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C3GPS

Antipodes in Great Circle

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Does anyone know of a tool that will allow me to enter a random waypoint and precise heading to create a great circle. Then, return to me the antipodes on that circle?

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Posted (edited)

There is no such a tool because it is not possible. Great circles exists only on spheres and Earth is not a sphere according to WGS84. That's why mapping tools can not draw a great circle from a random waypoint to the antipode. Meridian lines and the equator line are only antipodal great circles you can draw on this Earth model.

 

It is possible to draw an arc between antipodes but when you continue this arc around the globe you will find that it does not return back to the starting point. It is not a circle at all.

Edited by arisoft
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The antipodes of any place on Earth is the point on the Earth's surface which is diametrically opposite to it. Two points that are antipodal to each other are connected by a straight line running through the centre of the Earth.
So there is no sense to draw a line or circle across the surface of the Earth.

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Posted (edited)
1 hour ago, StefandD said:

So there is no sense to draw a line or circle across the surface of the Earth.

 

Diametrically opposite points belongs to opposite meridians that constitutes a great circle together. Whether it makes sense or not, cartographers tend to draw these lines on a map.

Edited by arisoft

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I know of a tool that will allow you to generate antipodes coords for any set of coords.

https://www.antipodesmap.com/

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2 hours ago, Isonzo Karst said:

I know of a tool that will allow you to generate antipodes coords for any set of coords.

https://www.antipodesmap.com/

I found that but I don’t know the coords of either antipode. I only know 1 point on the circle and the bearing of travel

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2 hours ago, C3GPS said:

I found that but I don’t know the coords of either antipode. I only know 1 point on the circle and the bearing of travel

 

Are you trying to solve a mystery cache?

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5 hours ago, C3GPS said:

 

As you see, the mystery is limited to spherical world which is not compatible to any real mapping tool. You have to make a special tool for this special case.

Edited by arisoft

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It is an interesting problem, as the CO gives a coordinate and direction, so the Great Circle can be calculated using the suggested tool, however, the cache could be anywhere on that great circle and without a specific distance measurement, how would you know where to stop?  Obviously, you have a general idea, such as city and state, but at what exact point on the circle do you stop?  Passing over the antipode is irrelevant as on any great circle, you will pass over a coordinate and its antipode.  Should one assume that the fake puzzle coordinate actually has some true meaning?

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Oh... the cache has not been found yet.:)

 

1 hour ago, BG2015 said:

Should one assume that the fake puzzle coordinate actually has some true meaning?

 

Do not assume anything. Try both ways!

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The only other thing I know is that only 2 of the bullet points are relevant. I don't know which 2 but my assumption is that one is the one that mentions antipode and the other is the coordinate and direction of travel.

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I should rephrase...I believe the only 2 bullets that matter are the one with coords and the one with direction of travel. Antipode seems to be irrelevant.

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On 1/5/2020 at 6:19 AM, C3GPS said:

I found that but I don’t know the coords of either antipode. I only know 1 point on the circle and the bearing of travel

 

If the bearing is not 0 or 180, then it's not the analog to a "great circle" in WGS-84.  You can project along a geodesic, which will eventually come back to the same point, but it is not guaranteed to pass through the antipodes unless you are heading due north or due south.

 

Also, without knowing a point, how can you get the antipodes?  I suspect the problem you are trying to solve is not what you asked.  Do you want to project from a point at a given azimuth?  And then what?  Find the point on the surface of the Earth at a maximum distance from the first point (which is the thing that roughly correspond to an "antipodes")?  I doubt that because it doesn't seem to have any use.  I suspect, rather, that your question is a straightforward projection question, such as "what is the closest distance from the geodesic to some given point?" which is doable, but not in closed form.

 

ETA:  Oh.  I looked at the puzzle.  The puzzle creator wants you to use a spherical approximation, in which case there are great circles and antipodes.

 

Edited by fizzymagic

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On 1/11/2020 at 11:32 PM, fizzymagic said:

 

The puzzle creator wants you to use a spherical approximation, in which case there are great circles and antipodes.

 

 

Indeed, there is a Great Circle in the spherical model, which can be determined by a singular point and a direction.  However, there is no distance given in the puzzle, so either the CO has made an error in information given, or the part of the puzzle that needs to be solved will give you that distance along the Great Circle to be traveled.

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I believe there is enough information in there to solve it.  I don't want to give too much away, but look at the sixth bullet. 

 

The key here is that there are an infinite number of great circles between any pair of antipodal points.  That is not true for an ellipsoid: where there are only 2 geodesics between antipodes. Which is why this puzzle requires the great circle calculation.

Edited by fizzymagic

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So I was drifting off to sleep and I suddenly had a flash of insight, and I knew the simple way to solve this puzzle.  There is indeed enough information to solve it.  I used my own program, FizzyCalc, to prove it could be done by hand.

 

Turned out to be a very interesting problem.  I am impressed!

 

ETA:  BTW, for what it's worth, it turns out that this problem could have been done using the ellipsoid, since the geodesic in question does pass through the antipodes!

Edited by fizzymagic
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That's a pretty cool puzzle.... Solved in a few minutes, my method involving hitting the great circle calculator a few times, but got the green tick on first guess.  I think I may "copy with pride" and make a version of it myself.

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Is this what this an example of what this thread is about? It's one I found a few years ago. I can't remember exactly how I solved it but I think I found a formula somewhere on Google.

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42 minutes ago, colleda said:

Is this what this an example of what this thread is about? It's one I found a few years ago. I can't remember exactly how I solved it but I think I found a formula somewhere on Google.

 

The puzzle was actually not about the antipodes at all, despite the original question.  But yes, the antipodes of a point is the point with the latitude inverted (S instead of N, for example) and the longitude different by 180 degrees.

