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    Some Piloting Formulas
    From: Dan Hogan
    Date: 1995 Jun 13, 17:39 PDT

    The following are a collection of piloting formulas and hints I have picked
    up over the years. I carry them in my Navigation Notebook. The "Distance Off
    Two Landmarks or Seamarks," was first shown to me by a friend that flew a
    TBM Avenger in WWII. He said %99 of his navigation to get back to the
    carrier was dead reckoning.
    
    ==========================================================================|
                  | First   | Second    |  Dist. Off    |     Dist. off       |
                  | Bearing | Bearing   | Sec'd Bearing |       Abeam         |
    ====================================================|=====================|
    Bow & Bearing |  45     |   90      | Dist. Run     | Dist. Run           |
    --------------------------------------------------------------------------|
    Double Angle  |  ...    | 2 * First | Dist. Run     |                     |
    --------------------------------------------------------------------------|
    Seven Eights  |  30     |   60      | Dist. Run     | 7/8(.875) * Dist Run|
    --------------------------------------------------------------------------|
    Seven Tenths  |  22.5   |   45      | Dist. Run     | 7/10(0.7) * Dist Run|
    --------------------------------------------------------------------------|
    Prediction    |  26.5   |   45      | 1.4 * Dist Run|  Dist. Run          |
    ==========================================================================|
    
    TIME/SPEED/DISTANCE ( In Hours, Knots , Nautical Miles)
    	Distance = Speed * Time * ((60, for minutes) or (3600, for seconds))
    
    THE RULE of 60
    	Distance Abeam * 60 = Bearing on the Bow * Range
    
    TIME at 1 Mile = 60 / Speed
    SPEED = (SQRT Bow Wave Length) * 1.33
    DISTANCE at 6 Minutes = 1/10 (0.1) * Speed
    
    DISTANCE TO HORIZON = 1.144 * SQRT of HE
    BREAKERS AT HORIZION = 3 Miles Off
    WINDOWS VISIBLE = 2 Miles Off
    COUNT TREES = 1 mile Off
    
    ECHO
      Distance Off = 1/2 Time in seconds * .18
    SOUND
      DISTANCE OFF = Time in Seconds * .18
    
    *************************
    FORMULAS FOR CALCULATORS
    	D = distance
    	R = distance run
    	A = First Bearing
    	B = Second Bearing
    
    Distance Off at Fiirst Bearing & Run Abeam
    	D = (R x sin A) / Cos A
    
    Distance Off Second Bearing Form Second Bearing on Bow & Run
    	D = (R x sin A) / (sin A [abs. diff.] B)
    
    Distance Off Abeam with Second Bearing on Bow and Run Betweeen
    	D1 = (R x sin 2B) / sin C
    	D2 = D1 x sin A (D2 = distance off abeam)
    
    Distance Run to a Given Bearing & Distance Off When on That Bearing
    	D1 = (R1 x sin B) / sin C
    	D2 = D1 x sin A (D2 equals distance off abeam)
    
    *************************************
    Sailing to Weather
    D = (200 / sine of the angle between tacks) * sine attack angle
    
    	D = percentage of the distance to the mark.
    	attack angle = the angle off the wind when sailing.
    
    *************************************
    Distance Off Two Landmarks or Seamarks
    	   s / SIN LS : a /sin LA : b / sin LB
    
    	s  = distance between marks
    	LS = angle diff between a & b
    	a  = distance from ship to b
    	LA = angle diff between bearing of a from ship & bearing of b from a
    	b  = distance from ship to a
    	LB = angle diff between bearing from b to a & Bearing b from ship
    
    ************************************
    Tacking Downwind
    
    Total distance sailed =
    2 * base distance * sin of the divergence angle/sin (divergence angle * 2)
    
    Chart:
    
    Divergence   |  Speed	 |  Distance	 |   Time
     angle       |	         |   to sail	 |   Hrs.
    -------------|-----------|---------------|-----------
    
    Dan Hogan
        dhhogan---.net  dhhogan{at}delphi.com
    
    ==========
    
    Date: Thu, 15 Jun 1995 22:26:39 -0400
    From: sgrant{at}BIONET.BIO.DFO.CA (Steve Grant)
    Subject: Navigation Formulae
    To: navigation{at}ronin.com
    
    Another useful 'trick' for estimating distance off is to hold up a pencil
    (or your finger) at arm's length aligned with the edge of an object whose
    length you know, say an Island, with one eye closed. Then switch eyes (i.e.
    close the open one and open the other); the pencil will appear to jump. The
    "Formula" is: "dist = 10 x length of object". e.g. if you are looking at an
    island that is 1 mile across, from the direction you are looking at it, and
    the pencil jumps exactly from one edge to the other you are about 10 miles
    away. The main assumption here is that the angle subtended between your
    finger at arm's length and your eyes is about one tenth of a radian i.e.
    5.7 deg.  The above equation should really be "dist = (57.3/A) x length of
    object" where A is the angle in deg. that the pencil (or finger) moves.
    Most people have an "A" of 6-7 deg. You sould 'calibrate' yourself and get
    more accurate results! But remember, if it takes two "blinks" to get from
    one side of the island to the other the distance is 'half' - not double
    because you are 'dividing' be 'A'! Enjoy! You will find this and dozens of
    other practical by-eye navigation techniques in "The Practical Pilot -
    Coastal Navigation by Eye, by Leonard Eyges, International Marine
    Publishing Co., Camden, ME.
    STG
    
    Steve Grant             sgrant{at}bionet.bio.dfo.ca
    Canadian Hydrographic Service,  Bedford Institute of Oceanography
    P.O. Box 1006,           Dartmouth, NS, Canada, B2A-4A2
    23 Garden Court Terrace, Dartmouth, NS, Canada, B3A-3S6
    (H) 902-463-5019        (W) 902-426-5409
    ==========
    
    From: Dan Pfeiffer 
    Subject: Re: Navigation Formulae
    To: navigation{at}ronin.com
    Date: Fri, 16 Jun 95 8:33:36 EDT
    
    >
    > Another useful 'trick' for estimating distance off is to hold up a pencil
    > (or your finger) at arm's length aligned with the edge of an object whose
    > length you know, say an Island, with one eye closed. Then switch eyes (i.e.
    .....snip....
    > because you are 'dividing' be 'A'! Enjoy! You will find this and dozens of
    > other practical by-eye navigation techniques in "The Practical Pilot -
    > Coastal Navigation by Eye, by Leonard Eyges, International Marine
    > Publishing Co., Camden, ME.
    > STG
    >
    Another great source for all sorts of tricks like this and lots
    of other facinating stuff is a book called "Emergency Navigation".
    I don't have the author's name (lent mine ...) but I recently
    saw it in a book store at Fisherman's Warf S.F. while visiting
    with AWS last month.  So the book is available.  And I think it's
    in the West Marine catalog.  Maybe someone knows the author.
    The book is all about how to navigate with next to no instrumentation
    (e.g. just your fingers) and how to fabricate simple instruments
    from items you might have abord.
    
    -DHP
    pfeiffer{at}humboldt.kent.edu
    
    

       
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