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    Re: Distance by Vertical Angle Adaptation
    From: Greg Rudzinski
    Date: 2009 Jul 29, 06:59 -0700

    Frank,
    
    Sometimes shortcuts become long splices ;-(  The over the horizon
    distance by vertical angle problem remains begging for a calculator/
    slide rule friendly formula.
    
    Greg
    
    On Jul 28, 10:36�pm,  wrote:
    > Hi Greg, you wrote:
    >
    > "Yes it does appear that small angles are not covered by this
    > compromise formula (below 8' for 250ft , 10' for 500ft, 12' for
    > 1000ft)."
    >
    > And since it doesn't work well without iteration on the calculation, it's 
    probably better to go for the more complete solution rather than using this 
    correction for the obscured portion of the object *after* calculating a 
    preliminary distance.
    >
    > Here's how we can get the "more complete solution":
    > We have two formulas from your original post. First there's the simple 
    relationship that applies to any measured angle,
    > � D = 0.566*H/a
    > when H the height (or length along the arc perpendicular to the line of 
    sight) is in feet, a in minutes of arc, and the result D is in nautical 
    miles. Next there's the formula based on the maximum range of visibility for 
    objects at height h and H which tells us that the remaining height visible 
    above the horizon is
    > � H = H0 - [D/1.17 - sqrt(h)]^2
    > with D in n.m. as above and h, the height of the observer in feet. Now we 
    take that first formula, invert it to give H in terms of D and a as
    > � H = a*D/0.566
    > and set that equal to the second:
    > � a*D/0.566 = H0 - [D/1.17 - sqrt(h)]^2.
    > This can now be solved for D. But D is inside that squared term on the right 
    so we have to expand that and then solve for D. Skipping the algebra, the 
    result is
    > � D = 1.21*[sqrt[a1^2+0.94*(H0-h)] - a1]
    > where a1=a-0.97*sqrt(h)) or a1=a-dip with the dip calculated for the 
    observer at height h (in feet). With only a very small difference, this is 
    identical to the equation used to generate Bowditch Table 15.
    >
    > -FER
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