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# 3-D Earth Geometry [Solved!]

### My question

Consider 2 points on the Earth's surface:-
Pt 1 defined by Latitude (in degrees),
Pt 2 defined by Initial Heading from Pt 1 (in degrees),
and Distance from Pt 1 given by Central Angle (in degrees) along a great Circle,
what (reasonably simple!) equations(s) will give me:-
the Latitude of Pt 2,
Neither Longitude of either point, nor the actual distance between them are wanted in the final results.

Thank you, Sooty

### Relevant page

3-D Earth Geometry

### What I've done so far

I have spent quite some time studying the material in the URL mentioned, and although most impressive, my math capabilities are simply overwhelmed by it all!

X

Consider 2 points on the Earth's surface:-
Pt 1 defined by Latitude (in degrees),
Pt 2 defined by Initial Heading from Pt 1 (in degrees),
and Distance from Pt 1 given by Central Angle (in degrees) along a great Circle,
what (reasonably simple!) equations(s) will give me:-
the Latitude of Pt 2,
Neither Longitude of either point, nor the actual distance between them are wanted in the final results.

Thank you, Sooty
Relevant page

<a href="https://www.intmath.com/vectors/3d-earth-geometry.php#top">3-D Earth Geometry</a>

What I've done so far

I have spent quite some time studying the material in the URL mentioned, and although most impressive, my math capabilities are simply overwhelmed by it all!

## Re: 3-D Earth Geometry

Sooty

I'm really sorry - your question got buried in my crazy inbox and I only just discovered it now.

Did you manage to solve your question?

Assuming you didn't, let's clarify things a bit.

(1) Let's use the example HKG -> NRT, as drawn here: Great Circle Mapper

(2) Latitude of HKG is 22.3080° N

(4) Central angle is along the great circle route (and can be verified using the formula in 3-D Earth Geometry)

(5) The final answer will be NRT's latitude (35.7720° N), and some final heading, which should be a little larger than 054° in this case.

Do I have it correct?

X

Sooty

I'm really sorry - your question got buried in my crazy inbox and I only just discovered it now.

Did you manage to solve your question?

Assuming you didn't, let's clarify things a bit.

(1) Let's use the example HKG -&gt; NRT, as drawn here: <a href="http://www.gcmap.com/mapui?P=HKG-NRT">Great Circle Mapper</a>

(2) Latitude of HKG is 22.3080° N

(4) Central angle is along the great circle route (and can be verified using the formula in <a href="/vectors/3d-earth-geometry.php">3-D Earth Geometry</a>)

(5) The final answer will be NRT's latitude (35.7720° N), and some final heading, which should be a little larger than 054° in this case.

Do I have it correct?

## Re: 3-D Earth Geometry

Hi Murray,

I thought my question had been just well buried in history by now!

Answer, yes, I did manage to derive an equation in the form I wanted. After spending quite a bit of time looking for 'an easy answer' while experimenting with a globe, a piece of string, and a cheap plastic ruler and potractor, I finally resigned myself to having to do some 'serious math'. I started googling 'Spherical Geometry'. Ouch! I found some equations including the Haversine Law, (of which I had never heard before!) and the Spherical Law of Sines. Neither of these were of the form I wanted, but using both of them I was able to derive a single equation that is. And to say the least, it does look kinda weird! But hey, all 'trivial solutions' I threw at it checked out perfectly, and 'non-trivial solutions' seemed to check out reasonably with my globe-and-string verification system!

Now to your question. Hong Kong to Tokyo. The little program I've built with my equation handles only integers.
Taking HKG-NRT as 3000 Kms gives an Internal Angle of ~30°. Entering HKG Lat as 22°, Initial Heading as 55°, HKG-NRT Internal Angle as 30°, my prog gives me Tokyo Lat as 36°, and Final Heading as 'a little larger than 54°' at 70°. So hey, your figures and mine aren't too different at all!

X

Hi Murray,

I thought my question had been just well buried in history by now!

Answer, yes, I did manage to derive an equation in the form I wanted. After spending quite a bit of time looking for 'an easy answer' while experimenting with a globe, a piece of string, and a cheap plastic ruler and potractor, I finally resigned myself to having to do some 'serious math'. I started googling 'Spherical Geometry'. Ouch! I found some equations including the Haversine Law, (of which I had never heard before!) and the Spherical Law of Sines. Neither of these were of the form I wanted, but using both of them I was able to derive a single equation that is. And to say the least, it does look kinda weird! But hey, all 'trivial solutions' I threw at it checked out perfectly, and 'non-trivial solutions' seemed to check out reasonably with my globe-and-string verification system!

Now to your question. Hong Kong to Tokyo. The little program I've built with my equation handles only integers.
Taking HKG-NRT as 3000 Kms gives an Internal Angle of ~30°. Entering HKG Lat as 22°, Initial Heading as 55°, HKG-NRT Internal Angle as 30°, my prog gives me Tokyo Lat as 36°, and Final Heading as 'a little larger than 54°' at 70°. So hey, your figures and mine aren't too different at all!

## Re: 3-D Earth Geometry

Sounds good, Sooty.

Would you mind sharing your solution here (at least some kind of pseudocode based on your program)?

It would help others who come here looking for the answer.

X

Sounds good, Sooty.

Would you mind sharing your solution here (at least some kind of pseudocode based on your program)?

It would help others who come here looking for the answer.

## Re: 3-D Earth Geometry

Hey, why not. :)

No guarantees, but I hope they are in order!

My equations...

1. - - - Lat2 = ArcCos(Cos(IntAng)*Cos(90 - Lat1) + Sin(90 - Lat1)*Sin(IntAng)*Cos(Dir1))

and

2. - - - Dir2:= ArcSin(Sin(Lat1)*Sin(Dir1)/Sin(Lat2))

where

Lat1 = Latitude of Point 1,
Dir1 = Initial Heading, (at Point 1),
IntAng = Internal Angle between Point 1 & Point 2,
Lat2 = Latitude of Point 2,
Dir2 = Final Heading (at Point 2).

Headings and Angles in Degrees. Latitude Notation - N Pole = -90°, S Pole = +90°.

I think that's it!

(If I remember correctly, some of what I found to be useful stuff came from this (rather complex!) site - Geographic Distance and Azimuth Calculations)

X

Hey, why not. :)

No guarantees, but I hope they are in order!

My equations...

1. - - - Lat2 = ArcCos(Cos(IntAng)*Cos(90 - Lat1) + Sin(90 - Lat1)*Sin(IntAng)*Cos(Dir1))

and

2. - - - Dir2:= ArcSin(Sin(Lat1)*Sin(Dir1)/Sin(Lat2))

where

Lat1 = Latitude of Point 1,
Dir1 = Initial Heading, (at Point 1),
IntAng = Internal Angle between Point 1 &amp; Point 2,
Lat2 = Latitude of Point 2,
Dir2 = Final Heading (at Point 2).

Headings and Angles in Degrees. Latitude Notation - N Pole = -90°, S Pole = +90°.

I think that's it!

(If I remember correctly, some of what I found to be useful stuff came from this (rather complex!) site - <a href="https://www.codeguru.com/cpp/cpp/algorithms/article.php/c5115/Geographic-Distance-and-Azimuth-Calculations.htm">Geographic Distance and Azimuth Calculations</a>)

## Re: 3-D Earth Geometry

Thanks a lot for sharing, Sooty.

X

Thanks a lot for sharing, Sooty.