The great-circle vertex

A flight from London to Tokyo crosses 70.9° north. Neither city is anywhere near it — London is at 51.5° and Tokyo at 35.7° — and the aircraft is not detouring.

That number is not a consequence of the route being flown. It is fixed by the departure bearing, it can be computed before the journey starts, and the quantity that fixes it is the only thing that stays constant along a shortest path on a sphere.

The invariant #

On a surface of revolution, a geodesic satisfies Clairaut’s relation: the product of the distance from the axis and the sine of the angle to the meridian does not change along the path. On a sphere of unit radius the distance from the axis is $\cos\varphi$, so

$$\cos\varphi\,\sin\alpha = \text{constant}$$

where $\alpha$ is the bearing. That constant is the only thing conserved along a great circle, and every other property of the route follows from it.

The vertex is where the path reaches its highest latitude. There it runs due east or west, so $\sin\alpha = 1$, and

$$\cos\varphi_v = \left|\cos\varphi_1\,\sin\alpha_1\right|$$

Departure latitude, departure bearing, one cosine. No integration and no path.

The check #

The site computes the vertex both ways and requires agreement.

One route is the formula above, evaluated at the departure point. The other generates the great circle by spherical interpolation at two thousand points and takes the maximum latitude.

Agreement to five decimal places, between three lines of trigonometry evaluated at one point and a maximum over two thousand generated points that knows nothing about it.

The constant itself is also re-measured along the path — at every one of those two thousand points, the local latitude and the local bearing must give the same product — and it varies by 1.0×10⁻¹³ over the whole route.

Why it explains polar routes #

The numbers in that table are the whole argument for high-latitude flying, stated without any reference to aviation.

Anchorage is at 61.2° and London at 51.5°. Neither is inside the Arctic Circle. The shortest path between them reaches 80.4° — within eleven degrees of the pole, a thousand kilometres from it.

That is not a route choice. It is where the geodesic goes, and the alternative — following the rhumb line, holding one bearing — costs 2,031 km, or twenty-eight per cent.

The general rule falls out of the formula. The vertex is high when $\cos\varphi_1\sin\alpha_1$ is small, which happens when the departure latitude is high, or the departure bearing is near due north or south, or both. So east–west journeys at high latitude have very high vertices, and that is precisely the class of route between northern Europe, northern Asia and North America.

Which is why the Arctic became strategically central in the twentieth century and had not been before. The shortest paths between the industrial regions of the northern hemisphere all pass close to the pole, and that fact is Clairaut’s relation rather than a matter of policy.

What the constant is, geometrically #

The formula has a picture behind it, and the picture makes it obvious rather than memorable.

A great circle is the intersection of the sphere with a plane through the centre. That plane makes some angle with the equatorial plane — call it the inclination $i$. The great circle’s highest latitude is exactly $i$, because that is where the plane is furthest from the equatorial one.

And Clairaut’s constant is $\cos\varphi_v = \cos i$. So the conserved quantity is the cosine of the great circle’s inclination: a property of the plane, fixed the moment the plane is chosen, and therefore fixed by any point on the circle together with the direction through it.

Stated that way the conservation is not a theorem so much as a restatement. What makes Clairaut’s relation useful is that it is expressed in quantities a navigator can measure — the local latitude and the local bearing — rather than in the inclination of a plane through the centre of the Earth, which nobody can observe.

That is the same move as elsewhere on this site: the useful invariant is the one expressible in what the instrument reads.

The vertex is not always on the route #

An honest qualification, and the site’s assertion has to handle it.

The vertex is a property of the whole great circle, which is a closed curve. The journey is an arc of it, and the arc may not include the vertex.

Cape Town to London runs almost due north. Its great circle has a vertex at 80.5° — the circle passes near the pole on the far side of the Earth — and the journey itself never goes above London’s own latitude of 51.5°. Sydney to Santiago is a southern-hemisphere route whose great circle has a northern vertex at 61.8° that the journey comes nowhere near.

So the formula gives the vertex of the circle, and whether the journey reaches it depends on whether the arc contains it. The check on this site compares the predicted vertex against the sampled maximum only when the sampled maximum exceeds both endpoints — which is the condition for the vertex to be interior to the arc — and reports the rest as not applicable rather than as agreement.

The rhumb line has no vertex #

The contrast with the other route sharpens what the vertex is, and it is worth drawing.

