Which way do I go

[Gun Nut]

Let’s confine the parameters to the continental United States. Pick a point towards the middle such that the point can get no further from the border. Then, draw a circle around the point such that it can be no larger without crossing a border. I don’t know what the radius would be in miles, but i’ll just use a nice round number and call it 500 miles. The distance from the most northern point to the most southern point would be 1000 miles. East to west would be 1000 miles. In fact, every point would be equal distance to its opposite.

Now, let’s say the entire area of the circle was paved and flat and we ran a car race where we both averaged 100 MPH. We would essentially tie every time we raced. What’s most important here is that it wouldn’t matter what direction we raced. We always tie.

Now, you decide to take to the skies while I remain below. We run 360 races (one for each degree of the circle). In fact, it’s like running 180 courses twice—one one way and one the other way. No matter what, we always average 100MPH.*

But, we don’t tie anymore with one exception. You beat me each and every time except for the one time we tie. If I want to run the race again (and not lose) where do I start, and where do I finish? Essentially, what direction do I go? Which of the 180 tracks do I choose and which way do I run it?

It sounds like you are asking, "if you jump straight into the air, how for to the West will you land". Right? You are asking how far the planet moves beneath you in a given amount of time while airborne, versus being in contact with the ground? It also sounds like you are trying to compare two ground-racers.. but why would either beat the other when they are both traveling the same distance on the ground? The question doesn't really make sense to me.

That is just not how it works. Gravity affects the air just as much as the ground beneath it. The air is just a less-dense part of the planet, in a manner of thinking. Just like the water in the oceans.... The air moves with the ground as the planet rotates. Other forces affect the motion of air (wind from weather) which is caused by uneven heating of the surface of the Earth.... The ground itself does not move at different speeds relative to any other piece of ground (they do not move relative to each other, otherwise our planet would have ripped itself in half long ago)

 

[Gun Nut]

Maybe it’s the arcs representing flight paths that is giving the illusion that the planes aren’t flying straight. Maps don’t rotate. If the planes fly straight, they’re not flying toward their destination but instead where their destination is gonna be. A map would have to be virtual to capture that. A straight line to represent the plane not turning and a rotating map to capture earths spin.

The "arcs representing flight paths" are not due to the earth's spin or the planes anticipating where their destination will be, they're by and large due to the fact that the shortest route between any two points is never along a parallel (except when the parallel is the equator), but rather along a  Great_circle, plus some following winds.

Also, the shortest route from a point 500 miles to your east to a point 500 miles to your west does *not* cross your position, and that's true for air routes as much as surface routes. Unless, again, you're at the equator - or if by "a point 500 miles to your east" you mean the point you'd reach in a great circle route starting at your position with an initial bearing of 90.

A Great Circle is not a curved path. It is only a curved path when drawn on a flat map... as maps are. It is a correction of the error imposed by drawing the surface of a sphere onto a flat square.
A path "due East" around the equator of the planet is a straight line. It is not curved. Space-time is curved by the gravity of the planet. Your path through space-time is straight. Think of a satellite... Is it constantly turning to stay at constant altitude? nope. It is traveling straight, just like Newton predicted.

 

[Jokodo]

Maybe it’s the arcs representing flight paths that is giving the illusion that the planes aren’t flying straight. Maps don’t rotate. If the planes fly straight, they’re not flying toward their destination but instead where their destination is gonna be. A map would have to be virtual to capture that. A straight line to represent the plane not turning and a rotating map to capture earths spin.

The "arcs representing flight paths" are not due to the earth's spin or the planes anticipating where their destination will be, they're by and large due to the fact that the shortest route between any two points is never along a parallel (except when the parallel is the equator), but rather along a  Great_circle, plus some following winds.

Also, the shortest route from a point 500 miles to your east to a point 500 miles to your west does *not* cross your position, and that's true for air routes as much as surface routes. Unless, again, you're at the equator - or if by "a point 500 miles to your east" you mean the point you'd reach in a great circle route starting at your position with an initial bearing of 90.

