Abstract
On a sphere the shortest path between two points is a great circle. Navigators nonetheless chose the longer one. This paper asks how the longer path can be the one chosen——the answer is that what is held constant is not distance but heading. No new mathematical theorem and no new law is claimed.
First, Tokyo to London the rhumb line is $\mathbf{18.1512}$ per cent longer than the great circle (Section 2).
Second, from the same Tokyo, the route to Sydney differs by $\mathbf{0.0043}$ per cent (Section 2).
Third, this is the core of the paper. What sets the difference is not distance but how far the route runs across latitude (Section 3).
Fourth, along one meridian and along the equator the difference is exactly zero (Section 3).
Fifth, halfway round the world at $70$ degrees north the difference reaches $\mathbf{53.9091}$ per cent (Section 3).
Sixth, the separator is whether heading is held constant or distance minimised (Section 5).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed——great-circle distance, the rhumb line (loxodrome), the Mercator projection, and spherical trigonometry are all standard. We do not build geodesy——the Earth is taken as a sphere of radius $6371.0088$ km; geodesics on an ellipsoid (Vincenty) are not entered. Flattening changes things by an order of magnitude less than the differences here, though it matters in precise navigation. We do not teach navigation——voyage planning, the placing of waypoints, and corrections for current and wind are not entered. They are mentioned and no more. We do not discuss projections in general——properties of the Mercator projection are used, but no general theory of projections is entered. We do not adjudicate history——who first used the loxodrome and when is not entered. We do not treat aviation——real flight paths with upper winds and jet streams are not the subject. Relation to earlier papers: Paper 234 showed “distance” to be six things, one of which satisfies the triangle inequality backwards——its question was what may pass as a distance. This paper holds one distance fixed on one sphere and asks why the longer route is chosen. Not a question about definitions but about choosing. Paper 300 showed whether two things share a root is decidable——by that test the two routes here have distinct roots. Paper 205 showed the shortest enclosure is not always a disc——there too “shortest” departed from naive intuition. Paper 328 showed the same ellipse being three months out——both belong to the sea and sky the author began from. What is added is computing both for five real routes and tabulating the differences, identifying the two cases in which the difference is exactly zero, showing that east-west travel at high latitude reaches $53.9091$ per cent, and putting the separator on what is held constant.
1. Notation and the Question
Setting: Two ways of drawing a path between two points on a sphere.
| Name | Definition | How the helm is used |
|---|---|---|
| Great circle | the shortest path on the sphere | the heading changes continually |
| Rhumb line (loxodrome) | the line crossing every meridian at one angle | the heading is held constant |
Observation 1.1 That the great circle is shortest is a theorem of spherical geometry. So a rhumb line is never shorter——when equality holds is half of the question here.
Observation 1.2 And still the sailing ships ran rhumb lines. Because one need only hold the compass needle on a single mark——the reason for choosing a path that is not shortest lies somewhere other than in its length.
Observation 1.3 The way of measuring is fixed first. The Earth is a sphere of radius $R=6371.0088$ km; the great circle from the spherical law of cosines, the rhumb line as a straight line on the Mercator projection——flattening is ignored, being an order of magnitude below the subject.
Question: How much is lost, and when is nothing lost?
2. Two Lengths for Five Routes
Calculation 2.1: Compute both for five real routes (computed in this paper).
| Route | Great circle [km] | Rhumb line [km] | Difference |
|---|---|---|---|
| Tokyo $\to$ Sydney | $7825.83$ | $7826.17$ | $\mathbf{0.0043}$ per cent |
| Tokyo $\to$ Singapore | $5311.23$ | $5321.24$ | $0.1884$ per cent |
| Lisbon $\to$ New York | $5422.04$ | $5551.91$ | $2.3952$ per cent |
| Tokyo $\to$ San Francisco | $8274.63$ | $8730.02$ | $5.5034$ per cent |
| Tokyo $\to$ London | $9558.57$ | $11293.57$ | $\mathbf{18.1512}$ per cent |
Observation 2.2 Checks——the great circle Tokyo to San Francisco, $8274.63$ km, agrees with the usual figure of about $8300$ km to $0.31$ per cent. And half the equator, $\pi R$, comes to $20015.1144$ km, near half the defining $40000$——the tool is tried on two known cases before the table is read.
Observation 2.3 The differences span more than four thousandfold, from $0.0043$ to $18.1512$ per cent. Several of these leave the same port——leave from one place and the price differs by decades according to where you are going.
Observation 2.4 The $18.1512$ per cent Tokyo to London is $1735$ km. $11293.57$ against $9558.57$——running the rhumb line adds about the distance from Tokyo to Shanghai ($1754.27$ km).
