A bearing on a sphere is decided by its latitude, not its radius

Four radii of the Earth priced the sphere that stands in for the ellipsoid, and found that the right radius depends on what is being kept: 6,371 kilometres is right for an area to half a part per million and wrong for a meridian distance by 559. An ellipsoid computed to a nanometre is known to a decimetre then set a meridian distance’s three error budgets side by side and left the next quantity for later: not a length, but the direction from one point to another.

A direction is what a bearing on a map stands in for, and every navigation formula that works on a sphere computes one. The question is which sphere serves it — and the first thing to notice is that the question the radius essay answered does not arise.

Enlarging a sphere moves no angle #

A sphere that stands in for the ellipsoid takes each point’s latitude and longitude, puts the point on a sphere at some latitude and the same longitude, and computes there. The azimuth from one point to another on that sphere is the initial direction of the great circle between them.

Scaling the sphere scales every distance on it and leaves every angle alone, so the azimuth does not depend on the radius at all. The 6,371-kilometre sphere, the authalic sphere of 6,371.007 and a sphere the size of a marble all give the same direction between the same two points. The four radii that matter so much for a distance say nothing about a bearing.

What does matter is the latitude. Geodetic against geocentric latitude is the essay about two of the angles called latitude, and the ellipsoid has six in common use — geodetic, geocentric, parametric, conformal, authalic and rectifying — each of which makes one property of the ellipsoid behave spherically. Put each on a sphere, compute the azimuth of a geodesic’s endpoints there, and compare it with the azimuth the geodesic actually sets out on.

The truth for that comparison is the ellipsoid’s own. Each geodesic is integrated out across WGS84 from its starting point on a stated azimuth, and the azimuth between its two ends is then solved back by Vincenty’s method, which shares no algebra with the integration; geodesics on the ellipsoid is the essay about why the problem needs either. The two agree to five billionths of an arcsecond in direction and ten micrometres in length over 3,000 kilometres.

Four latitudes are wrong before anything moves #

The expectation for a substituted sphere is an error that grows with distance, the way a flat map’s does. Four of the six latitudes do not behave like that at all.

One kilometre from the start, a sphere carrying the geodetic latitude gets the azimuth wrong by up to 693 arcseconds at the equator, 347 at 45° and 21 at 80°. At ten metres it is the same. At a hundred kilometres it has barely moved. The error is there before the destination is any distance away at all, and the shape of the curve says why.

A substitution is itself a map, from the ellipsoid to the sphere. At any point it stretches the north–south direction by one amount and the east–west direction by another, because the ellipsoid’s two radii of curvature there — the ones the radius of curvature is two numbers separates — are different, and a sphere’s are the same. A map that stretches two directions differently turns every other direction, and the most any direction is turned is half the difference between the two stretches. For the geodetic latitude that difference is $e^2\cos^2\varphi$, a sixty-seventh of a per cent at the equator, and the largest turn is half of it: $e^2\cos^2\varphi/2$ radians, which is the curve in the figure.

That number has appeared before, from the other side. Web Mercator is not conformal found that putting geodetic latitudes into a spherical formula leaves a map with a maximum angular deformation of 0.3848 degrees at the equator. Angular deformation counts the angle between two directions each turned the opposite way, so it is twice the largest turn of one — and twice 693 arcseconds is 0.3848 degrees. The web map and the substituted sphere are the same mistake measured two ways.

A half, a third and a quarter #

The other three latitudes that fail at the start fail in a proportion that is too tidy to be an accident.

At 45° the parametric latitude is wrong by 174 arcseconds, the authalic by 116 and the rectifying by 86.8: the geodetic error over two, over three and over four, to within 0.55 per cent. The closed form $e^2\cos^2\varphi/2$ is 345 arcseconds and the geodetic error measured is 347, the difference being the terms in $e^4$ the first-order account leaves out.

Each of those latitudes is built to make one property spherical, and each gets part of the way to conformality as a side effect. The rectifying latitude stretches the meridian to match its length, the authalic balances area, the parametric projects the point onto a circle of the equatorial radius, and each of those removes a different fraction of the mismatch between the two stretches. None of them removes all of it, because none of them was defined by the angle. That is the sense in which the six latitudes are not six estimates of one thing: they are six answers to six different questions, and on the question of direction four of them are wrong from the first metre.

In position, the difference is not small. An azimuth wrong by 347 arcseconds puts the end of a thousand-kilometre route 1.68 kilometres to one side.

Why the fractions are those fractions #

The tidiness has a short explanation, and it is worth having because it predicts the two latitudes that start right as well as the four that do not.

