line somewhere, so basically you just carry on in the same direction until you get to C.

Hope that helps.

[edby]

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.

Still don't follow. Suppose I travel down the meridian (0 longitude) from London A to some point on the equator B. Then suppose I travel along the equator (which is another great circle) to somewhere else C. It clearly doesn't follow that ABC is 'one and the same great circle'. Clearly there is a third great circle AC, but B does not lie on it.

Perhaps I misunderstand.

I think it's more likely that I'm not explaining this very well. I'll have another go.

In your example you have points A, B and C, but you don't have two antipodal points and that difference is crucial. So let's fix that and make A and C antipodal.

Travel down the meridian (0 longitude) as you did before to get to B (on the equator). Now you need to head for C, but C is constrained to be directly opposite A, which means it must also be on the 0 longitude line somewhere, so basically you just carry on in the same direction until you get to C.

Hope that helps.

Yes it does. I was reading 'antipodal' as meaning 'in the antipodes'. You see my mistake.

[Tom Bishop]

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

Perhaps.

Perhaps maybe you should just try to answer a question honestly.

Mercator methods are un-zetetic, because they forfeit basic calculus to find their answers. Perhaps that is why you do not like them.

So rather than shine the lampshade-Sun onto models you don't believe in, would it not be more productive to emphasize the models you do champion?

I invite you.

Draw the lines of the images presented in this thread on any of the maps or models we present and you will see your error.

[inquisitive]

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

Perhaps.

Perhaps maybe you should just try to answer a question honestly.

Mercator methods are un-zetetic, because they forfeit basic calculus to find their answers. Perhaps that is why you do not like them.

So rather than shine the lampshade-Sun onto models you don't believe in, would it not be more productive to emphasize the models you do champion?

I invite you.

Draw the lines of the images presented in this thread on any of the maps or models we present and you will see your error.

Please give links to some of your maps.

[Tom Bishop]

Please give links to some of your maps.

You have been registered on this website for almost 5 years. Why are you guys resorting to delay tactics?

[QED]

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

Perhaps.

Perhaps maybe you should just try to answer a question honestly.

Mercator methods are un-zetetic, because they forfeit basic calculus to find their answers. Perhaps that is why you do not like them.

So rather than shine the lampshade-Sun onto models you don't believe in, would it not be more productive to emphasize the models you do champion?

I invite you.

Draw the lines of the images presented in this thread on any of the maps or models we present and you will see your error.

Continue to dodge the question, if you must. But if you wish to be honest; if the truth matters to you, then you will draw the lines as shown above, compare them to data, and will immediately see your error.

But you probably won't do this. Such an act is too zetetic for you to swallow. But others reading this post might do it. When they do, they will see a result that I have transparently offered, but which you have tried to hide.

Ouch!

True zetetic process leads to one conclusion: modern scientific thought.

The extent to which you deny this is the extend to which you depart from true zetetic practice.

[robinofloxley]

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.

Still don't follow. Suppose I travel down the meridian (0 longitude) from London A to some point on the equator B. Then suppose I travel along the equator (which is another great circle) to somewhere else C. It clearly doesn't follow that ABC is 'one and the same great circle'. Clearly there is a third great circle AC, but B does not lie on it.

Perhaps I misunderstand.

I think it's more likely that I'm not explaining this very well. I'll have another go.

In your example you have points A, B and C, but you don't have two antipodal points and that difference is crucial. So let's fix that and make A and C antipodal.

Travel down the meridian (0 longitude) as you did before to get to B (on the equator). Now you need to head for C, but C is constrained to be directly opposite A, which means it must also be on the 0 longitude line somewhere, so basically you just carry on in the same direction until you get to C.

Hope that helps.

Yes it does. I was reading 'antipodal' as meaning 'in the antipodes'. You see my mistake.

Yes indeed, sorry - I could probably have explained better.

Now we're on the same page, do you see why I think this is an interesting challenge for the flat earth?

On a flat earth it makes no sense to me that flying from London, one route could start out heading north-west, another due east, with stopovers thousands of miles apart and yet they meet up at the same place and take the same amount of time. These routes exist.

[AATW]

Draw the lines of the images presented in this thread on any of the maps or models we present and you will see your error.

Try and construct a map on a flat plane using known distances between places and you will see yours 

[stack]

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

Perhaps.

Perhaps maybe you should just try to answer a question honestly.

Mercator methods are un-zetetic, because they forfeit basic calculus to find their answers. Perhaps that is why you do not like them.

So rather than shine the lampshade-Sun onto models you don't believe in, would it not be more productive to emphasize the models you do champion?

