Here’s a quick one from today’s practice race. On the downwind leg from Ile d’Aix to Ile Madame, the direct line put you at a twa of 148°. Normally with no shift, the straight line is the fastest, but looking at the polar we can see that 148° is in the middle of a “dent” in the curve.
The black arrow on both the above map and the polar diagram below shows the direct line. We need to avoid that dent to sail the leg as fast as possible, so we “bridge” the gap with the two angles that are tangent on either side. In this case they work out to around 139°. and 167°. We can sail the higher angle until we reach the layline that will take us to the turn on the deeper one, shown by the pink arrows.
How much faster is it? You can find the average speed over and two directions on the polar by connecting the points with a straight line. The yellow line above bridges the “humps” on the polar (but is offset a bit for clarity) you can see that the two-angle strategy gains about a 1/3 of a knot made good. Gain on the leg is around 80 seconds or 0.16 nm!