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Re: Closest point of approach.
From: Gary LaPook
Date: 2012 Jan 3, 03:49 -0800
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:
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