A Great Circle is not a curved path. It is only a curved path when drawn on a flat map... as maps are. It is a correction of the error imposed by drawing the surface of a sphere onto a flat square.
A path "due East" around the equator of the planet is a straight line. It is not curved. Space-time is curved by the gravity of the planet. Your path through space-time is straight. Think of a satellite... Is it constantly turning to stay at constant altitude? nope. It is traveling straight, just like Newton predicted.

Thanks for asking! Cars and planes are not satellites and great circle routes not orbital trajectories. Next question?

Other than that, I fail to see a Connection between my post and your response.

 

[Gun Nut]

A Great Circle is not a curved path. It is only a curved path when drawn on a flat map... as maps are. It is a correction of the error imposed by drawing the surface of a sphere onto a flat square.
A path "due East" around the equator of the planet is a straight line. It is not curved. Space-time is curved by the gravity of the planet. Your path through space-time is straight. Think of a satellite... Is it constantly turning to stay at constant altitude? nope. It is traveling straight, just like Newton predicted.

Thanks for asking! Cars and planes are not satellites and great circle routes not orbital trajectories. Next question?

Other than that, I fail to see a Connection between my post and your response.

orbital trajectories (which you never mentioned in your post) have nothing to do with great circles. and my response was adding to yours, not opposing it. "you" was the poster you were responding to... sorry that was confusing.

I still think the OP was more about a failure to understand that the air (for the most part) moves with the planet... and momentum keeps things moving that are already being moved by ground's rotation.

 

[bilby]

Maybe it’s the arcs representing flight paths that is giving the illusion that the planes aren’t flying straight. Maps don’t rotate. If the planes fly straight, they’re not flying toward their destination but instead where their destination is gonna be. A map would have to be virtual to capture that. A straight line to represent the plane not turning and a rotating map to capture earths spin.

The "arcs representing flight paths" are not due to the earth's spin or the planes anticipating where their destination will be, they're by and large due to the fact that the shortest route between any two points is never along a parallel (except when the parallel is the equator), but rather along a  Great_circle, plus some following winds.

Also, the shortest route from a point 500 miles to your east to a point 500 miles to your west does *not* cross your position, and that's true for air routes as much as surface routes. Unless, again, you're at the equator - or if by "a point 500 miles to your east" you mean the point you'd reach in a great circle route starting at your position with an initial bearing of 90.

A Great Circle is not a curved path.

Say that again. Slowly.

Any straight path that starts on the surface of a sphere leaves that surface (either it's a tangent that heads off into space, or a chord that tunnels into the ground).

All paths that are bound to the surface of a sphere are curved.

 

[fast]

So, if the lines running east and west are misleading, then running a perfectly straight steel ladder from the east coast to the west coast (or let’s say, perfectly east and west with no northern or southern bend), then besides the vertical bend, as it will concave down, a representation of that on a map will not appear straight. I guess.

 

[skepticalbip]

So, if the lines running east and west are misleading, then running a perfectly straight steel ladder from the east coast to the west coast (or let’s say, perfectly east and west with no northern or southern bend), then besides the vertical bend, as it will concave down, a representation of that on a map will not appear straight. I guess.

A 500 mile long straight object running E/W along a latitude line (following the curvature of the Earth) will look straight on the projected flat map but appear shorter on the map than it actually is (except at the equator). For it to connect two points at the same latitude on the Earth that are 500 miles apart, it will look like an arc on the projected flat map,

ETA:
OOPS, brain fart. If the object is straight (except for following the Earth's curvature) then it can't stay on the same latitude for 500 miles. It has to look like an arc on the flat map with either the center or the ends not on the latitude line. So you are right, a straight ladder will appear curved on the flat map.

 

[bilby]

Other than the equator, a line of latitude cannot have a centre of curvature that coincides with the centre of the Earth. That is, even after accounting for the curvature needed in the plane of the gravitational pull, there must be an additional curvature in the plane normal to the pull of gravity. Only a Great Circle - a circle centred on the centre of the Earth - can curve only in the plane of the gravitational pull. And the shortest distance between two points on a sphere is the shorter of the two segments of a Great Circle that passes through both points. So a line of latitude is not the shortest distance between points of equal latitude. The shortest route will always entail moving closer to the nearer pole than the centre of the latitude line.



