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    Re: Closest point of approach.
    From: Gary LaPook
    Date: 2012 Jan 3, 03:49 -0800
    I hope I didn't make that sound too complicated because it is quite easy with a calculator.

    Using my example we will call the side of the first observation "a."
    The side of the second observation is 'b."
    The relative movement line between the two observation is "c."
    The angle between the two observation is "C."

    Then, using the law of cosines we find the length of side c.

    c^2 = a^2 + b^2 -2 (ab cos C)
    c^2 = 4.0^2 + 2.2^2 -2 (4 x 2.2 cos 17°)
    c^2 = 4.0^2 + 2.2^2 -2 (4 x 2.2 x 0.956)
    C^2 =4.009
    c = 2.002

    Next, using the law of sines we calculate angle "A."

    sin A /a = sin C/c
    sin A = sin C a/c
    A = arc sin (sin C a/c)
    A = arc sin (sin 17° 4.0/2.002)
    A = arc sin (0.292 x 4.0/2.002)
    A = arc sin (0.584)
    A = 35.7°

    Angle "A" is actually 144.3° which has the same sine as 35.7° so your calculator returns that value but that is a good thing. Since for a calculation of point of closest approach angle A will always exceed 90° because if it were less than that the other vessel would have already passed the point of closest approach and would be going away already. So the value returned by the calculator is actually the angle inside the second triangle, the one we use to find the distance of closest approach. Let's call this angle "D" so the side that represents the distance of closest approach is side "d." The last side is the movement of the other vessel from the second observation to the POCA, let's call that side "e" making the angle at your vessel between the second observation and the POCA angle "E." We know the size of angle "E" because it is 90° - angle "D" so it is 54.3° because the angle at the POCA is 90° and we call this angle BB (since it is opposite side b.)

    So to calculate the distance to the POCA we again use the law of sines to solve for side "d."
    Sin D /d = sin BB/b

    Since BB is always 90° its sine is always 1
    So sin D/d =1/b
    d = b  sin D
    d =  2.2 sin 35.7°
    d = 2.2 x 0.584
    d = 1.28

    Next we solve for side "e" using the law of sines then divide it by side "c" and multiply by the time between the two observation to find the time to POCA.

    e = b sin E
    e = 2.2 sin 54.3°
    e = 2.2 x 0.81
    e= 1.786

    Time to POCA = e/c (delta time between first two observations)
    Time to POCA = 1.786/2.002 x 6 minutes
    Time to POCA = 5 min and 21 sec

    It takes a lot less time than it does to type it up!

    gl


    --- On Tue, 12/27/11, Gary LaPook <garylapook@pacbell.net> wrote:

    From: Gary LaPook <garylapook@pacbell.net>
    Subject: [NavList] Re: Closest point of approach.
    To: NavList@fer3.com
    Date: Tuesday, December 27, 2011, 1:11 AM

    Sure, it's easy. I have attached several photos of a sample plot to show how the calculation proceeds.
    At 0120 the vessel bears 060° relative, distance 4.0 NM.
    At 0126 the vessel now bears 043° R, 2.2 NM.
    The difference in bearings is 017°.
    With this information we use the law of cosines to find the length of the third side which is 2.0022 NM.
    With the lengths of all three sides we use the law of sines to find the angle between the relative bearing to the second position and the direction of relative movement which is 35.7°.
    At the point of closest approach the angle between the realative bearing and the direction of relative movement is 90° the sine of which is 1.
    Now using the law of sines again we compute the distance at the point of closest approach which is 1.28 NM.
    Using the law on sines again we compute the distance the other vessel moves from the second position to the point of closest approach which is 1.78 NM.
    Next we divide this distance by the distance traveled between the first two positions and then multiply by the time between the first two positions to find the time to the point of closest approach which is 5 minutes and 21 seconds.

    gl

    --- On Mon, 12/26/11, Amrin Shahziya <ammushaz@gmail.com> wrote:

    From: Amrin Shahziya <ammushaz@gmail.com>
    Subject: [NavList] Re: Closest point of approach.
    To: NavList@fer3.com
    Date: Monday, December 26, 2011, 10:45 PM

    Instead of plotting is there any other way to calculate the closest point of approach using speeds and courses?

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