The fastest way is not the straight one

The mark is directly upwind. The boat cannot sail there — the no-go zone is in the way — so it must sail some course to one side, then tack and sail the other. The question is which course, and the intuitive answer is “as close to the mark as the boat can manage”.

That answer is wrong, and it is wrong by a wide margin. The best course is nearly three times further off the wind than the closest the boat could point, the boat sails about seventy per cent further than the straight-line distance, and it arrives first.

The quantity that matters is a component #

The previous rung produced the polar: the boat’s speed in every direction, as

$$\frac{V_b}{V_t} = \frac{\sin(\beta - \lambda)}{\sin\lambda}$$

with $\beta$ the course off the true wind and $\lambda$ the sum of the two drag angles.

Racing does not reward speed. It rewards progress towards the mark, and if the mark is dead upwind that is

$$V_{mg} = V_b\cos\beta$$

Two competing effects, and the whole essay is their product. Bearing away — increasing $\beta$ — makes the boat faster, because $\sin(\beta - \lambda)$ grows. It also makes each unit of speed count for less, because $\cos\beta$ shrinks. Somewhere between them is a best course, and it is nowhere near either end.

The closed form, which is one line #

Maximise $\sin(\beta - \lambda)\cos\beta$ over $\beta$. Differentiating,

$$\cos(\beta - \lambda)\cos\beta - \sin(\beta - \lambda)\sin\beta = 0 \quad\Longrightarrow\quad \cos(2\beta - \lambda) = 0$$

which gives

$$\beta_{\text{best}} = 45° + \frac{\lambda}{2}$$

Running downwind is the same calculation with the sign of the cosine reversed, and gives

$$\beta_{\text{best}} = 135° + \frac{\lambda}{2}$$

Both are startlingly simple, and both are worth reading as instructions. The optimum beat is at forty-five degrees, corrected by half of whatever the boat’s foils cost it. A perfect boat with no drag at all would beat at exactly forty-five degrees to the wind — not at zero, not at the closest it could point, but at forty-five — and every degree of drag angle pushes it half a degree further off.

For the reference boat, $\lambda = 20°$, so the answers are 55° and 145°.

What was computed, and why it is a search #

The optima are located by golden-section search on the computed polar, over the whole legal range of courses, with no knowledge of the answer. They come out at

• beating: 55.000000°, against $45° + \lambda/2 = 55.000000°$;
• running: 145.000000°, against $135° + \lambda/2 = 145.000000°$.

That is the site’s habitual arrangement — find it, then check it against the closed form — and it is doing more work here than it looks. The polar is built from a force balance and a velocity triangle, neither of which knows anything about $\cos(2\beta - \lambda)$. Two entirely separate pieces of trigonometry have to agree, and if the speed formula had a sign wrong somewhere the search would find a maximum in the wrong place while still producing a perfectly reasonable-looking curve.

The assertion also checks that the best speed made good is positive, which catches the failure where the optimiser has locked onto the wrong branch and is confidently sailing backwards.

What the extra distance costs, and why it is worth it #

At 55° off the wind, the distance sailed to a mark dead upwind is $1/\cos 55° = 1.74$ times the direct distance. Seventy-four per cent further.

The boat is making 2.29 times the wind speed on that course, of which the component towards the mark is 0.962 times the wind speed. Compare it with two alternatives:

A boat sailing at 25° is pointing nearly at the mark and takes four times as long to get there. That is the essay’s refutation, in a table, and it is why “pointing” and “footing” are the perennial argument in a racing fleet: the curve near the optimum is flat, so both are nearly right, and either taken to an extreme is disastrous.

A second thing the table settles is the shape of the penalty. The curve of speed made good against course is not symmetric about its maximum: falling twenty degrees below the optimum costs 20 per cent, while rising fifteen degrees above it costs under three. So a crew unsure of the right angle should err by bearing away, and that asymmetry — rather than any general preference for speed over height — is the defensible version of the “when in doubt, foot” advice.

The reason for the asymmetry is visible in the polar. Below the optimum the boat is approaching the no-go zone, where the speed collapses to zero over a few degrees; above it the boat is heading towards its fastest point of sail, where the speed is still climbing and only the cosine is punishing. One side of the maximum has a cliff on it, and the other does not — the same lopsidedness a lift curve has near the stall, and for a related reason.