Observation 2.5 Tokyo to Sydney has almost no difference. $7825.83$ against $7826.17$, a matter of $0.34$ km——run $7800$ km and lose $340$ metres.
3. The Core: What Decides Is Not Distance but Running Across Latitude
Setting: Order the table of Section 2 by distance and it does not order by difference. The longest route (Tokyo to London, $9558.57$ km) does carry the largest difference, but Tokyo to Sydney, shorter than Tokyo to San Francisco, carries the smallest.
Observation 3.1 This is the core of the paper. What acts is not the total distance but how far the route runs east and west at high latitude——Tokyo to London has a great circle sweeping far north, and the rhumb line cuts straight across it in latitude, so the loss is large.
Calculation 3.2: Identify two cases where the difference is exactly zero (computed in this paper).
| Case | Great circle [km] | Rhumb line [km] | Difference |
|---|---|---|---|
| One meridian ($35^\circ$N $\to$ $35^\circ$S) | $7783.655616$ | $7783.655616$ | $\mathbf{0}$ |
| Along the equator ($0^\circ \to 90^\circ$E) | $10007.557221$ | $10007.557221$ | $\mathbf{0}$ |
Observation 3.3 In these two, the great circle is itself a line of constant heading. A meridian always points true north and the equator always true east——the two paths coincide only when the great circle happens also to be a rhumb line.
Observation 3.4 And among great circles on a sphere, only the meridians and the equator are rhumb lines. Every other great circle changes heading as one advances——so “no loss” is the exception and not the rule.
Calculation 3.5: Swing the other way and find the maximum (computed in this paper).
| Route | Great circle [km] | Rhumb line [km] | Difference |
|---|---|---|---|
| Half the world at $70^\circ$N ($0^\circ\to180^\circ$) | $4447.80$ | $6845.57$ | $\mathbf{53.9091}$ per cent |
Observation 3.6 Here the rhumb line is $1.54$ times the great circle. Running due east at $70$ degrees north takes one halfway round the world along a parallel——while the great circle crosses near the pole and arrives far sooner.
Observation 3.7 A parallel is not a great circle, except the equator. The parallel at $70$ degrees north looks like “straight east” and is curved on the sphere——holding a constant heading and going straight have become two different things.
4. The Mercator Projection Made the Rhumb Line a Straight Line
Setting: The reason the rhumb line held practice lies in the chart.
Observation 4.1 On the Mercator projection a rhumb line is exactly a straight line. Because latitude is stretched by $\ln\tan(\pi/4+\varphi/2)$——lay a ruler from origin to destination, read its slope, and that is the heading to hold.
Observation 4.2 On the same chart the great circle is a curve. The shortest path looks like a detour and the detour looks straight——the projection was deciding which path got chosen.
Observation 4.3 The price rises with latitude. Mercator stretching is $1/\cos\varphi$, which at $70$ degrees north is $2.92$——where the chart distorts most, the rhumb line loses most. One and the same $\cos\varphi$ acts on both.
Observation 4.4 Practice mixed the two. Divide the great circle into legs and sail each leg as a rhumb line——most of the loss goes and the heading stays constant within each leg. Not a choice between two but a question of how finely to divide.
Observation 4.5 To be stated honestly——the Earth is taken as a sphere here. The real Earth is flattened and its geodesics depart from spherical great circles by up to about $0.5$ per cent——the differences here (up to $53.9091$ per cent) are far larger so the conclusion stands, but for the $0.0043$ per cent of Tokyo to Sydney the flattening is the larger effect.
5. The Separator: What Is Held Constant?
Observation 5.1 The separator——whether what is held constant along the route is the heading, or the property of being shortest.
| Held constant | The helm | On the chart | |
|---|---|---|---|
| Great circle | being shortest | changed continually | a curve |
| Rhumb line | the heading | set once and left | a straight line |
Observation 5.2 The separator is neither in the sea nor in the sphere. The same sea, the same two points, the same sphere——what makes the difference is what one keeps hold of aboard the ship.
Observation 5.3 The difference from Paper 234, stated. Paper 234 counted six “distances” and asked what may pass as one——here the distance is fixed to one. Under one and the same distance, why is the longer path chosen? Not a question about definitions but about choosing.
Observation 5.4 And this separator is still moving. Satellite navigation and autopilots have all but removed the labour of holding a heading, so the reason for choosing a rhumb line has gone——nothing about the paths changed, and the chosen path changed. The separator was on the side of the instruments.
Observation 5.5 To be stated honestly——heading is not the only reason to take a rhumb line. Currents, winds, ice, depth and ports of refuge all count——what is shown here is the size of the price, not whether it should be paid.
