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    Re: Closest point of approach.
    From: George Huxtable
    Date: 2000 Aug 11, 3:29 PM

    Russell Sher asks-
    >Does anyone know a simple way of calculating the closest point of approach
    >between two vessels on given courses at given speeds?
    >I would imagine that one method is to plot the courses and calculate which
    >vessel will reach the point of intersection of the projected tracks first.
    >The distance to the other vessel should give the CPA ( correct ?)
    Well, no, Russell, not at all correct! This becomes obvious if you consider
    the case of two vessels approaching each other along parallel tracks,
    courses differing by 180 degrees. At some point, they will pass each other
    (hopefully port-to-port) and clearly that will be their closest point of
    approach. But their tracks, being parallel, will never intersect. So
    Russell's proposed solution doesn't work in that situation. Indeed, it
    doesn't work at all.
    Instead, it's necessary to work out the relative velocity between the two
    vessels. Draw a plot (North-up is perhaps simplest but not essential)
    showing your own vessel as a stationary dot at the centre. You can think of
    this as the scene viewed from a helicopter flying directly above your own
    vessel and at exactly the same speed, so from its point of view your own
    vessel is not moving.
    Now put in a dot showing the position of the other vessel, relative to you,
    at a certain moment. Draw a vector (a line with an arrowhead) from that dot
    showing the course and speed of the other vessel. You can scale the vector
    as you like, so that it shows where the other vessel will get to in 5
    minutes time, or 1 minute, or an hour, depending on the scale of your
    diagram and the urgency of the situation.
    Now comes the crucial bit. What you want to get is the velocity of the
    other vessel relative to yours. So from the head of the arrow vector you
    have just put in, draw a second arrow, length corresponding to your own
    vessel's speed on the same scale as before, and direction 180 DEGREES
    OPPOSITE to the course of your own vessel. Join the dot which represented
    the initial position of the second vessel to the head of this second arrow.
    That line represents the course and speed of the vessel relative to your
    own. This is the relative velocity vector. It shows where the other vessel
    will have got to, relative to your own, after the time interval you have
    Clearly, if that new vector is pointing directly at the fixed dot which
    represents your own vessel, you are in some degree of trouble, and you
    would be able to calculate with some accuracy when the collision will occur
    (though you might perhaps be better employed doing something about it,
    With better fortune, the relative velocity of the other vessel will not be
    pointing directly toward you, and it can be extrapolated to show where and
    when it will come closest to your dot at the centre. This will show the
    relative distance and bearing of the other vessel at that moment of closest
    All of this applies only to the situation in which both vessels maintain
    course and speed, which I think was what Russell was asking for. Once
    either vessel starts to make any manoeuvre which may be required, a new
    situation is created.
    Russell's question doesn't mention radar, so I have presumed that he is
    referring to the situation of a vessel without radar assistance. In that
    case, the relative velocity plot described above will often be of academic
    interest only, because the course and speed of the other vessel will seldom
    be known. If it is known, perhaps as the result of a dialogue on VHF, then
    such a relative velocity plot is called for.
    Instead of making a physical plot on paper, it is of course possible to
    work the same calculation all out by trig, but my view is that the
    resulting loss of immediacy and feel for what is going on would be
    Russell adds-
    >I'm a bit suprisesd that few, if any, small-boat navigation books discuss
    Yes, I agree. Perhaps it's really because collision-avoidance techniques
    are considered to be more a matter of pilotage than of navigation. Dutton's
    Navigation and Piloting (mine is 12th ed., 1969) devoted a whole chapter to
    "Graphic solutions for Relative Motion problems", which was highly
    radar-related, and aimed at the special needs of Naval operations.
    Yours, George Huxtable.
    George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    Tel. 01865 820222 or (int.) +44 1865 820222.

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