(A Rhumb line is a line of constant bearing; All straight lines on a Mercator Projection are Rhumb lines. These were important to navigation in the age of sail, when fuel costs were non-existent, and ships were steered by seamen who had varying degrees of skill or literacy - following a constant bearing is easy, while navigating a Great Circle, using just a compass, is hard).

 

[steve_bank]

Maybe it’s the arcs representing flight paths that is giving the illusion that the planes aren’t flying straight. Maps don’t rotate. If the planes fly straight, they’re not flying toward their destination but instead where their destination is gonna be. A map would have to be virtual to capture that. A straight line to represent the plane not turning and a rotating map to capture earths spin.

The "arcs representing flight paths" are not due to the earth's spin or the planes anticipating where their destination will be, they're by and large due to the fact that the shortest route between any two points is never along a parallel (except when the parallel is the equator), but rather along a  Great_circle, plus some following winds.

Also, the shortest route from a point 500 miles to your east to a point 500 miles to your west does *not* cross your position, and that's true for air routes as much as surface routes. Unless, again, you're at the equator - or if by "a point 500 miles to your east" you mean the point you'd reach in a great circle route starting at your position with an initial bearing of 90.

A Great Circle is not a curved path. It is only a curved path when drawn on a flat map... as maps are. It is a correction of the error imposed by drawing the surface of a sphere onto a flat square.
A path "due East" around the equator of the planet is a straight line. It is not curved. Space-time is curved by the gravity of the planet. Your path through space-time is straight. Think of a satellite... Is it constantly turning to stay at constant altitude? nope. It is traveling straight, just like Newton predicted.

Having flown small planes, you fly a circular path above the Earth maintaining constant altitude. In vusual flight you keep the horizon in a fixed position to the aircraft.

If you flew in a straight line you either hit the ground or if ]f you have enough thrust leave the planet.

 

[fast]

Wow. If I took out a large flat map of the US and placed a pin on the map that marks the point I’m at in South Carolina — and then — place another pin on the map that marks a point 1000 miles west of my location — and then — use a straight edge ruler to draw a straight line between the two pins, then not only would the path marked on the map not reflect the closest path to take to get between the two points, but the actual distance between the points (as the crow flies) would actually be less than 1000 miles. That means walking due west (or due east or due anything) is not a straight walk. The curved line IS the straight path.

 

[steve_bank]

The Mercator Projective gives a linear distance scale on a flat map. The scaled distance between to points is the distance you would walk ion a spherical surface.

https://en.wikipedia.org/wiki/Mercator_projection

The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because of its unique property of representing any course of constant bearing as a straight segment. Such a course, known as a rhumb or, mathematically, a loxodrome, is preferred by navigators because the ship can sail in a constant compass direction to reach its destination, eliminating difficult and error-prone course corrections. Linear scale is constant on the Mercator in every direction around any point, thus preserving the angles and the shapes of small objects and fulfilling the conditions of a conformal map projection. As a side effect, the Mercator projection inflates the size of objects away from the equator. This inflation starts infinitesimally but accelerates with latitude to reach infinite at the poles. So, for example, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator, such as Central Africa.

The spherical model[edit]

Although the surface of Earth is best modelled by an oblate ellipsoid of revolution, for small scale maps the ellipsoid is approximated by a sphere of radius a. Many different methods exist for calculating a. The simplest include (a) the equatorial radius of the ellipsoid, (b) the arithmetic or geometric mean of the semi-axes of the ellipsoid, (c) the radius of the sphere having the same volume as the ellipsoid.[14] The range of all possible choices is about 35 km, but for small scale (large region) applications the variation may be ignored, and mean values of 6,371 km and 40,030 km may be taken for the radius and circumference respectively. These are the values used for numerical examples in later sections. Only high-accuracy cartography on large scale maps requires an ellipsoidal model.

 

[fast]

A Great Circle is not a curved path. It is only a curved path when drawn on a flat map... as maps are. It is a correction of the error imposed by drawing the surface of a sphere onto a flat square.
A path "due East" around the equator of the planet is a straight line. It is not curved. Space-time is curved by the gravity of the planet. Your path through space-time is straight. Think of a satellite... Is it constantly turning to stay at constant altitude? nope. It is traveling straight, just like Newton predicted.