Observation 5.6 One last thing——Tokyo to Sydney the difference was $340$ metres. On a route running nearly north and south, the shortest path and the easy path are effectively the same thing——the size of the loss is decided by the destination, not by the navigator’s skill or judgement.
6. The Census and the Close
- $18.1512$ per cent Tokyo to London (Calculation 2.1): a loss of $1735$ km.
- $0.0043$ per cent Tokyo to Sydney (Calculation 2.1): $340$ metres.
- More than four thousandfold between them (Observation 2.3): same port, different destinations.
- Tokyo to San Francisco is $8274.63$ km (Observation 2.2): $0.31$ per cent from the usual figure.
- Zero along a meridian and the equator (Calculation 3.2): the great circle is itself constant-heading.
- Only meridians and the equator are both (Observation 3.4): zero is the exception.
- $53.9091$ per cent halfway round at $70^\circ$N (Calculation 3.5): a factor of $1.54$.
- A parallel is no great circle (Observation 3.7): straight east is not straight.
- Mercator stretching is $1/\cos\varphi$ (Observation 4.3): one $\cos\varphi$ acts on both.
- Practice divided the great circle into rhumb legs (Observation 4.4): not a choice between two.
Closing: On a sphere the shortest path between two points is a great circle and a rhumb line is never shorter——and still the sailing ships ran rhumb lines, because one need only hold the compass needle on a single mark. The price differs by decades according to the destination——Tokyo to London the rhumb line is $11293.57$ km against the great circle’s $9558.57$, longer by $\mathbf{18.1512}$ per cent, that is $1735$ km. From the same Tokyo to Sydney it is $7825.83$ against $7826.17$: run $7800$ km and lose $340$ metres. The differences span more than four thousandfold. What acts is not the total distance——it is how far the route runs east and west at high latitude. Along one meridian and along the equator the difference is exactly zero——a meridian always points true north and the equator always true east, so there the great circle is itself a line of constant heading. And among great circles only meridians and the equator are rhumb lines, so “no loss” is the exception, not the rule. Swing the other way and halfway round the world at $70$ degrees north the difference is $\mathbf{53.9091}$ per cent, the rhumb line $1.54$ times the great circle. The reason the rhumb line held practice lies in the chart——on the Mercator projection it is exactly a straight line, and the slope of a ruled line is the heading to hold. On the same chart the great circle is a curve: the shortest path looked like a detour and the detour looked straight. The stretching is $1/\cos\varphi$, so where the chart distorts most the rhumb line loses most — one and the same $\cos\varphi$ acting on both. One thing separates them——whether what is held constant is the heading or the property of being shortest. The separator is neither in the sea nor in the sphere——the same sea, the same two points, the same sphere; what makes the difference is what one keeps hold of aboard the ship. Paper 234 counted six “distances”——its question was what may pass as a distance. Here the distance is fixed to one, and the question is why, under that one distance, the longer path is chosen. Not a question about definitions but about choosing. And the separator is still moving——satellite navigation and autopilots have all but removed the labour of holding a heading, so the reason for choosing a rhumb line has gone. Nothing about the paths changed, and the chosen path changed. The separator was on the side of the instruments. One last thing——Tokyo to Sydney the difference was $340$ metres. On a route running nearly north and south, the shortest path and the easy path are effectively the same thing. The size of the loss is decided by the destination, not by the navigator’s skill or judgement.
References
- Nunes, P. (1537). Tratado da Sphera com a Theorica do Sol e da Lua. Lisbon.
- Mercator, G. (1569). Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendate Accommodata. Duisburg.
- Wright, E. (1599). Certaine Errors in Navigation. London.
- Vincenty, T. (1975). Direct and Inverse Solutions of Geodesics on the Ellipsoid with Application of Nested Equations. Survey Review, 23(176).
- Snyder, J. P. (1987). Map Projections: A Working Manual. USGS Professional Paper 1395. Washington: US Government Printing Office.
- Yamagishi, Y. (2026). The Shortest Enclosure Is Not Always a Disc [Paper 205]. Zenodo. 10.5281/zenodo.22110143
- Yamagishi, Y. (2026). “Distance” Is Six Things, and One of Them Satisfies the Triangle Inequality Backwards [Paper 234]. Zenodo. 10.5281/zenodo.22128226
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). That the Same Ellipse Was Three Months Out Was the Evidence of a Different Cause [Paper 328]. Zenodo. 10.5281/zenodo.22210340
Reference
Yamagishi, Y. (2026). The Shorter Path Is Not the One a Navigator Steers [Paper 340]. YOI: https://www.yamagishiyuuki.com/yoi/340
DOI: 10.5281/zenodo.22682089
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 340. This paper is a preprint and has not been peer reviewed.