Hello everyone. I am lost in question . This question is that An aircraft takes off in New York(40 degree 50 minutes North, 74 degree West) with the azimuth 7 degree 22 minutes and flies 9190 km along the orthodrome. R=6378 km. Calculate the geographical coordinates of the destination point. (Round up to minutes)?ÿ
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Can you explain me how to solve it ?
Solve it with an airplane, and a tank of gas! 😉
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I learned a new term - didn't know the great circle was called an orthodrome. As opposed to a loxodrome (rhumb line) that I had heard of.
Do you have to have the calculation done on a sphere? You need to consult a book on spherical trigonometry for that.
It's harder on an ellipsoid model of the earth.?ÿ The NGS Toolkit program FORWARD will do this on any of several standard earth ellipsoid models, and you can define your own user ellipsoid (e.g. exact sphere).?ÿ The program is VERY fussy about input format and likes decimal points.
https://geodesy.noaa.gov/cgi-bin/Inv_Fwd/forward2.prl
But I suspect it is is for a class.?ÿ If so, you should do the spherical trig and only use the tool as a check.?ÿ You don't learn much about the process by using the tool.
I learned a new term
I learned a bunch of new terms, thanks ?????ÿ
That's a big tank to go so far, and no guarantee you can refuel if you get there.
This is the nasty one from spherical trig. The problem is that the azimuth changes continuously along the course. The departure azimuth is sort of northeast, but the arrival azimuth will be sort of southeast. You're flying on a circle concave to the equator; draw a frowny face and the left mouth corner will be the departure and the right one will be the arrival.?ÿ
Look for a differential equation solution. Likely, its solution will be in the form of a series.
Here's the output from NGS Forward. Note that you use a custom ellipsoid with equal major and minor radii. The program can't handle equal, so one is 6,378,000.0000 meters and the other is 6,379,999.9999 meters. You're going to end up pretty far north, so wear a coat!
Output from FORWARD
Ellipsoid : User defined.
Equatorial axis, a = 6378000.0000
Polar axis, b = 6377999.9999
Inverse flattening, 1/f = ****************
First Station : New York
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LAT = 40 50 0.00000 North
LON = 74 0 0.00000 West
Second Station : Destination
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LAT = 55 58 13.48380 North
LON = 92 52 6.66593 East
Forward azimuth FAZ = 7 22 0.0000 From North
Back azimuth BAZ = 350 1 1.7658 From North
Ellipsoidal distance S = 9190000.0000 m
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But you're the math teacher. I thought you'd do the spherical trig. 😉
Interesting that FORWARD won't take a prolate ellipsoid (longer between the poles). I tried adding the b dimension a cm longer than a, and it objects, substituting the default ellipsoid instead. And you have to be paying attention to even notice that it did.?ÿ It's a somewhat fragile program with minimal error diagnostics.
I learned a new term - didn't know the great circle was called an orthodrome. As opposed to a loxodrome (rhumb line) that I had heard of.
One of my favorites; Bagels and lox washed down down with a pint of Rhumb. ?????ÿ
No telling what kind of line you'd get after a whole pint.
If ingested all at once, then a greeting line at the wake.
Sorry for the delay. I've been off forecasting covid-19 for North Carolina (scary as hell!) creating polyconic projections for Pittsburgh (amazing!), and complicating spheres by confusing them with ellipsoids. Your problem can be solved by first applying the spherical law of cosines to the given data and then using the result in the spherical law of sines to get the final answer. This solution agrees with the "black box" solution from NGS Forward.
Best of luck with your studies.
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So I had to do the spherical trig, using a spreadsheet to do the actual calculations, but was interrupted by supper (Central Standard Time).?ÿ Turns out when I got back here to post, @mathteacher had already posted the method.?ÿ We used the same formulas and got the same answers.
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Show offs!!!
Many of the tools in the NGS Toolkit were created by NGS employees for their and their coworkers use.
In the case of Forward and Inverse (and their 3D versions) they were to deal with real world ellipsoids. The link for this toolkit item includes references to algorithms and articles about the problems.?ÿ
The determination that the earth was oblate through actual observations (1700s) played an important part in the history of geodesy. See this 2002 paper:?ÿ https://www.fig.net/resources/proceedings/fig_proceedings/fig_2002/Hs4/HS4_smith.pdf
Many of the toolkit items have links to their code or technical papers on how they work. See T. Vincenty's paper from 1975 on the Direct and Inverse problem: ?ÿ https://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf ?ÿ
On the issue of teaching students/surveyors about distances and azimuths between points on the surface of the Earth, I like the exposition in Elithorp and Findorf??s, ??Geodesy for Geomatics and GIS Professionals.? ?ÿThey start with computations on a sphere, then using great circles and then on an ellipsoid.
I was teaching using the Elithorp book at the same time a colleague was using Mathematical Techniques in GIS. Both contained the great circle computation with mine using co-latitudes and his using latitudes. Some students thought one must be wrong. Always a struggle to get some folks to think.
Cheers,
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My reference is Plane and Spherical Trigonometry by Ballou and Steen, 1943, bought in a used book store for $1.95 sometime in the late 1980s. It includes instruction on plane sailing, parallel sailing, middle-latitude sailing, and dead reckoning, all of which can be useful on dry land as well.
Occasionally, I refer to Plane and Spherical Trigonometry originally written by William Granville in 1908. I have the revised 1941 edition by Smith and Mikesh and owned by a Mr. Buchanan of Clemson College in 1942-43. I also bought it in a used book store, but the price was not recorded in pencil on the flyleaf as it was in the other book.
The problems involving ships "steaming" across the ocean, haversines, and logarithms are priceless reminders of how routine the solving of difficult problems was in the past. The math, of course, is as sound now as it was then, so solving those problems with spreadsheets and CAS using that math is a marvelous exercise.
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My reference for the formulas was CRC Standard Mathematical Tables.
My H.S. trig teacher in 1968 recommended that college-bound students buy it from a clearance sale for $1.?ÿ Best dollar I ever spent on a book. Although I used it much less after calculators, I still refer to it for some things, especially the less-used trig identities, and rarely the integral formulas.?ÿ It was my go-to for normal probability values until spreadsheets, because most calculators didn't have that function.