To first order in the eccentricity, every one of the six latitudes is the geodetic latitude less some fraction of the same small term:

$$\theta \approx \varphi - c\, e^2 \sin\varphi \cos\varphi ,$$

with a different $c$ for each. For the geodetic latitude $c$ is nothing. For the parametric latitude it is a half, for the authalic two thirds and for the rectifying three quarters; for the conformal and the geocentric latitudes it is one.

Put that into the two stretches. On the ellipsoid the ratio of the north–south radius of curvature to the east–west one is, to the same order, $1 - e^2\cos^2\varphi$. On a sphere carrying $\theta$, the corresponding ratio of stretches differs from one by $c\,e^2\cos^2\varphi$ — the sliding of the latitude makes up that much of the mismatch. What is left is $(1 - c)\,e^2\cos^2\varphi$, and the largest turn of a direction is half of it. So the error at the start is the geodetic error times $1 - c$: all of it, a half, a third, a quarter, and nothing.

That is why the measured fractions at 45° are 1.000, 0.500, 0.333 and 0.250, and why the geocentric latitude — whose first-order term is exactly the conformal latitude’s — starts right to within the $e^4$ terms. It also says what each auxiliary latitude is for, in the same currency: each takes back the fraction of the direction error that its own defining property happens to require, and only the latitude defined by angles takes back all of it.

The two that start right #

The conformal latitude is defined so that the map from the ellipsoid to its sphere stretches every direction equally at every point. So it turns no direction at the start, by construction, and one kilometre out the measured error is 54 milliarcseconds.

The geocentric latitude is the surprise. It is the latitude a naive spherical formula is usually accused of using, and it is wrong at the start by only 620 milliarcseconds at 45° — five hundred times less than the geodetic latitude that is actually a coordinate. The reason is that to first order in the eccentricity the geocentric and conformal latitudes are the same function: both are $\varphi - (e^2/2)\sin 2\varphi$ plus terms in $e^4$. So the geocentric sphere is conformal to first order, and its error at the start is the second-order remainder. By a hundred kilometres the two are indistinguishable in the hero figure, both near five and a half arcseconds, because what remains for both is the growth with distance.

The conformal sphere drifts, and at the equator it drifts slowly #

A conformal substitution keeps angles at every point, and a geodesic is still not a great circle on its sphere. The conformal sphere’s scale varies with latitude, so the image of a geodesic curves away from the great circle between its ends, and the azimuth of that great circle differs from the geodesic’s own by an amount that grows as the two separate.

From 30° and from 60° the error grows exactly in proportion to the distance — a slope of 1.00 on these axes between 100 and 1,000 kilometres — and reaches 47 arcseconds at a thousand. From the equator it grows as the square of the distance and is only 2.85 arcseconds at a thousand kilometres. The difference is where the conformal sphere’s scale changes fastest. Its departure from uniform is symmetric about the equator and has no slope there, so a geodesic leaving the equator starts in a region where nothing bends it and only picks up curvature as it moves away.

That is the conformal sphere’s budget, and it is the reason the conformal sphere is used where it is. A projection that goes from the ellipsoid to a plane by way of a sphere wants the first step to keep angles, and the double stereographic grids — the Dutch national grid is one — pass through exactly this sphere. Transverse Mercator and the series that computes it passes through the conformal latitude for the same reason. What they pay is this drift, over the size of the grid.

Bessel’s sphere changes the longitude #

None of the six substitutions is exact at any distance above zero. There is a sphere that is.

In 1826 Bessel showed that a geodesic on an ellipsoid of revolution can be mapped onto a great circle of an auxiliary sphere, with every angle along it preserved, if two things are changed. The latitude becomes the parametric latitude. And the longitude is replaced by a spherical longitude $\omega$ that differs from the geographic one by a correction that depends on the geodesic itself. The geodesic that sphere straightens is the curve the normal section is not the geodesic shows a theodolite does not sight along; Vincenty’s method, which provides the truth for every figure here, is an iteration for $\omega$.

The correction is small. From 45° it is 77 milliarcseconds of longitude at one kilometre, 7.7 arcseconds at a hundred and 77 at a thousand, growing in proportion to the distance. Leave it out and the sphere is the parametric sphere, wrong by 174 arcseconds at a kilometre and 210 at a thousand. Put it in and the error falls to the arithmetic’s floor, $1.2 \times 10^{-7}$ arcseconds, at every distance tried.

So the whole of the parametric sphere’s failure is the longitude it was not given. That is not the conclusion the setup invites. Every substitution tried above changed the latitude and kept the longitude, on the natural assumption that a latitude is where an ellipsoid differs from a sphere and a longitude is the same on both. For a distance or an area that is enough. For a direction it is not. A correction of 77 milliarcseconds of longitude is under two metres on the ground at 45°, and it is worth 174 arcseconds of azimuth, because a direction over a short distance is decided by a few metres.

