The start line · Nordic Stars

Two pings, a wind direction and a position. Everything the pre-start tells you — distance to the line, which end pays, whether you are over — comes out of those four numbers.

What the line is made of

A start line is two points: the committee boat and the pin. You record them by pinging each end — sailing to it and pressing a button. From that moment the platform stores the line as a segment between two latitude/longitude pairs, and everything else is geometry.

The wind direction is the third input, and it is the one that turns geometry into tactics. A line has no favoured end until you know where the wind is coming from.

1 Flat water, flat maths

The Earth is round; a start line is 100 metres long. Over that distance the curvature is irrelevant, so the platform works in metres on a flat plane, converting from degrees with one constant and one correction:

north = Δlatitude × 111 320 east = Δlongitude × 111 320 × cos( latitude )

Read as a sentence: a degree of latitude is the same length everywhere; a degree of longitude shrinks as you go toward the poles.

The cosine is taken at the mean latitude of the two ends. Skip it and a line measured in Stockholm comes out nearly twice as long as it is, east–west, which would be a spectacular way to lose a start.

2 How far you are from the line

Distance to the line is the perpendicular distance to the infinite line through both ends — not the distance to the nearer end, and not the distance to the midpoint. It is what you would sail if you pointed straight at the line and went.

distance = ( line_east × boat_north − line_north × boat_east ) ──────────────────────────────────────────────── line length

Read as a sentence: how far off the line’s own axis the boat has strayed, measured square to it.

The vectors are pin-to-committee and pin-to-boat. The numerator is a cross product, and a cross product does something useful for free: it comes out signed. Which side of the line you are on is contained in whether the answer is positive or negative.

Which sign means over

A raw cross product only knows left and right, and left and right depend on which end you happened to ping first. That is no use at all when the answer means over the line.

So the wind decides. The platform checks which side of the line the upwind direction points toward, and orients the sign so that:

Sign	Meaning
positive	On the pre-start side — the side the fleet sits on, facing the line. The number is how far you have to go.
negative	On the course side. Over early: the number is how far back you have to come.

Being over the line is not a separate calculation. It is the sign of the distance you were already being shown.

Which is why the readout never disagrees with itself: there is no threshold to cross, no second rule that could be tuned differently, and no moment where the distance says one thing and the status says another. Two ends pinged at the same spot give no line at all, and the platform answers with nothing rather than a number.

3 Which end pays

Line bias is the answer to a single question: is one end of this line closer to the windward mark than the other? Project the pin-to-committee vector onto the upwind direction and you have it, in metres:

upwind = ( sin TWD , cos TWD ) bias metres = east × sin TWD + north × cos TWD

Read as a sentence: how much further upwind the committee end is than the pin — negative if it is the pin that is further upwind.

A bias of +12 metres means starting at the committee boat puts you twelve metres closer to the top mark than starting at the pin. That is a real, cashable advantage, and it is why the boat end of a square-looking line can still be worth fighting for.

Bias in degrees

Metres are the exact unit but degrees are the sailing one, because degrees are comparable between a 100-metre dinghy line and a 400-metre grand-prix one. The conversion is the angle whose sine is that ratio:

bias degrees = asin( bias metres / line length )

Read as a sentence: how far the line is tilted away from square to the wind.

A perfectly square line is 0°. Twelve metres of bias on a 100-metre line is about 7°, and 7° is the same tactical fact on any line length — which is what makes it the number to say out loud in the pre-start.

The ratio is clamped to the range a sine can take before the arcsine is applied. It is a guard against floating-point arithmetic wandering a hair past 1 on a line that is exactly square-on, where the answer would otherwise be no number at all — a wholly avoidable way to lose a readout at the worst possible moment.

The favoured end, and the neutral band

bias > +1 m committee end favoured bias < −1 m pin end favoured otherwise neutral

Read as a sentence: inside a metre either way, the line is square and the platform says so.

That one-metre band is not timidity — it is honesty about the input. A ping is a GPS fix taken from a moving boat, and a metre is comfortably inside its uncertainty. Without the band, a square line would flicker between “pin favoured” and “boat favoured” on noise alone, all through the sequence, and a readout that changes its mind every few seconds teaches a crew to ignore it.

4 Which ping counts

Ends get re-pinged. The committee boat drags its anchor, the race officer moves the pin, or somebody pings the wrong end and does it again properly.

So for any moment being asked about, each end independently uses the most recent ping at or before that moment. Pings after it are not consulted — which is what lets the debrief replay a start against the line as it stood at the gun, rather than against the line as it was eventually corrected. An end nobody pinged before that moment has no position, and there is no line.

5 Two surfaces, one formula

This geometry runs in two places: on the tablet during the sequence, and in the debrief afterwards when you replay the start. Those are different programs, in different languages, on different devices.

Both implementations are tested against one shared set of cases — the same inputs, the same expected outputs, agreeing to within a tenth of a metre and a tenth of a degree.

The expected numbers in that set were not derived twice; they were printed once by the original implementation and copied verbatim into both. A formula may only change in every home at once, and the shared cases exist precisely to fail when somebody tries to fix one of them on its own.

The point of all that machinery is a single sentence you should be able to rely on: the distance-to-line you were shouting from the rail at T−30 is the same distance-to-line the debrief shows you that evening. If it ever were not, that is a failing test rather than a discovery on the water.

6 What the start report draws on top

The start report prints a plan view of the last three minutes with the line on it. Two rules govern what is drawn, and both follow from this page rather than adding to it:

The line is the one that was pinged. The report draws the recorded pin and committee-boat positions and the line between them. A line re-pinged after the gun is not applied backwards to a start that was sailed against the old one.
The distance behind the line at the gun is the recorded one. It is read from the gun event as it was logged, not recomputed from positions. The two would agree to a tenth of a metre, and the recorded value is the number the crew saw.

A race with no line ping and no recorded distance gets neither: the report says so in a line of its own rather than drawing a line nobody laid. The plan view still draws the track, because the boat was there.