 

What I didn't know until I did the puzzle is that on the ellipsoid, unlike a sphere, most directions you go you won't come back to the same point after going around the world, and you will not pass through the antipodes on your way.

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52 minutes ago, fizzymagic said:

What I didn't know until I did the puzzle is that on the ellipsoid, unlike a sphere, most directions you go you won't come back to the same point after going around the world, and you will not pass through the antipodes on your way.

 

I have made a mystery using exactly this phenomenon. ;)

It is somehow surprising when you realize it the first time.

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2 hours ago, fizzymagic said:

 

The puzzle was actually not about the antipodes at all, despite the original question.  But yes, the antipodes of a point is the point with the latitude inverted (S instead of N, for example) and the longitude different by 180 degrees.

 

What I didn't know until I did the puzzle is that on the ellipsoid, unlike a sphere, most directions you go you won't come back to the same point after going around the world, and you will not pass through the antipodes on your way.

 

Correct me if I'm wrong, but you also won't end up back where you started, right?

 

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15 hours ago, funkymunkyzone said:

 

Correct me if I'm wrong, but you also won't end up back where you started, right?

 

 

Yes.  Sorry if I was not clear enough about that.  You don't end up back where you started.

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9 hours ago, fizzymagic said:

 

Yes.  Sorry if I was not clear enough about that.  You don't end up back where you started.

 

Oh, actually you did say it.  Sorry.

 

I had a puzzle out years ago that told finders they should walk x km east, x km north, x km west and x km south to find the cache.  The number of people that went to the published coords (which was a horrible patch of thorns) thinking that's where they would end up, was astonishing. :)

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No you all got me confused. Surely, if you keep on the great circle, i.e. follow planar route around the globe, you end up where you started.

If you keep the same heading, you end up on the pole.

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4 hours ago, funkymunkyzone said:

 

Oh, actually you did say it.  Sorry.

 

I had a puzzle out years ago that told finders they should walk x km east, x km north, x km west and x km south to find the cache.  The number of people that went to the published coords (which was a horrible patch of thorns) thinking that's where they would end up, was astonishing. :)

I'm going to use this idea if you don't mind.

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On 1/22/2020 at 4:39 AM, papu66 said:

No you all got me confused. Surely, if you keep on the great circle, i.e. follow planar route around the globe, you end up where you started.

If you keep the same heading, you end up on the pole.

 

The first is true for a sphere, but the Earth is not a sphere, so it is not true for the Earth.  The second is true for any rhumb line, which is  curve of constant azimuth.

Turns out that from any point on the Earth, there are only 4 directions that you can go that will bring you back to your starting point.  Two of them are due north and due south.

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6 hours ago, fizzymagic said:

 

The first is true for a sphere, but the Earth is not a sphere, so it is not true for the Earth.  The second is true for any rhumb line, which is  curve of constant azimuth.

Turns out that from any point on the Earth, there are only 4 directions that you can go that will bring you back to your starting point.  Two of them are due north and due south.

I'm sorry, but these are the only two definitions of "going in certain direction" that I can understand. Either you keep constant azimuth of you keep on the plane that is set by center of earth, your current position and your current velocity. The intersection of that plane with the globe surface will give a closed loop which is circle, ellipse or something in between. (Inasmuch as circle is also an  ellipse I guess that amounts to saying something between ellipse and ellipse, which I assume is also ellipse).

Edited by papu66

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21 hours ago, papu66 said:

I'm sorry, but these are the only two definitions of "going in certain direction" that I can understand. Either you keep constant azimuth of you keep on the plane that is set by center of earth, your current position and your current velocity. The intersection of that plane with the globe surface will give a closed loop which is circle, ellipse or something in between. (Inasmuch as circle is also an  ellipse I guess that amounts to saying something between ellipse and ellipse, which I assume is also ellipse).

 

"Going in a certain direction" on an ellipsoid means following a geodesic, which is the shortest distance between points on the surface.  The geodesic, in general, does not follow a plane that goes through the center of the ellipsoid.  A geodesic is a "straight line" in the sense that a geodesic between two points follows the points on  the ellipsoid's surface that are closest to the straight line (in Cartesian space) connecting the two points.  So, in the sense that following a geodesic is going straight along the surface of the Earth, with curving either to the left or the right, it is the closest thing there is to going a constant direction.  However, for the true shape of the Earth, a geodesic (except for two special cases) does not return to its starting point after a single trip around the Earth, and it does not lie on a plane that goes through the center of the Earth.

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5 hours ago, fizzymagic said:

"Going in a certain direction" on an ellipsoid means following a geodesic, which is the shortest distance between points on the surface.

Hmm... I think of "going in a certain direction" as maintaining a constant bearing. If I'm going northeast (a bearing of 45°), I will follow a spiral towards the north pole. And if I'm going east-northeast (a bearing of 67.5°), I will follow another spiral towards the north pole, but one that is longer, approaching the north pole more slowly.

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13 hours ago, niraD said:

Hmm... I think of "going in a certain direction" as maintaining a constant bearing. If I'm going northeast (a bearing of 45°), I will follow a spiral towards the north pole. And if I'm going east-northeast (a bearing of 67.5°), I will follow another spiral towards the north pole, but one that is longer, approaching the north pole more slowly.

 

Those are known as "rhumb lines."  I was trying to articulate the closest thing the ellipsoid has to what on a sphere are called "great circles."  It's difficult to explain because, while there are "great circles" on an ellipsoid that behave like those on a sphere, there are only two from any given point instead of an infinite number.

 

Maybe this will help.

 

Also this.

Edited by fizzymagic
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