Of Laylines and Beats

A Google Earth track of the virtual boat attempting an 'Olympic' course.
A Google Earth track of the virtual boat attempting an "Olympic" course. The course is to go from the leeward mark upwind to the windward (or weather) mark, sail across to the gybe mark, gybe (obviously!) and sail down to the leeward mark. After that, sail up to the windward mark again and then straight downwind to the leeward mark.

On the left, you can see the track left by simulating the Beoga Beag navigation software. It’s a short, olympic course suitable for dinghies and smaller boats. For a dinghy race, the whole thing should take less than an hour so the legs are quite short.

This time, I set the simulation granularity a bit smaller, so the updates are more regular and there are more data points, which explains the curved route in some cases. You can click on the image for a slightly larger version. An Olympic course is known to sailors as a “triangle and sausage”, because you sail upwind to the top mark, turn onto a broad reach to the gybe mark, gybe around, and broad reach down to the leeward mark. From there you beat back up to the windward mark, and then turn dead downwind to the finish line at the leeward mark. A true Olympic course would put the start and finish line about a third of the way up the beat, shortening the first leg somewhat, but also making for an upwind finish as you would have to sail around the leeward mark and beat to the finish. I skipped all that!

You can see that the boat took off to Starboard (as is it’s tendency when all else is equal) and headed up on the beat. About a quarter of the way up, the boat decided the other tack was favoured. Let’s discuss this a little bit. From my previous postings (here and here) you can see I’ve been playing with different algorithms. The first algorithm just computed the best VMG to the mark, and that resulted in an upwind leg comprised of many, many small tacks. Not good! My second attempt was to only tack if the opposite tack was favoured by more than 5 degrees. This introduced a weird error I mentioned previously. This version poisons the VMG slightly. It takes the tack error (as discussed) and multiplies it by 0.1VMG and adds that to the VMG. This favours (by 20% of the VMG) the existing tack. As you can see, those initial legs up the ladder are quite good. Not perfect, but pretty good. Let’s take a moment to discuss the “laylines”. Imagine that the boat is at the gybe mark, and inexplicably it’s trying to get to the windward mark. The course is 45 degrees, or North-East. The wind is coming from the North in our diagram (that’s why the windward mark is due North of the leeward mark), so our TWA is -45. In other words, close-hauled on a port tack. The thing about it is, ignoring things like tide and leeway (which we’ll save for another day), the boat can reach the mark without needing to tack back and forth. The sailing term for this is “laying the mark.” The lay lines then are lines which stretch away from the windward mark in either direction, at opposite 45 degree angles (South-West and South-East). When we hit the layline, we need to tack. If we sail over the layline, we’ve overshot the mark. In other words, we’ve gone too far. It’s worth always having a mental picture of the laylines when you’re sailing a boat. A rule of thumb used by dinghy sailors, who don’t have any electronics on board, or even a compass, is to look over your backward-facing shoulder, and see if you can see the mark. If you can, you’re on the layline.

If you stare at the track above, you should be able to see the laylines in your head. What you’ll notice is that the virtual boat doesn’t go anywhere near the laylines. It sails quite conservatively up the middle of the course. There’s nothing wrong with this per se, but excessive tacking is a bad thing. The boat will slow to a crawl, and possibly stop during a tack, so you need speed going into the tack, and you need to minimize how many tacks you perform. Speaking of which, take a look at the track up towards the top of the beat. You’ll notice a lot of small tacks, as the error difference swings wildly, close to the mark. This is to be expected, unfortunately. The obvious solution to what we’re doing here, is to increase the poisoning percentage to the VMG. Instead of +/-10% of VMG, we need to make it closer to 30%. I will try this later today, and see how it improves things. It should bring the boat closer to the laylines and should remove a lot of the excess tacking at the top of the course.

On the down side, however, when the legs are extremely long (on the order of 1,000nm), we don’t want the boat sailing off to the layline on either side because the ocean isn’t big enough. Henry suggested using more waypoints to reduce the leg distance, and that’s worth doing. However, I also want to add a VMG adjustment which is related to the distance-to-mark (or DTM). Up close to the mark, I’d like the boat to stay on one tack a bit more, but don’t shoot past the layline. Further from the mark, I’d like it to not be too aggressive in terms of how much of the course it uses. The way to do this is to take the DTM and use it as the percentage adjustment. For example, if we take modified formula (VMG + 1)KTe, K in the above example was 0.1. If we make it 1/Log10(DTM) for example, then as we get closer to the mark, K increases, which poisons the opposite tack a lot more but it avoids dramatic and wide tacks on longer distances.

What’s also interesting in the picture is the curve in the downwind leg. It’s actually correct! But more about that, later…