Which latitude outweighs which ellipsoid #

The previous essay’s three budgets for a distance put the arithmetic far below the conventions and the conventions below the uncertainty of the ellipsoid. The same comparison for a direction comes out in a different order.

Changing the ellipsoid barely touches a direction. From WGS84 to the ellipsoid that best fits the Earth, the azimuth of a thousand-kilometre geodesic moves by at most 1.86 milliarcseconds; from WGS84 to GRS80, by 3.4 microarcseconds. Both are far below any instrument that measures a bearing.

Changing the latitude put on a sphere moves it by 347 arcseconds, five orders of magnitude more. So for a direction the modelling choice — which sphere, and which latitude on it — is the whole budget, and the choice of ellipsoid is a rounding error on it. A flat picture has one direction between two places, and the Earth has two found that a flat picture of world cities cannot get their bearings right to within tens of degrees; a sphere with the wrong latitude on it is wrong by a few arcminutes, which is nothing on a world map and a great deal on a survey.

Put in position, an arcsecond is 4.85 metres of sideways miss at a thousand kilometres. The geodetic sphere’s 347 arcseconds is 1.68 kilometres; the conformal sphere’s 54 arcseconds at that distance is 263 metres; the best-fitting ellipsoid’s 1.86 milliarcseconds is nine millimetres.

What that means for the ordinary bearing formula #

The spherical bearing formula — the one the great-circle vertex reads a route’s highest latitude from, and the one every spherical route computation starts with — is almost always fed the latitude and longitude a receiver reports, which are geodetic. That is the first row of the budget. Near the equator it turns a bearing by up to eleven and a half arcminutes before the destination is any distance away, and it does so on a sphere of any radius.

Feeding it geocentric latitudes instead, which sounds like the cruder choice, improves the start by a factor of five hundred at 45°. Feeding it conformal latitudes makes the start exact and leaves only the drift with distance. None of those is a change of radius, and a formula documented as using “the mean radius of the Earth” has said nothing about the one choice that decides its bearings.

What each number was checked against #

The truth is two methods that share nothing. A geodesic integrated across the ellipsoid by Runge–Kutta and the same geodesic solved by Vincenty’s iteration agree to five billionths of an arcsecond in azimuth.

Four failures have a closed form, and meet it. At one kilometre from 45°, the geodetic, parametric, authalic and rectifying spheres must be wrong by $e^2\cos^2\varphi/2$ over one, two, three and four, and are, to 0.55 per cent — the $e^4$ terms the closed form omits.

The conformal sphere must start right and grow with distance. Its error at one kilometre must be under a tenth of an arcsecond, and it must grow between eight and twelve times from a hundred kilometres to a thousand at 45°. It grows 10.0 times.

Bessel’s sphere must be exact. At every distance from one kilometre to five thousand, under a millionth of an arcsecond.

And on a sphere every substitution must vanish. Run on an ellipsoid with no flattening, where all six latitudes are the same angle, every sphere’s azimuth error must be zero to the arithmetic, and is. A comparison that found a latitude wrong on a sphere would be measuring its own arithmetic rather than the ellipsoid.

What the six spheres were not asked #

Only the direction a geodesic sets out on. The azimuth at the far end, and the direction along the way, are different quantities with their own errors; a sphere that is right at the start is not thereby right at the destination.

Twenty-four starting directions, from each latitude. The fan includes every multiple of fifteen degrees, which catches the peak of an error that varies as the sine of twice the azimuth; routes over a pole are left out, because the azimuth of a meridian that crosses the pole is ambiguous by a half-turn.

WGS84, and nothing beneath it. A geoid, a deflection of the vertical, or an astronomically observed azimuth are all different directions again, and each of them is larger than several rows of the budget above.

Still open: the angles of a triangle #

A surveyor does not measure one azimuth. A triangulation measures the three angles of a triangle, and for two centuries those triangles were computed on a sphere by Legendre’s theorem, which subtracts a third of the spherical excess from each angle and solves the rest as a plane triangle.

A triangle’s angles are differences of directions at its three corners, and every error measured here is a turn of one direction. A turn that is the same for every direction at a point cancels in a difference, and a turn that varies with the direction — as all four of the latitudes that fail at the start do, in proportion to the sine of twice the azimuth — does not. Which of the six spheres keeps the angles of a triangle, whether Bessel’s sphere is still the only exact one when three geodesics meet, and how large a triangle has to be before the choice of sphere is worth more than the excess Legendre’s theorem corrects for, are questions a single direction cannot ask.