I commend those who look deeper into these issues. I prefer to rely on Vincenty.
I provide links below to Excel implementations of his formulae, a short bio, and a short discussion from ESRI.?ÿ
https://github.com/tdjastrzebski/Vincenty-Excel
https://en.wikipedia.org/wiki/Thaddeus_Vincenty
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?ÿI realize many like to use Excel for computations. My preference is Matlab which I??ve used too long to consider alternatives.
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Vincenty is one of my heroes, along with Oscar Adams, Rod Deakin, and Elon Musk.
Speaking of Dr. Deakin, this paper of his provides an excellent presentation of the problem. From the differential rectangle on page 4 and Clairaut's equation on page 5, an iterative method using Euler's method for solving differential equations can be derived. It's a cool teaching exercise, but its fatal flaw is ultimate deterioration of accuracy because of accumulated rounding errors. I would imagine that Matlab could overcome that. but Excel can't. On the other hand, generating a thousand or two rows of Excel-calculated positions one meter apart along a twisted line is somewhat pleasing to an old numerical rebel like me.
I also like to hammer in screws!
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http://www.mygeodesy.id.au/documents/Geodesics%20-%20Pittman%20method.pdf
@geeoddmike?ÿ I would like to point out that Vincenty's?ÿ method of the Direct and Inverse solution for the Geodetic distance and Azimuth
came from Rainsford's work (see Bulletin Geodesique, No. 37, 1955). This is Vincenty's 3rd reference to his paper
"Direct and Inverse Solutions of Geodesics on the Ellipsoid with application of?ÿNested Equations, Survey Review XXII, 176, April 1975.
Mr. T. Vincenty gave?ÿFULL?ÿcredit to Mr. Rainsford (see 3rd par. down from Introduction, pg. 88 of Survey Review, 1975).
Where the real genius of Vincenty came in was his use of?ÿNested Equations to keep the solution short but very accurate.
If you want to read an outstanding article on the subject that includes a short history read?ÿ"Algorithms for Geodesics" by?ÿ
Dr. C.F.F. Karney?ÿin Journal of Geodesy, 2013, 87, pp. 43-55.
Note: The first poster wanted the distance on a?ÿSPHERE not an ellipsoid.
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JOHN NOLTON
@mathteacher?ÿ I have many book on Spherical Trigonometry but the first one I used in H.S. (1959 or 1960) was
Plane and Spherical Trigonometry by Kells, Kern and Bland.?ÿ Did you ever look at this one?
JOHN NOLTON
I may have, John. When I was teaching, I loaned many of my old books to other teachers and some never found their way back to me. The Ballou and Steen book is so well written that I would never lend it to anyone, though.
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@mathteacher?ÿ ?ÿMT?ÿ I also have Ballou and Steen book. Had to go down to my library to double check. I have also learned the?ÿ
hard way about lending books.
Yup, my High school made me buy one in my senior year.?ÿ The biggest book I ever bought, and worth every penny, the "let's look that up" kinda book.
Ballou and Steen is a great book for self study. Pages 3 - 120 on plane trig and pages 123 - 172 on spherical trig, no trig tables, but answers to odd problems. And it's about the size of a tablet computer, so it's ok for reading while you're driving.
But you never have enough books. The Granville, Smith, Mikesh book has a section on finding the trig functions of angles near 0?ø and 90?ø without using tables. The values of sine and tangent change too fast here for a linear interpolation to be accurate. Instead, for four decimal place accuracy, the authors recommend using the radian measure of the angle in the place of the function.
Their example is sin(2?ø 12') = 0.0384, tan(2?ø 12') = 0.0384, cos(87?ø 48') = 0.0384) and radians(2?ø 12') = 0.0384. That determines their cutoff values for interpolating functions.
Today, we wouldn't consider 4 decimal places, interpolation, or substituting radians for function values. Black boxes have made things easier and more accurate, but some of the richness of applying the math is lost. I wouldn't go back, but it's good reading.
Heck, I was lost when the distance was given in kilometers.
