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    Re: Refraction near the Horizon ? Observation vs. Calculation
    From: Brad Morris
    Date: 2013 Apr 9, 09:27 -0700


    In order to have a civil discussion, we must have a common basis for that discussion.

    Lets begin by defining some terms and units

    Rrefr = (Rgeom/(1-k) ; defines the equivalent radius of the earth for refraction
    Rgeom =6400000 ; Geometric (true) radius of the earth
    nominally 6400000 meters
    k =.164 ; unitless value describing the alteration
    from Rgeom to Rrefr; nominally .164
    HoE ; height of eye, meters
    pi() = 3.14159265 ; magic number defining the relationship of the
    circumference of a circle to its diameter
    180 ; magic number for 1/2 circle or 180 degrees

    Next we have the equation Dip,deg=acos(Rrefr/(Rrefr+HoE))*180/pi(). That is, in words, dip, whose units are degrees, can be found by taking the inverse cosine of a number close to one. That number is defined as the refracted radius of the earth divided by the sum of the refracted radius of the earth and the height of eye. If the output of your arccosine function is in radians, you multiply this output by 180/pi() to convert from radians to degrees.

    Additionally, the equation Dip,min = 3438*sqrt(2*HoE/R) was offered. In this equation, R was not defined and was assumed to be Rgeom. Upon further contemplation, it has become evident that R is meant to be Rrefr. Since this output is in minutes, we must divide by 60 to get degrees. Let me restate the equation then to be Dip,deg=3438*sqrt(2*HoE/Rrefr)/60. That is, in words, Dip, in degrees is equal to the number 3438 (3437.746=180/pi()*60 minutes per degree) multiplied by a number close to zero. The number is the square root of twice the height of eye divided by the refracted radius of the earth. Since this output is in minutes, divide by 60 minutes per degree to obtain degrees for the comparison.

    Both equations have unit conversions.
    180/pi() = 57.29578
    3438/60 = 57.30000
    3437.746/60 = 57.29578
    Both equations have the same conversion!
    That is 180/pi()=180/pi()*60/60

    Practical marine navigation has the HoE less than 157 feet (47.7 meters). HoE, even for large vessels such as Jeremy's, are on the order of 31 meters.

    Dip,deg=57.29578 *acos(Rrefr/(Rrefr+HoE))
    Dip,deg=57.29578 *sqrt(2*HoE/Rrefr)
    over the range of practical marine navigation

    The delta between the two equations is less than 0.1 seconds of arc. The equations are equivalent for practical marine navigation.

    So no one misses it, I will retype that in caps. Please do note that I am not yelling or mad, I'm just emphasizing the statement [THE EQUATIONS ARE EQUIVALENT FOR PRACTICAL MARINE NAVIGATION]. Again, the caps are just for emphasis.


    Much misunderstanding could have been prevented by the simple definition of terms. Stating R, without clearly identifying that it was the equivalent refracted radius of the earth, led to confusion. If it confused me, it may have confused others.

    Using a value like 3438, without showing that it was approximately 180/pi(), is a Magic Number. When writing software, we often create a variable to avoid Magic Numbers. For example, we could state MinutesConversion = 180/pi()*60. Then Dip,min=MinutesConversion*sqrt(2*HoE/Rrefr) makes more sense. When Magic Numbers, as opposed to variables, appear in code, they often lead to programming errors.

    I want to state unequivocally that I do like and respect you Frank. You are clearly knowledgeable and a terrific asset to NavList. That said, of late you have demonstrated what some may consider to be rather coarse behavior, to me and to others. Rather than presenting you with a list of these obvious transgressions, I would urge you to think for a moment, when you are rude to folks, after a while, they become rude in return. When you say nasty things about someone's work product, they will eventually begin to return the favor, and say nasty things about your work product. You probably didn't mean for it to come across that way, but it did.

    I do hope that we can return to civil discourse. I have offered this up quite carefully and upon thoughtful consideration. The ball is in your court now, how will you return it?

    Best Regards
    Brad Morris
    NavList message boards and member settings: www.fer3.com/NavList
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