Dip of the horizon
How far below eye level the visible horizon lies from a height, with refraction, and the navigator's 1.76′√h rule with its difference.
Planning and education aid. Not for primary navigation. Full disclaimer
The horizon dips 5.68′ below eye level.
- This assumes a smooth Earth: terrain, buildings, and trees can block the view or the signal.
- Rule of thumb 1.76′√h
- Rule difference
Provenance
- Computed by
- navigation.los.dip 1.0.0, core 0.1.0
- Model
- Spherical Earth with refraction k = 0.13 (effective radius 7322989 m)
- Accuracy
- Exact for the model; abnormal refraction over water can change dip by several arcminutes
- Notes
- 1 shown with the answer
- Cites
- National Geospatial-Intelligence Agency, Pub. No. 9, The American Practical Navigator (Bowditch)
Something look off?
How we got thisFormula, worked example, sources, and proof
Model: Spherical Earth with effective radius R/(1 − k)
Accuracy: Exact for the model; abnormal refraction over water can change dip by several arcminutes
When to use this: Use this to correct a sextant altitude taken against the sea horizon: the horizon is below true horizontal by this angle, so it is subtracted from every observed altitude. It also answers how far below level the horizon sits from a bridge wing, a lookout, a lighthouse, or a cabin window, which is the angle a camera or a sight has to be depressed to put the horizon on the crosshair.
Limitations: It assumes a clear sea horizon and a uniform atmosphere. Over land the visible horizon is terrain rather than the sea, and this answer does not apply; the tool says so rather than guessing. Abnormal refraction over water — a temperature inversion, a cold sea under warm air — moves the real dip by several arcminutes, which is far larger than any rounding here, and no coefficient known in advance captures it. Read the rule difference carefully: most of it is the refraction coefficient, not the rule. The default k is 0.13, while the navigator's 1.76′√h assumes about 0.169, and at that coefficient the rule is within a quarter of a percent of the exact dip.
Worked example: Height of eye 10 m. Source: Bowditch (NGA Pub. 9) Table 12 and the 1.76′√h dip rule, which are the same rule: together they fix the effective radius at 7,666,208 m, and the exact dip on it is 5.5526′ against the rule's 5.5656′. It is golden test vector v001, and every build checks the tool still gives its answer within its tolerance.
You enter
- Height of eye
- 10 m
You get
- Dip
- 5.68′
- Rule of thumb 1.76′√h
- 5.57′
- Rule difference
- -0.12′
Review: Not yet independently reviewed by a geodesist.
Last verified: 2026-09-19, when a maintainer last confirmed this tool's sources at the issuer. See the sources ledger.
Status: version 1.0.0, core 0.1.0. See this tool in the verification report.
Checked against: 23 golden test vectors (download the test vectors, each with its source and tolerance). See how results are checked and every source.
Sources
- The American Practical Navigator (Bowditch), National Geospatial-Intelligence Agency, Pub. No. 9, 2024 edition. Volume II, Table 12 (Distance of the Horizon) and Table 14 (Dip of the Sea Short of the Horizon).