A rhumb line holds $\alpha$ constant by definition. So Clairaut’s product $\cos\varphi\sin\alpha$ is not constant along it — it varies as $\cos\varphi$ does — and the curve has no vertex at all. It spirals toward the pole, crossing every meridian at the same angle, approaching the pole asymptotically without reaching it and without ever turning back.

That is a genuinely different shape of curve. A great circle rises to a maximum latitude and descends; a rhumb line at any bearing other than due east or west rises forever. On Mercator the two look equally straightforward — the rhumb is a straight line and the great circle bows — and the difference between a curve with a maximum and a curve with an asymptote is invisible in that picture.

It also explains the shape of the excess between them. The rhumb line stays at lower latitudes than the great circle for most of the journey and therefore travels further, and the gap is largest exactly when the great circle’s vertex is furthest from both endpoints — which is to say when the journey is east–west and at high latitude, which is the pattern the excess chart shows.

What the constant is worth knowing before departure #

A short practical list, because the number does more work than its derivation suggests.

Whether the route is polar. $\cos\varphi_v$ below about 0.3 means a vertex above 72°, which is a route through Arctic airspace with everything that implies for diversion airports, fuel reserves and communications.

Whether the route is symmetric. The vertex is the great circle’s axis of symmetry, so a journey whose vertex falls at its midpoint is symmetric and one whose vertex is off the arc is not. That decides where the fuel-critical point sits.

Which latitudes the route visits, and at what angle. Inverting the relation gives the bearing at any latitude the route reaches, and states that latitudes above the vertex are unreachable on that circle — one formula answering a question and delimiting its own domain.

All three are available from a departure latitude and a bearing, before a single point of the route is computed. That economy is what makes an invariant worth finding.

The two useful corollaries #

Beyond the vertex itself, the relation answers two questions navigators actually ask.

Can this route stay below a given latitude? A route with $\cos\varphi_v$ below $\cos\varphi_{\max}$ cannot, and the answer is available from the departure bearing. Composite sailing — the classical practice of following a great circle until a limiting latitude, running along that parallel, then rejoining a great circle — exists precisely because some vertices are further north than a ship can safely go, and the limiting latitude is chosen from ice or weather rather than from geometry.

Where does the route cross a given latitude? Clairaut’s relation inverts: at latitude $\varphi$, the bearing is $\sin\alpha = C/\cos\varphi$, which has a solution only for $\cos\varphi \geq C$. So the relation both gives the crossing bearing and states that latitudes above the vertex are unreachable on that circle — one formula answering a question and delimiting its own domain.

That second property is the useful one for planning. A route’s Clairaut constant is a single number that says everything about which latitudes it visits and at what angle, and it is available before anything is plotted.

The same relation on an ellipsoid #

The result generalises, and the form it takes is the reason ellipsoidal geodesics are computable at all.

On an ellipsoid of revolution, Clairaut’s relation still holds — it is a consequence of rotational symmetry rather than of the sphere — but in terms of the parametric latitude $\beta$ rather than the geodetic one:

$$\cos\beta\,\sin\alpha = \text{constant}$$

with $\tan\beta = (1-f)\tan\varphi$. That is the same statement with one substitution, and it is why Vincenty’s method for ellipsoidal geodesics begins by converting both endpoints to parametric latitude: in that variable the problem becomes spherical trigonometry with a correction, and in the geodetic one it does not.

So the auxiliary latitudes are not bookkeeping. Each is the change of variable that makes one hard problem look like an easy one, and the parametric latitude is the one that makes geodesics tractable.

Composite sailing, and why the vertex forced it #

The classical response to a vertex too high to sail through is worth describing, because it is a design decision made against exactly the quantity this essay computes.

A shortest route from Britain to Japan has a vertex above 70°, which in the nineteenth century meant pack ice for much of the year. The shortest sailable route is therefore not the great circle; it is the shortest path that stays below some limiting latitude $\varphi_L$.

That path has a known shape. Follow the great circle from the departure point until it reaches $\varphi_L$; run along the parallel $\varphi_L$; then follow a second great circle down to the destination. This is composite sailing, and the two great-circle arcs are each tangent to the limiting parallel, so each has its vertex exactly at $\varphi_L$.

Clairaut’s relation gives both arcs immediately: the departure bearing for the first is the one whose constant is $\cos\varphi_L$, and likewise for the arrival bearing of the second. So the whole route is determined by one number — the limiting latitude, chosen from ice charts rather than from geometry — and the rest is two applications of the formula.