I invite you.

Draw the lines of the images presented in this thread on any of the maps or models we present and you will see your error.

The two main FE "models" use as a placeholder AE Globe Projection maps as FET does not have a map, as we all know. The mono-pole, AE Globe Projection centered on the North Pole and the bi-polar, AE Lambert Globe Projection centered on 0° N 0° E.

The Mercator Globe Projection, is probably the more widely used version when it comes all forms of goods and people transport because of it's simplicity; all verticals point N & S, all horizontals point E & W.

Rhumb Line navigation is fine for short haul navigation.
Great Circle is what is used for long haul navigation, e.g. an ocean passage.

Both can be plotted on either an AE or Mercator map. The routes and distances are the same.

B/c Tom prefers the AE Globe Projection view, here's an example of a Rhumb Line vs a Great Circle on a North Pole centered one (same as the FE mono-pole model). You'll notice the Great Circle route, even that far north, is better than the Rhumb Line route and that's why airlines fly the Great Circle path.



Fast forward 150 years from Rowbotham's contention in ENAG regarding the superiority of Rhumb Line navigation, it just doesn't apply to modern, even less than modern, long haul transport.

[AATW]

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.

Right. Yes, you are correct. Again, taking my example because it's easier to think about. If you're at the North Pole then you can, as I said above, go down any line of Longitude to get to the South Pole. And since you must go through every point of Latitude as you do so you can go to any place of Longitude and Latitude on earth on your way between the poles.
That remains true for any antipodal points on earth. The way to think about it is imagine a ring around the earth going through the two points A and B which are antipodal. That ring is a great circle. You can pivot that ring around points A and B so it goes through any point on earth.

Note to certain people. This is how grown ups discuss things, actually conceding points and admitting error when shown to be incorrect. It's not that difficult...

[robinofloxley]

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.

Right. Yes, you are correct. Again, taking my example because it's easier to think about. If you're at the North Pole then you can, as I said above, go down any line of Longitude to get to the South Pole. And since you must go through every point of Latitude as you do so you can go to any place of Longitude and Latitude on earth on your way between the poles.
That remains true for any antipodal points on earth. The way to think about it is imagine a ring around the earth going through the two points A and B which are antipodal. That ring is a great circle. You can pivot that ring around points A and B so it goes through any point on earth.

Note to certain people. This is how grown ups discuss things, actually conceding points and admitting error when shown to be incorrect. It's not that difficult...

I like the pivoting ring analogy and I love the "grown ups" comment, but in the spirit of this forum, perhaps we should really have thrown at least a few insults at each other along the way ;-)

[inquisitive]

Please give links to some of your maps.

You have been registered on this website for almost 5 years. Why are you guys resorting to delay tactics?

Surely you have been delaying in drawing a map using measured distances?

[HorstFue]

Please give links to some of your maps.

This question is valid.  Where are the flat earth maps?
So far I've seen none. I've seen some maps which clearly are projections of a globe, so these are globe earth maps.
You cannot navigate without maps. So if you want to debate about navigation, Great Circles, Rhumb Lines etc. you need a map.
How should we discuss issues of flat earth navigation, when there's no map?
We could discuss about globe earth navigation, there are plenty of maps in all resolutions, with multiple projections.

[markjo]

After pondering the title of this thread for a bit, I can't help but to wonder if the concept of antipodes even applies in the context of a flat earth map.  What exactly does it mean to have opposite points on a flat earth?

[AATW]

After pondering the title of this thread for a bit, I can't help but to wonder if the concept of antipodes even applies in the context of a flat earth map.  What exactly does it mean to have opposite points on a flat earth?

Well, quite. It would be a pretty meaningless concept. But the fact that you can get flights from London to New Zealand via multiple routes is pretty much the point. On a flat earth there is only one shortest line between points A and B. On a globe if the points are antipodal then there are multiple shortest lines and you can put those lines through any point C on the globe.
When I've got more time I'll have a look at flight times between antipodal points via different "C" points and do some comparisons.
In theory they should be reasonably consistent and that wouldn't make any sense on a flat earth.

[robinofloxley]

After pondering the title of this thread for a bit, I can't help but to wonder if the concept of antipodes even applies in the context of a flat earth map.  What exactly does it mean to have opposite points on a flat earth?

Well, quite. It would be a pretty meaningless concept. But the fact that you can get flights from London to New Zealand via multiple routes is pretty much the point. On a flat earth there is only one shortest line between points A and B. On a globe if the points are antipodal then there are multiple shortest lines and you can put those lines through any point C on the globe.
When I've got more time I'll have a look at flight times between antipodal points via different "C" points and do some comparisons.
In theory they should be reasonably consistent and that wouldn't make any sense on a flat earth.

