Hi All!

I am new to the MHF so greetings everybody. I am a Search and Rescue Mission Coordinator for the UK Coastguard but I am currently working with the British Antarctic Survey as a Boating Officer in the Antarctic. Part of my job is to implement a Search and Rescue framework for the maritime operations undertaken here and as part of this I am writing a Maritime Search Planning handbook. The purpose is to allow personnel with little experience plan a quantified "rapid response" search using the "Expanding Square" search pattern. The planner will calculate and plot a square area on a chart within which he or she will expect the missing person or craft to be drifting. The search and rescue unit will proceed to the exact centre of this square and begin navigating an expanding square pattern with an exact initial leg length and subsequent leg lengths and 90 degree alterations of course to the right or left. The Initial leg length is worked out using tables of existing data. The spacing between adjacent legs will be equal in distance to the length of the first leg and is called the Track Spacing. The pattern goes like this: Lifeboat proceeds North from the middle of the search area for 1 x track spacing, alters course right by 90 degrees, proceeds east for 1 x track spacing, alters course by 90 degrees right, proceeds South for 2 x track spacing, 90 degrees right, west for 2 x track spacing, 90 degrees right, North for 3 x track spacing and so on....... The result is an expanding square pattern with fills the entire square search area leaving no areas within the area further from the lifeboats track than the track spacing (known as a coverage factor of 1)

The problem I would like you guys to help with is this:Is there a formula which will tell you how many legs are required to complete the search area to a coverage factor of 1 having been given the Area of the Search Box and the track spacing (aka the initial track spacing)?

I have worked for weeks on this problem and in practice all my theories are marginally inaccurate or only work for certain size area and track spacings. My favoured theory was that the answer lays in the length of one side of the area divided by the track spacing, multiplied by the number of legs it takes to achieve one more advancement along the length of the side (which appears to be 4, except the first 360 degree rotation which takes 5 legs to complete) however this appears to simplistic as when I draw test areas to scale on graph paper the formula does not work and gives inaccurate answers.

Some common parameters would be:

Search Areas: 2nm2, 5nm2, 10nm2, 12nm2, 18nm2

Track Spacings: 0.1nm, 0.3nm, 0.5nm, 1nm, 1.25nm

Use the above figures in any combination as they are mutually exclusive and may be combined.

I am no Maths expert so, I am hoping this may be quite simple to someone with a good level of knowledge in Mathematics.

Kind Regards,

Matthew Kenney

KEP Research Station,

Antarctica.