Having flown small planes, you fly a circular path above the Earth maintaining constant altitude. In vusual flight you keep the horizon in a fixed position to the aircraft.

If you flew in a straight line you either hit the ground or if ]f you have enough thrust leave the planet.

“Straight” is perhaps not the best word for me to use. For my purposes, I’m okay with altitude adjustments to maintain flight. It’s those other directional changes I was talking about, but now it seems a true straight flight path (save altitude adjustments) are not proportionally represented on flat maps, unless maybe the volume of space on the map to represent the volume of land mass were adjusted to keep straight paths taken to be represented by straight lines on maps.

 

[skepticalbip]

Wow. If I took out a large flat map of the US and placed a pin on the map that marks the point I’m at in South Carolina — and then — place another pin on the map that marks a point 1000 miles west of my location — and then — use a straight edge ruler to draw a straight line between the two pins, then not only would the path marked on the map not reflect the closest path to take to get between the two points, but the actual distance between the points (as the crow flies) would actually be less than 1000 miles. That means walking due west (or due east or due anything) is not a straight walk. The curved line IS the straight path.

It is a bit more obvious if you look at the entire planet on a flat map.

If you are in Antarctica, walk 10 meters due north from the south pole, then start a walk due east or west, a walk of 62.8 meters will get you back to where you started. On the map, this walk would be from one side of the map to the opposite side.

 

[steve_bank]

Wow. If I took out a large flat map of the US and placed a pin on the map that marks the point I’m at in South Carolina — and then — place another pin on the map that marks a point 1000 miles west of my location — and then — use a straight edge ruler to draw a straight line between the two pins, then not only would the path marked on the map not reflect the closest path to take to get between the two points, but the actual distance between the points (as the crow flies) would actually be less than 1000 miles. That means walking due west (or due east or due anything) is not a straight walk. The curved line IS the straight path.

I think it depends on the map. Take a map of the USA with a distance scale on the bottom in miles per inch. If you were to walk from downtown NYC to downtown San Francisco the scale would give you an accurate distance. If not then the map is useless.

Remember when you walk, sail, or fly you are actually on close to a sphere. Linear is only a local phenomena.

When Boeing built the huge 747 assembly plant in Everett Wa they had to take into account Earth's curvature to make the floor level. The Alaska Pipeline is not straight, over distance it curves with the Earth.

 

[fast]

Wow. If I took out a large flat map of the US and placed a pin on the map that marks the point I’m at in South Carolina — and then — place another pin on the map that marks a point 1000 miles west of my location — and then — use a straight edge ruler to draw a straight line between the two pins, then not only would the path marked on the map not reflect the closest path to take to get between the two points, but the actual distance between the points (as the crow flies) would actually be less than 1000 miles. That means walking due west (or due east or due anything) is not a straight walk. The curved line IS the straight path.

It is a bit more obvious if you look at the entire planet on a flat map.
View attachment 22132
If you are in Antarctica, walk 10 meters due north from the south pole, then start a walk due east or west, a walk of 62.8 meters will get you back to where you started. On the map, this walk would be from one side of the map to the opposite side.

No, I get it. If I walk the 10 meters and then realize my destination is 31.4m due East, it doesn’t matter if I walk east or west; it’ll take 31.4m of walking to get there either way. Of course, when someone asks what’s the shortest distance you can walk to get to the destination, the answer is 20m.

We need to make this a cube planet. All this spherical oblong circling stuff is for the curves.

 

[steve_bank]

A map is a projection not reality. From the Mercator link. If you have a flat map 'projection' and are at point x and want to go to poiny y you lay your compass on the map, set the compass oriented properly to North and read the bearing in degrees to point y off the compass.

Walking and maintain the compass heading will get you where you want to go.

When I was flying a small plane I'd look on the map which had a compass printed on it. From that I'd fly a bearing on the plane/s compass to the destination.

From the link the maps do all the calculations of a sphere for you. As was just posted looking at the world map projection it is not linear, it is distorted so you can make linear measurements off the map.

If you have a USA map and read the distance between NYC and SF it is the distance you would measure with a flexible ruler placed on a sphere.

