# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**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 chosen. 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, instead). 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 approach. 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 counter-productive. Russell adds- >I'm a bit suprisesd that few, if any, small-boat navigation books discuss >this. 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.u-net.com George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. Tel. 01865 820222 or (int.) +44 1865 820222. ------------------------------