by Lester Gilbert

VMG is the theory behind finding the pointing angle that maximizes your progress towards the windward mark. It isn’t the angle at which your boat sails the fastest, and it may not be the highest angle at which you can point, if you want to. We’ll need it, along with the course theorem, to find out the optimum pointing angle for maximum VMG.

To start, we’ll plot a graph of our boat speed at various headings relative to the true wind. We’ll do this as a “radar” or “polar” chart (hereinafter referred to as just polar) and an example is shown in Figure 1. At a heading of 30 degrees, or 330 degrees, it is the same, our boat makes about 1 m/s. At a heading of 90 degrees (and 270 degrees) our boat makes almost 1.5 m/s. At angles closer to the wind than 30 degrees, our boat speed decreases precipitously and goes to zero. The polar of our boat’s performance has characteristic lobes of highest speeds at reaching headings.

Figure 1. Polar chart of boat speed for various headings

This polar becomes very useful in telling us the best heading, which gets us to the windward mark in the least time; we simply find that point on the polar that is closest to the mark. This is illustrated in Figure 2, where the dotted line is the line we can draw that is closest to the mark, while just touching the polar.

Figure 2. Optimum heading for highest VMG

In our example, the point at which this closest line touches the polar corresponds to a heading of 39 degrees. This is the optimum heading, which gives us the best “velocity made good,” or VMG, towards the windward mark. At this heading, reading off the polar, our boat speed is 1.13 m/s, and our VMG is 0.88  m/s.

The polar also tells us what happens if we pinch and sail closer to the wind than the heading that optimizes the VMG. For example, if we point at a heading of 30 degrees, the polar tells us that boat speed drops to 0.95 m/s, and that although we are pointing better towards the mark, our VMG has dropped to 0.82 m/s. Figure 3 illustrates this and also illustrates what happens if we foot, that is, sail lower but with higher boat speed.

Figure 3. Sailing high and sailing low

In our example, let’s suppose we foot to a heading of 50 degrees, where the boat sails faster at 1.28 m/s. It turns out that our VMG has dropped to 0.82 m/s, the same VMG as we’d get sailing at 30. Although the boat is sailing much faster, it is just as slow getting to the mark.

It is interesting to look at the dotted VMG line of Figure 2 and to notice that in fact there is a region of the polar where VMG is within 0.05 percent of maximum, somewhere between headings of 35 and 42 degrees. If the optimum heading under perfect conditions is 39 degrees, and pinching in order to point at 30 degrees is significantly slower, we now know we are finely balanced on a knife edge. Maybe we could pinch to 35 degrees, but after that we are losing out. Those here, who can tell if they are sailing at 30 instead of 35 degrees—hands up? I thought not…

On the other hand, the knife edge is somewhat blunter when footing and sailing low. Maybe we could foot to 42 degrees instead of trying to maintain 39 degrees, particularly if we have waves or gusty wind. Footing at 50 degrees is as bad as pointing at 30 for VMG, but we are much less likely to lose out, and may even gain. A wave or a heading gust at 30 can trip us over the cliff and kill both boat speed and VMG, whereas the same event at 50 would merely marginally increase boat speed and marginally lower VMG.

Our final challenge is to trim the sails to give us the balance of lift and drag we need at the heading we wish to sail. Are we going to maximize drive? Maximize efficiency? Or something else?