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    Re: Great Circle Course via calculator & HO 208
    From: Gary LaPook
    Date: 2016 Oct 1, 16:14 +0000
    To answer one of your questions, the number of waypoints you need  for your great circle course (actually an approximation of it) depends on the latitude and on the amount of extra distance you can accept. For example, Amelia Earhart was flying from Lae New Guinea, 6°47' south latitude, to Howland Island, 0° 38' north latitude, a distance of 2222 nautical miles. The rhumb line distance is only one-tenth of a nautical mile longer than flying a perfect (with an infinite number of waypoints) great circle course and the two tracks are never more than 10 nautical miles apart so it would be difficult using celestial navigation in a plane, or even on a boat,  to tell the difference because the course was so close to the equator. 

    I gave the following on the TIGHAR website as an explanation to a question about whether Earhart had followed a great circle or the much simpler to calculate and fly rhumb line:
    -----------------------------------------------------------------------------------------------------------------------------------------------
    
        There are times when computing and flying a great circle is worth the additional effort, not
    because is is more accurate but because the length of the great circle is shorter than the rhumb
    line. Try this on Google Earth. As a starting point use 43̊° 00.0' North, 88̊° 00.0' West. For the
    end point use 43̊° 00.0' North, 85̊° 00.0' East and have Google Earth calculate the distance in
    nautical miles. You will notice that the initial course takes you almost straight north and the final
    course is almost straight south. I picked those two spots because it is easy to calculate the rhumb
    line between them. Since they are both at the same latitude, the true course is straight east,
    90.00000000000000̊°. To compute the rhumb line distance figure the difference in longitude, 88°
    + 85° = 173 degrees. At the equator each degree of longitude is 60 nautical miles so the space
    between these two longitudes, at the equator, is 173 x 60 = 10,380 NM. At the latitude of 43̊° the
    length of a degree of longitude is equal to the cosine of the latitude times the 60 NM distance at
    the equator. So to find the distance between our two points we just multiply the 10,380 NM times
    the cosine of 43̊°, which is 0.731353, so the distance on the rhumb line between these two points
    is 7,591.45 NM. Compare that to the Google Earth result for the great circle course that passes
    near the north pole.

    ----------------------------------

    Here is the answer to my little experiment. The great circle course is only 5,644.18 NM long
    while the rhumb line between the two points is 7,591.45 so you would save almost 2,000 miles
    using the great circle. This is obviously an extreme case that I chose to illustrate this point. The
    initial great circle course leaving Milwaukee is almost straight north, 005̊° T and the final course
    is almost straight south, 175̊° T. On the way the course changes through all the intermediate
    directions, 6° then 7 then 8°.....then 90°.....then 120°....then, finally 175.°
    The rhumb line never changed, being 90.000000000000000000̊°.

    gl



    From: Bruce J. Pennino <NoReply_Pennino@fer3.com>
    To: garylapook---.net
    Sent: Saturday, October 1, 2016 8:23 AM
    Subject: [NavList] Great Circle Course via calculator & HO 208

    Hello:
    A friend recently gave me Dutton's, 13 edition.  As I was flipping through the pages I noticed the section on great circle courses, calculating waypoints etc. I've done some surveying and staked out many circular curves/courses using a theodolite. So the math was interesting.  Surveyors basically use  deflection angles from a back tangent and chord distances to locate points (waypoints) along the arc.  A surveyor moves up on the curve(course) and then can layout points ahead using subdeflection angles. Of course there is no concern with curvature and non-parallel meridians. 
    Anyway using HO 208 for initial course and distance, and using Dutton's mid latitude calculation methods with a hand calculator, I tried to do a great circle course from Charleston, SC to the Lizard Point, UK.  With only 4 intervals (5 waypoints), my results did not agree very well with an online calculator. With 6 waypoints, results were somewhat better. But the course was too far north.  The rhumb line distance and course is only 3-4 % longer more than great circle.  My first question: What is the best method/equations  for locating waypoints when the initial course and great circle distance is known? I assume latitude change = distance * cosinecourse, but longitude change?
    Second question: How many waypoints would I practically need to establish a proposed  course from Charleston, SC to Lizard Point? Based on the online calculators, 6 or 7 waypoints are sufficient?  I would update my course with actual CN positioning and using HO 208 . I assume no onboard computer, GPS, etc.  I imagine back in the sailing and steamship days navigators using previous courses  and just repeated their previous successful crossings. Interesting.
    Thanks and best regards,
    Bruce


       
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