Rhumb Lines Info

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A ‘rhumb line’ (or loxodrome) is a path of constant bearing, which crosses all meridians at the same angle.

Sailors used to (and sometimes still) navigate along rhumb lines since it is easier to follow a constant compass bearing than to be continually adjusting the bearing, as is needed to follow a great circle. Rhumb lines are straight lines on a Mercator Projec­tion map (also helpful for naviga­tion).

Rhumb lines are generally longer than great-circle (orthodrome) routes. For instance, London to New York is 4% longer along a rhumb line than along a great circle – important for avia­tion fuel, but not particularly to sailing vessels. New York to Beijing – close to the most extreme example possible (though not sailable!) – is 30% longer along a rhumb line.

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Distance Formula

Since a rhumb line is a straight line on a Mercator projec­tion, the distance between two points along a rhumb line is the length of that line (by Pythagoras); but the distor­tion of the projec­tion needs to be compensated for.

On a constant latitude course (travelling east-west), this compensa­tion is simply cosφ; in the general case, it is

$Δφ/Δψ where Δψ = ln( tan(π/4 + φ2/2) / tan(π/4 + φ1/2) )$ (the ‘projected’ latitude difference)

Formula: $Δψ = ln( tan(π/4 + φ2/2) / tan(π/4 + φ1/2) )$ (‘projected’ latitude difference) $q = Δφ/Δψ (or cosφ for E-W line)$ $d = √(Δφ² + q²⋅Δλ²) ⋅ R (Pythagoras)$ where φ is latitude, λ is longitude, Δλ is taking shortest route (<180°), R is the earth’s radius, ln is natural log

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Bearing Formula

A rhumb line is a straight line on a Mercator projection, with an angle on the projec­tion equal to the compass bearing.

Formula: Δψ = ln( tan(π/4 + φ2/2) / tan(π/4 + φ1/2) ) (‘projected’ latitude difference) θ = atan2(Δλ, Δψ) where φ is latitude, λ is longitude, Δλ is taking shortest route (<180°), R is the earth’s radius, ln is natural log

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Destination Formula

Given a start point and a distance d along constant bearing θ, this will calculate the destina­tion point. If you maintain a constant bearing along a rhumb line, you will gradually spiral in towards one of the poles.

Formula: δ = d/R (angular distance) φ2 = φ1 + δ ⋅ cos θ Δψ = ln( tan(π/4 + φ2/2) / tan(π/4 + φ1/2) ) (‘projected’ latitude difference) q = Δφ/Δψ (or cos φ for E-W line) Δλ = δ ⋅ sin θ / q λ2 = λ1 + Δλ where φ is latitude, λ is longitude, Δλ is taking shortest route (<180°), ln is natural log, R is the earth’s radius

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Mid-point Formula

This formula for calculating the ‘loxodromic midpoint’, the point half-way along a rhumb line between two points, is due to Robert Hill and Clive Tooth1 (thx Axel!).

Formula: φm = (φ1+φ2) / 2 f1 = tan(π/4 + φ1/2) f2 = tan(π/4 + φ2/2) fm = tan(π/4+φm/2) λm = [ (λ2−λ1) ⋅ ln(fm) + λ1 ⋅ ln(f2) − λ2 ⋅ ln(f1) ] / ln(f2/f1) where φ is latitude, λ is longitude, ln is natural log