Two lights in one line — leading lights as a geometric diagram

One set of leading lights, seen from three places · a model, not a map

upper (farther) light
lower (nearer) light
scale corner in the frame: 20′ horizontally and 20′ vertically. The vertical is compressed approx. 3.2× to fit both lights.

Leading lights are two marks on one straight line. Along its extension they appear one above the other. From the side they move apart horizontally, and the nearer one shifts faster. Each frame is aimed at the upper mark, so only the difference in bearings is visible.

Without JavaScript the initial state is shown: an observer 25 m to the right of the axis. The controls need JavaScript; the explanation and sources below work without it.

Offset from the axis, |y|

Offset: 25 m

Left and right lie as many metres from the axis as you set. You choose the position with the buttons under the frames. The slider also works with the arrow keys, Home and End.

Plan view from above, the same axis as in the frames. Orange marker: the selected position; outlines: the other two. Bar by the marks: spacing d. Ticks at the bottom every 30 m from the axis. The plan is not to a single scale: across the axis it is magnified approx. 33× relative to the run of the axis. The lines of sight (solid to the lower mark, dashed to the upper one) therefore diverge more strongly in the drawing than in reality.

Angular separation Δθ 12.8′ Position +25 m

You are 25 m to the right of the axis. The lower (nearer) light lies 12.8′ to the left of the upper one. That is more than the conventional threshold ε = 1.5′; on a diagram with this threshold it would pass as separated.

You can change the threshold ε in the model parameters. This is the geometry of two points, not guidance for steering a vessel.

The same side, three mark spacings

Selected position: 25 m to the right of the axis. The angular scale is identical in all three frames; only the distance between the marks changes.

Short spacing · d = 120 mΔθ = 3.7′ · off the axis, above threshold εSensitivity: 1.5′ per 10 m of offset · threshold ε corresponds to ≈ 10.0 m

Your spacing · d = 500 mΔθ = 12.8′ · off the axis, above threshold εSensitivity: 5.1′ per 10 m of offset · threshold ε corresponds to ≈ 2.9 m

Long spacing · d = 1000 mΔθ = 20.7′ · off the axis, above threshold εSensitivity: 8.3′ per 10 m of offset · threshold ε corresponds to ≈ 1.8 m

Sensitivity is the increase in Δθ for every 10 m of offset. “Threshold ε corresponds to” is the offset at which Δθ reaches the threshold ε (small-angle approximation). A short baseline gives a smaller Δθ, so the same threshold covers a wider band of positions. This is a property of a diagram with a conventional threshold, not an assessment of a route.

Three relationships from the geometry of two points

Alignment is a shared bearing. Along the extension of the axis both marks lie in the same horizontal direction and stand one above the other in the frame. Being on the axis is geometric: the position y = 0 either holds or it does not.
The nearer mark shifts faster. An offset of y changes the bearing to the lower mark by about y/D₁ and to the upper one by y/D₂. To the right of the axis the lower mark shifts to the left relative to the upper one. The difference Δθ ≈ y·d/(D₁·D₂) grows with the offset and with the spacing.
Tolerance is a convention, not the axis. The threshold ε is a value of the diagram. When Δθ fits within ε, the position still lies off the axis. This page calls only y = 0 “the axis”.

Model parameters and formulas

Four synthetic values; each changes something visible in the frames, on the plan or in the captions. The height of the lower light above eye level is fixed (h₁ = 16 m).

Distance D₁ to the lower mark1600 m

Mark spacing d500 m

Upper mark higher by Δh18 m

Conventional diagram threshold ε1.5′

Sensitivity S₁₀ for your spacing: 5.1′ per 10 m of offset
Offset corresponding to threshold ε: ≈ 2.9 m
Upper mark higher than the lower by (Δφ): 21.3′
Smallest Δh for the upper mark to be higher: > 5.0 m

Δθ = |arctan(y/D₁) − arctan(y/D₂)| ≈ y·d/(D₁·D₂)difference of horizontal bearings to the two marks; D₂ = D₁ + d
S₁₀ = 10 m · d/(D₁·D₂) · 3437.75′/radincrease in Δθ per 10 m of offset (1 rad = 3437.75′)
Δφ = arctan(h₂/√(D₂²+y²)) − arctan(h₁/√(D₁²+y²))difference of angular heights; h₂ = h₁ + Δh; heights measured from eye level
y_ε ≈ ε · D₁ · D₂ / doffset at which Δθ equals the threshold ε (ε in radians)

Real pairs of marks and sources

Material from the earlier version of the page states that the list of maritime aids to navigation describes leading lights as pairs: a lower (front) beacon and an upper (rear) beacon. The same material names the Brzeźno, Westerplatte and Port Północny pairs in the Gdańsk area, according to the list of the Maritime Office in Gdynia. To be checked in the source

The model above is not fitted to any of them. The data for these pairs was not in the material, so no zeros or approximate values were entered:

Beacon spacing d: no data
Distances and heights: no data
Azimuth of the axis: no data
Current light characteristics: no data

The photograph of the upper Brzeźno beacon from the earlier version’s material (author according to that material: t.przechlewski, 19 October 2008) is not embedded here because the page works offline. Links: file page on Wikimedia Commons and CC BY 3.0 licence. The links were not checked in this step

Sources from the earlier version’s material

IALA · Design of Leading Lines (2005). According to the material: a definition of a leading line as a straight line defined by the alignment of lights or marks. No address in the material · to be checked
Urząd Morski w Gdyni · spis obiektów BON (2024) [Maritime Office in Gdynia · register of BON objects]. According to the material: the names and existence of the beacon pairs Brzeźno, Westerplatte, Port Północny. No address in the material · to be checked
Kapitanat Portu Gdańsk · informacje nawigacyjne [Harbour Master’s Office, Port of Gdańsk · navigational information]. According to the material: a mention of leading-lights marking in the description of the port area. No address in the material · to be checked

The descriptions of the sources above were not re-verified when this version was built.

Limits of the model

Distances, spacings and heights are synthetic parameters, not measurements of any existing leading lights.
The threshold ε is conventional; we do not attribute to it any normative or perceptual basis.
Simplifications: flat water, no curvature of the Earth, refraction or dip of the horizon; the horizon lies at eye level; Δθ calculated as the difference of horizontal bearings; the size and range of the lights are ignored.
The drawings are diagrams with openly different scales (compressed vertical in the frames, magnified side on the plan). They are not a map or a view from a real place.
This page is not a navigation aid and does not say where or how to steer a vessel. Current marking and rules are given only in official maritime publications.

What changed relative to the earlier version

View tabs were replaced by three frames at once, with the plan on the same axis beneath them.
Preset configurations were replaced by three spacings side by side.
Metric tiles and the table: the numbers were moved to the captions beside the drawings, and the formulas to the expandable parameters.
Saving in the browser and the clear button were removed because the state does not need to survive a refresh.
Kept: the formulas, the distinction between the axis and the threshold ε, the sensitivity, the height difference and the limits of the model.