Notice what the figure does not show: how many tacks. The total distance sailed is the same whether the boat makes one long tack and one short one or twenty of each, because every leg is at the same angle to the wind and the geometry of a zig-zag between two points depends only on the angle. Tacking more costs the manoeuvres themselves, and nothing else in this model. Everything a tactician does about when to tack is therefore about wind shifts, current and other boats — none of which is here.

Downwind, where the model has just given out #

The running optimum at $135° + \lambda/2$ is the more surprising of the two, because dead downwind looks like the one course a sailing boat should have no trouble with.

The previous rung showed why it does. At $\beta = 180°$ the apparent wind goes to zero, there is no force, and the model’s apparent answer of exactly wind speed is nothing divided by nothing. The solver refuses that course. So a boat wanting to go downwind must sail across the wind and gybe, for the same reason it must tack upwind, and the optimum is 145° rather than 180°.

The extra distance is only 22 per cent, because 145° is much closer to 180° than 55° is to 0°. And the speed made good — 1.96 times the wind — is twice what the dead-downwind artefact suggests. A boat that gybes downwind is not making a small tactical gain; it is doubling its progress.

This is why fast boats no longer run. A displacement yacht with a spinnaker at 170° is close to the old picture of downwind sailing; a foiling catamaran gybes through 120° and is never within fifty degrees of dead downwind at any point in the leg.

The two constructions belong together: the polar says where the best course is and the triangle says what the boat is doing on it, and neither is derivable from the other by looking.

The same shape of problem, three times on this site #

Maximising a product of two things that fight each other is not a sailing result. It is the shape of half this field, and putting the cases side by side makes the pattern visible.

Betz’s limit is exactly this. Slowing the air more takes more energy per kilogram and lets fewer kilograms through, so the power is a product with a maximum in it — pinned at zero at both ends, with the best a third of the way along. The algebra is a cubic rather than a trigonometric identity, and the reasoning is identical.

A glider’s best glide is the same again. Flying slowly costs induced drag and flying fast costs friction; the sum has a minimum, and the speed at which the lift-to-drag ratio is greatest is the speed that gets furthest. A glider pilot flying “as slowly as the aircraft will go” is making exactly the mistake the boat sailing at 25° is making, and the correct answer is again well away from the intuitive end of the range.

A wind-turbine blade’s tip-speed ratio is a third instance. Gearing up reduces the swirl loss and increases the tip’s own drag and noise, so the useful band is an interior optimum rather than an asymptote.

The geometric version in that caption is worth having, because it makes the closed form obvious after the fact. Speed made good is the projection of the polar’s radius onto the wind axis, so the best course is where a line perpendicular to the wind touches the polar. For a curve as simple as $\sin(\beta - \lambda)$ that tangency has a closed form; for a measured polar with kinks in it — a real boat that starts foiling at some speed — it does not, and the tangent construction is what a computer does instead.

Where the optimum moves #

The two closed forms are worth reading as a sensitivity, because they say what improving a boat buys.

$\beta_{\text{best}} = 45° + \lambda/2$ moves by half a degree for every degree of $\lambda$. So a boat with a 50° combined drag angle beats at 70° off the wind, and one with 8° beats at 49°. The better boat sails closer to the mark as well as faster, and its advantage compounds: at $\lambda = 8°$ the speed made good is 3.2 times the wind against 0.42 for the 50° boat, a factor of seven and a half.

That steepness is the same one the previous rung found in $1/\sin\lambda$, and it is worth stating in the general form because it is what makes this whole model useful: everything about a boat’s performance is a steep function of one angle, and that angle is a lift-to-drag ratio.

What a tide does to all of it #

The whole calculation above is conducted in one frame, and the water it is conducted in has been assumed to be still. On most racing courses it is not, and a tide of two knots against a boat making six is not a correction — it is a third of the answer.

The first thing to get right is which frame each quantity belongs to. The polar belongs to the water. A hull’s resistance is set by its speed through the water and a sail’s force by the wind over the water, so the model’s $V_t$ is not the wind a shore station measures: it is the ground wind minus the current, as vectors. A two-knot tide running with a ten-knot wind makes eight knots over the water; running against it, twelve; running across it, ten and a half from a noticeably different direction. Every number in the tables above is a function of that wind and not of the other one.