 

[fast]

It seems that it’s not a map one needs but a globe.

Let’s say I walked the 10 meters (marked as point A) and that’s where I find you. Your destination is 31.4/2 m east. I could show you on a globe which way to walk so you could walk straight there. I could point and say “go that way and don’t veer left or right.” The problem is that’s not a typical direction—more like a continuous set of changing directions.

 

[bilby]

A map is a projection not reality. From the Mercator link. If you have a flat map 'projection' and are at point x and want to go to poiny y you lay your compass on the map, set the compass oriented properly to North and read the bearing in degrees to point y off the compass.

Walking and maintain the compass heading will get you where you want to go.

When I was flying a small plane I'd look on the map which had a compass printed on it. From that I'd fly a bearing on the plane/s compass to the destination.

From the link the maps do all the calculations of a sphere for you. As was just posted looking at the world map projection it is not linear, it is distorted so you can make linear measurements off the map.

If you have a USA map and read the distance between NYC and SF it is the distance you would measure with a flexible ruler placed on a sphere.

Constant bearing is a characteristic of Mercator and related projections.

Don't try reading a bearing off any other kind of projection; You will end up lost.

All projections are a compromise between a flat map and a spherical Earth. Which is 'best' depends on your purpose - and Mercator is a shithouse choice for almost every purpose.

I would suggest that Mercator's projection is only good for two things - constant bearing navigation; And serving as an example of a terrible choice of projection that is nevertheless bizarrely popular.

 

[steve_bank]

It seems that it’s not a map one needs but a globe.

Let’s say I walked the 10 meters (marked as point A) and that’s where I find you. Your destination is 31.4/2 m east. I could show you on a globe which way to walk so you could walk straight there. I could point and say “go that way and don’t veer left or right.” The problem is that’s not a typical direction—more like a continuous set of changing directions.

It depends on the map. I lived in the Idaho panhandle in the 90s surrounded by federal and state forests. I bought uSGS topological sectional maps. They covered maybe a section 20x20 miles with altitude curves. The distance scale is linear. It is important to be able to see landmarks on the map in actual local relationships. The section is treated as a flat surface.

Whenever I went out in the back country by foot or jeep I carried a map and compass.

A USA map is a flat projection of a sphere. If you look at the distance scale at the bottom it is not linear, it varies. 12 inches is not 12 x 1 inch. on the map is not twice the distance of one inch on the map. It accounts for curvature of the surface.

To navigate between two points you only need a bearing relative to north to navigate. That is what a compass does. For long distance navigation a map projections allows you to derive a compass bearing to follow, a Great Circle path. It is not a line on a flat surface.

If you are flying from Europe to the USA there are multiple paths you can take.

 

[Jokodo]

It seems that it’s not a map one needs but a globe.

Let’s say I walked the 10 meters (marked as point A) and that’s where I find you. Your destination is 31.4/2 m east. I could show you on a globe which way to walk so you could walk straight there. I could point and say “go that way and don’t veer left or right.” The problem is that’s not a typical direction—more like a continuous set of changing directions.

It depends on the map. I lived in the Idaho panhandle in the 90s surrounded by federal and state forests. I bought uSGS topological sectional maps. They covered maybe a section 20x20 miles with altitude curves. The distance scale is linear. It is important to be able to see landmarks on the map in actual local relationships. The section is treated as a flat surface.

Whenever I went out in the back country by foot or jeep I carried a map and compass.

A USA map is a flat projection of a sphere. If you look at the distance scale at the bottom it is not linear, it varies. 12 inches is not 12 x 1 inch. on the map is not twice the distance of one inch on the map. It accounts for curvature of the surface.

To navigate between two points you only need a bearing relative to north to navigate. That is what a compass does. For long distance navigation a map projections allows you to derive a compass bearing to follow, a Great Circle path. It is not a line on a flat surface.

If you are flying from Europe to the USA there are multiple paths you can take.

Constant bearing is going to produce a suboptimal (longer) route in all but 2scenarios:

When you are travelling due North/South

When travelling due East/West at The equator.

A constant bearing is not a feature of great circle paths outside of thise edge cases, and of all the multiple routes between a point in the US and one in Europe, only one is the shortest.