Yep, that's exactly the point I was trying to make, thank you. And it doesn't matter what the actual map looks like either, so hopefully we avoid arguments about whether we should be looking at unipolar, bipolar etc. or the "we're working on it" map. So long as it's flat then these flights make no sense. If it's a globe, then they do.

[stack]

After pondering the title of this thread for a bit, I can't help but to wonder if the concept of antipodes even applies in the context of a flat earth map.  What exactly does it mean to have opposite points on a flat earth?

Well, quite. It would be a pretty meaningless concept. But the fact that you can get flights from London to New Zealand via multiple routes is pretty much the point. On a flat earth there is only one shortest line between points A and B. On a globe if the points are antipodal then there are multiple shortest lines and you can put those lines through any point C on the globe.
When I've got more time I'll have a look at flight times between antipodal points via different "C" points and do some comparisons.
In theory they should be reasonably consistent and that wouldn't make any sense on a flat earth.

Yep, that's exactly the point I was trying to make, thank you. And it doesn't matter what the actual map looks like either, so hopefully we avoid arguments about whether we should be looking at unipolar, bipolar etc. or the "we're working on it" map. So long as it's flat then these flights make no sense. If it's a globe, then they do.

Agreed. I would say at this point, FET doesn't have an explanation as to how travel as described and happens everyday can occur. 

[robinofloxley]

After pondering the title of this thread for a bit, I can't help but to wonder if the concept of antipodes even applies in the context of a flat earth map.  What exactly does it mean to have opposite points on a flat earth?

Well, quite. It would be a pretty meaningless concept. But the fact that you can get flights from London to New Zealand via multiple routes is pretty much the point. On a flat earth there is only one shortest line between points A and B. On a globe if the points are antipodal then there are multiple shortest lines and you can put those lines through any point C on the globe.
When I've got more time I'll have a look at flight times between antipodal points via different "C" points and do some comparisons.
In theory they should be reasonably consistent and that wouldn't make any sense on a flat earth.

Just to kick things off then, looking at bookable options, I picked a few extreme examples, all on the same day, starting from London Heathrow (LHR), one via Los Angeles (LAX), the other via Tokyo (NRT). The ultimate destination in both cases is Auckland NZ (AKL).

LHR and AKL are not ideal because the antipode of LHR is actually about 1600km (1000mi) S of AKL. There are other pairs of locations in the UK and New Zealand which are a better match, but we need major international airports with lots of flight options and these two are the main ones. What this means is there will naturally be some variation in distance (and hence flight time) depending on route and prevailing winds, but when we're talking of flights approaching 24h and 12500 miles, the differences aren't that great as we shall see.

Having done a bit more research, I think Madrid (MAD) in Spain and Auckland (AKL) are actually a closer match as the antipode of Madrid is only 400km (245mi) from Auckland and Madrid is an international airport, so although there are fewer flight options compared to London Heathrow, I've also included a couple of flights between the two, one via Beijing (PEK) and the other via LAX.

I looked them all up on flightradar24 to find actual route and flight times. Distances given are great circle distances from the flightradar24 pages:

A) 31st Oct BA269 LHR -> LAX flight time 10h 4m, distance: 8780km, followed by AA83 LAX -> AKL flight time 12h 11m distance 10467km (total flight time 22h 15m, total distance 19247km)
B) 31st Oct BA5 LHR-> NRT flight time 10h 46m, distance 9615km, followed by NZ90 NRT -> AKL flight time 10h 2m, distance 8806km (total flight time 20h 48m, total distance 18421km)
C) 31st Oct DY7743 MAD->LAX flight time 11h 25m, distance 9406km, followed by NZ1 LAX->AKL flight time 12h 1m, distance 10467km (total flight time 23h 26m, total distance 19873km)
D) 31st Oct CA908 MAD->PEK flight time 10h 30m, distance 9266km, followed by CA783 PEK->AKL flight time 11h 58m, distance 10402km (total flight time 22h 38m total distance 19668km)

The London flight via LAX is longer than the flight via Tokyo. The difference between the two is 1h 27m in the air and 826km (513mi). This is a relatively small discrepancy, explained by the fact that they are not exactly antipodal, but they are close enough for the two routes to make economic sense.

As expected, the two Madrid options are closer still, 48mins and 5km.

On different days with different winds and routing, timings and distances will vary somewhat, but not substantially. Bear in mind that these are also extreme examples of routes spread a long way apart.