The mark, meanwhile, belongs to the ground. So in the frame the boat is actually sailing in, the mark is moving — drifting at the current’s velocity reversed — and the direction to it changes over the course of the leg. Sailing to a moving target is a different problem from sailing to a fixed one, and it is the reason a tidal beat is planned rather than steered.

Two consequences follow, and the second is the interesting one.

The tacks stop being equivalent. The model above is symmetric about the wind, so port and starboard are mirror images and the choice between them is free. Add a cross-tide and it is not: the two tacks carry the boat towards a mark drifting sideways, so one of them closes on it and the other does not, and the asymmetry is a property of the course rather than of the boat.

And the folklore about it is a frame error. The received wisdom is that a tide on the lee bow is fast, because it pushes the boat to windward. In a uniform stream that cannot be right: a current that carries the boat sideways carries the mark sideways with it by exactly as much, and nothing about the geometry between them has changed. What is real is the first effect above — the tide alters the wind over the water, and on one tack that alteration is favourable — together with anything the stream does non-uniformly, which is most of what a real estuary does.

One thing does survive intact. The claim that the number of tacks makes no difference holds in a uniform current too, because the water-frame track’s endpoint is still fixed once the crossing time is known. It is a varying current that makes the timing of a tack matter, and that is a tactical question rather than one this model contains.

Where the model stops #

Everything the previous rung could not do, this one cannot either: no hull speed, no heeling, no waves, nothing unsteady. Three more that belong specifically to the racing question:

The polar is symmetric and real ones are not. A real boat’s best beating angle on port and starboard tack differ, and its polar changes with wind strength — a boat overpowered in a breeze must depower and its drag angle worsens. What that does to the optimum is not what $45° + \lambda/2$ suggests, because the worsening depends on the course: solved on every course, the best beat moves closer to the wind while the rig is being flattened, and away from it only once the sail is nearly flat.

Nothing here is about tactics. Wind shifts are the entire subject of upwind racing and this model has a steady uniform wind. A boat sailing the “wrong” angle to be on the lifted tack is doing something this calculation cannot see and which usually matters more than the calculation does.

And the optimum is flat. Between 50° and 62° the speed made good varies by under two per cent, which is smaller than the error in anything a crew can measure. The right reading of this model is not “sail at 55.000°”; it is that the answer is nearer 55 than 30, that the flatness is real, and that a boat kept anywhere in the broad band is giving away nothing worth having.

That last point deserves the same treatment the site gives every other instrument. A displayed speed made good is computed from a measured boat speed and a measured wind angle, each of which has its own error, and the quantity being maximised is flat near its maximum — so the displayed optimum wanders by several degrees from one moment to the next while the true one does not move at all. It is a milder version of the trouble an airspeed indicator has: the instrument is faithfully reporting something, and what it reports is not quite the thing the reader believes they are looking at.

The same caution applies to using this model on a real boat at all. Its single parameter is a combined drag angle that nobody measures directly; it is inferred from a measured polar, which means the model is being fitted to the data it is then used to explain. What saves it from circularity is that one number predicts the whole polar and both optima — so it is falsifiable, and stating what has been assumed and at what conditions is what makes the difference between a model and a curve fit.

Who found it, and when #

The optimisation is not attributable to anybody in particular; it is what happens when a polar diagram and a mark are put on the same page, and polar diagrams for yachts date from Curry and Davidson in the 1920s and 30s. The phrase velocity made good and its computation belong to the instrument era — the 1970s, when a boat could measure its own speed and the wind’s direction fast enough to display the product.

That gap is the interesting part. The optimum beat angle is a piece of nineteenth-century trigonometry that nobody wrote down because nobody had a polar to apply it to, and once instruments made the polar measurable the answer became a number on a display in front of the helmsman. It is one of the few cases in this subject where a result was practical before it was famous: generations of sailors found the right angle by feel, and the calculation arrived afterwards to explain why the feel was right.

Where the ladder goes #

The sailing anchor closes on an unusual note for this field: a result with nothing borrowed in it at all. No correlation, no measured separation angle, no fitted constant — two angles, a triangle, and two derivatives.

The field turns next to a place where the same style of argument is used on something alive. A blood vessel or a tree branch carries a flow and costs two things to own: the power to push fluid along it and the price of the tissue itself. Minimising the sum gives a best radius, the best radius makes the flow proportional to the cube of it, and the rule that follows — that the cube of a parent vessel’s radius equals the sum of the cubes of its branches — is a statement about a photograph that came out of a cost function.