A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2018 Jan 19, 10:27 -0800
Stan mentioned a textbook problem that is often described in articles on great circle sailing. The idea is that you are sailing in moderately high latitude, like from Tokyo to Seattle, and you decide that you want to limit yourself to latitudes below 47°N because of icebergs or submarines or sea monsters or something... This is a typical, typical textbook problem, and the solution in textbooks is usually something like this: you draw out the great circle from start to end, then you excise that portion that passes north of 47 and replace it with parallel sailing along that latitude. And there ya go. Problem solved. But is there any reason mathematically to use this construction even for the absolute problem as given? Is that a rational solution or a ritual solution? And does the absolute problem with a perfect go/no-go condition at 47°N have any practical meaning at all?
Here's a little coding project for those of you who enjoy coding:
Suppose I create a cost function for this trans-pacific voyage based on latitude. South of 47°N, I can sail at my normal speed, let's say 24 knots. North of that latitude I reduce speed at the rate of 1 knot for every 10' north of that latitude, bottoming out at 1 knot (if I am in latitude 47°30' N, my speed is 21 knots... if I reach latitude 49°N, my speed is reduced to 12 knots and above 51° my speed is dead slow, one knot). Note that the textbook case has a cost function where the vessel's speed drops essentially to zero the instant we cross 47°N. Given a more general cost function like this, what is the quickest passage you can make? One approach to coding this would be to start with pure great circle waypoints and then randomly adjust those points north or south until the net time settles to a minimum. How "stable" and how sensitive is the optimal path? What are the waypoints if plotted for every five degrees of longitude, and how do those compare with other solutions to this sort of problem?
The nice thing about a model like this is that it's very easy to extend. For example, if there's a major storm on your route, you can create a region with reduced speed centered on the storm at its forecast positions. An optimization algorithm will automatically route around the storm to save time. Note that the reduced speed in calculations like this does not have to equate literally to the speed of the vessel. It is a measure of cost. Time is money in shipping, so there is generally a simple proportionality of any expense to time (and therefore inversely proportional to effective speed).
Route optimization is real navigation. It's a function that mariners pay for...