That is a good illustration of what an invariant buys in practice. The optimisation problem “shortest path subject to a latitude constraint” sounds like a calculus-of-variations question, and Clairaut’s relation reduces it to reading two bearings off a table.

What survives, and what does not #

Worth naming the pattern, since it recurs across this site.

The bearing along a great circle changes continuously — 124.6° over the London–Tokyo route — and is a property of the path together with the local meridian. The latitude changes. The distance travelled changes. The Clairaut product does not.

An invariant is a quantity that stays fixed while the description varies, and finding one converts a curve into a number. That is exactly the move the principal scale factors make for distortion and that the Euler characteristic makes for curvature: the useful content of a problem is what survives the parameterisation.

Here the payoff is unusually direct. The invariant is one number, it is measurable at departure, and it determines the single most consequential feature of the route.

The vertex is determined by the departure bearing, so a route whose departure bearing is undetermined has no vertex either.

What was computed here #

Great circles are generated by spherical interpolation between the endpoints, which stays on the sphere by construction rather than by correction, at two thousand points per route.

Two assertions hold the essay. Clairaut’s product, re-measured at every point along each generated path from the local latitude and the local bearing to the destination, must be constant — it varies by 1.0×10⁻¹³ across five journeys. And the vertex predicted from the departure bearing alone must equal the maximum latitude the generated path reaches, where the vertex falls inside the arc, which it does to 1.3×10⁻⁵ degrees.

The two computations share no code. One is three lines of trigonometry at a single point; the other is a maximum over two thousand points produced by a different formula. Agreement between them is evidence about both, which is the only reason to compute a quantity twice.

The bearing figures are computed rather than quoted, and one of them corrected a number this site had previously printed: the London–Tokyo departure bearing is 31.7°, not the 27° an earlier essay stated.

The vertex figure is drawn with the route generated first and the vertex marked afterwards from a formula that never saw the route, so a discrepancy between the mark and the top of the curve would be visible rather than absorbed. On every journey drawn here the mark sits on the curve to within the width of the line, which is the visual form of the assertion the caption states as a number.

What the pictures cannot show #

The invariance. Clairaut’s constant is a number that does not change, and a figure of a number not changing is a figure of one number. The vertex is markable, and the reason it is where it is lives in the algebra.

The orthographic panel also cannot show a whole great circle, only the near hemisphere of one, so the vertex of a route whose circle passes round the far side is off the picture entirely — which is the same qualification the essay makes in prose about arcs that do not contain their vertex.

Who found it, and when #

Alexis Claude Clairaut published the relation in 1743, in his treatise on the figure of the Earth, as part of the analysis of geodesics on a spheroid. He was working on the same question the Lapland and Peru expeditions had been sent to settle, and the relation is a by-product of asking how a shortest path behaves on a flattened body.

That it applies on a sphere is the easy case; Clairaut’s interest was the hard one. The relation is the earliest general result about geodesics on a surface of revolution, and it predates the general theory of geodesics by a century.

Great-circle sailing as routine practice arrived much later, with the marine chronometer and reliable longitude, and composite sailing — following a great circle to a limiting latitude and no further — is a nineteenth-century refinement made necessary by exactly the vertices this essay computes.

The number to compute before plotting anything #

One practical use is worth separating out, because it turns a plotting exercise into a formula evaluated once.

The vertex latitude is decidable from the two endpoints alone. It falls out of the invariant without constructing the route, so a planner knows the highest latitude the great circle will reach before drawing a single leg of it.

And that is exactly the number the decision needs. The question a navigator is really asking is whether the shortest route trespasses on something — an ice limit, a weather regime, a political boundary at a stated parallel — and the answer is a comparison between two latitudes. If the vertex is below the limit, the great circle is available and the whole composite apparatus is unnecessary. If it is above, the route must be a composite one and the limiting parallel is where it breaks.

The order matters because plotting is the expensive step. The classical method draws the great circle on the gnomonic sheet, transfers a series of points to the Mercator, and joins them by rhumb legs — and a planner who does all that only to find the track crosses the ice edge has done the work twice. One evaluation of the invariant sorts the two cases first.

Where this goes next #

The routes the vertex explains are the shortest route is not straight. The version of the relation that survives on a flattened body is geodesics on the ellipsoid. And the projection that draws these paths as straight lines is the gnomonic companion.