Geodetic against geocentric latitude

A latitude is an angle. On a sphere there is only one angle it could be. On an ellipsoid there are two obvious candidates, they differ by up to eleven and a half arcminutes, and almost every treatment of projections says “latitude” without saying which.

The two angles #

Geodetic latitude is the angle between the equatorial plane and the normal to the surface at the point. Geocentric latitude is the angle between the equatorial plane and the line to the centre.

On a sphere those are the same line. On an ellipsoid they are not, because the surface is not perpendicular to the radius anywhere except at the equator and the poles.

The relation between them is one line:

$$\tan\varphi_c = (1 - e^2)\tan\varphi$$

with $e^2 = 2f - f^2$ the squared eccentricity, about 0.00669 for WGS84. To first order in $e^2$ the difference is

$$\varphi - \varphi_c \approx \tfrac12 e^2 \sin 2\varphi$$

which is zero at the equator, zero at the poles, and maximal at 45°.

The size of it #

The peak is at 45.1° and its value is 11.545 arcminutes — 0.1924 degrees.

That is 21.4 kilometres of ground distance. Not 21 metres. Twenty-one kilometres, at the same point, between two numbers both of which are correctly called the latitude.

The number is worth holding onto because it is far larger than most people expect from a flattening of one part in three hundred. The reason is that the flattening enters through the slope of the surface rather than through its radius: the radius changes by 21 km over the whole quarter-circle from equator to pole, while the direction of the normal swings by a fifth of a degree relative to the radius over the same range.

Which one the coordinate means #

Geodetic, always, in every practical system.

That is not a convention chosen for tidiness. It is what could be measured. Before satellites, latitude was determined astronomically — by the elevation of a star, or of the Sun at noon — against an instrument levelled by a plumb line or a spirit level. A levelled instrument is perpendicular to the local equipotential, which is very nearly perpendicular to the ellipsoid, so the angle it yields is the geodetic one.

Nobody could measure the direction to the centre of the Earth. The geocentric latitude was never observable, and it is the one the mathematics of a sphere assumes.

Which is a good example of a general pattern: a coordinate system’s definition follows the instrument, and reading it as though it followed the geometry is where the errors start.

Where the confusion actually bites #

Three places, in increasing order of how quietly they fail.

Converting to Cartesian coordinates. Turning latitude and longitude into $x$, $y$, $z$ requires the radius of curvature in the prime vertical, $N = a/\sqrt{1 - e^2\sin^2\varphi}$, and the $z$ component carries an extra factor of $(1 - e^2)$. Treating the geodetic latitude as geocentric and using $R\sin\varphi$ instead puts the point kilometres out. This one fails loudly.

Satellite and astronomical geometry. A satellite’s sub-point, a sun angle, a shadow direction — these are computed from the geometry of the sphere-like body and want geocentric latitude, while the ground station’s coordinates are geodetic. Mixing them produces errors of tens of kilometres in position and a fraction of a degree in angle. This one fails quietly, because the answers stay plausible.

Conformality. The subtlest of the three, and the one this site was built to catch. A spherical Mercator applied to geodetic latitudes does not fail visibly at all: the map looks right, the shapes look right, and the projection is simply no longer conformal, by 0.3848°.

The four other latitudes #

Geodetic and geocentric are not the whole list. An ellipsoid supports at least six angles that a formula might call latitude, and the other four are constructions with jobs.

Each is defined so that one property of the ellipsoid behaves spherically. Parametric latitude projects the point onto a sphere of radius $a$ and is the natural variable for the ellipse. Authalic latitude makes equal increments enclose equal areas, so substituting it into a spherical equal-area formula yields the ellipsoidal one. Rectifying latitude makes meridian distance proportional to angle. Conformal latitude does the same for angles, and is the substitution that turns a spherical Mercator into a correct ellipsoidal one.

The four are all between the geodetic and geocentric values, and they arrive in a fixed order. Writing the gap from the geodetic value as a multiple of $e^2\sin 2\varphi$:

The ordering is forced by those coefficients and holds at every latitude, which is why the curves in the hero figure never cross. The site asserts the ordering rather than the values, because a sign error in any one conversion breaks the ordering while leaving each individual number looking perfectly reasonable.

The pair that nearly coincide #

The last two rows of that table are the interesting ones, and they are the reason the assertion is on the ordering rather than on the numbers.

Conformal and geocentric latitude have the same first-order expansion. Both are $\varphi - \tfrac12 e^2\sin 2\varphi$, and they separate only at order $e^4$. The gap between them peaks at 60° and is half an arcsecond — fifteen metres on the ground.

Fifteen metres is a strange size of error. It is far too large for a survey and far too small to look wrong, and a computation that used one for the other would produce results that passed every sanity check anybody would think to apply. The site checks that the conformal latitude is strictly above the geocentric one at 60° for exactly that reason: an assertion on a quantity that is nearly zero is worth more than an assertion on one that is obviously large.

A worked case: where the twenty-one kilometres go #

The gap is easiest to believe when it is followed through one computation.

Take a point at 45° north on the WGS84 ellipsoid. Its geodetic latitude is 45.000000° by construction. Its geocentric latitude is 44.807577°, a difference of 0.192423°.

Now suppose a program treats that coordinate as spherical: it computes a position as $R\cos\varphi$ outward and $R\sin\varphi$ up, with $R$ the mean radius. The direction it produces is the geocentric one, so it has placed the point at the location whose geocentric latitude is 45° — which is a point whose geodetic latitude is 45.193°.

That is 0.193° of latitude, and a degree of latitude is about 111.2 km, so the point has moved 21.4 kilometres north. The program has not crashed, produced a negative number, or returned anything outside the valid range. It has produced a coordinate in Germany for a point in Switzerland.

The displacement is north everywhere in the northern hemisphere and south everywhere in the southern, it vanishes on the equator and at the poles, and it peaks in the mid-latitudes where most of the world’s mapping is. A dataset corrupted this way looks systematically shifted rather than randomly wrong, which is the pattern most likely to be attributed to a datum problem and fixed with the wrong correction.

The parametric latitude ends up inside the distance calculation. Vincenty’s method for ellipsoidal distances works in it throughout, and the distances it produces differ from the spherical ones by a few tenths of a per cent — in both directions, which is what tells a reader the difference is a shape rather than a scale.

What the parametric latitude is doing in the geodesic #

One of the four constructions earns a mention on its own, because it appears in the middle of a calculation this site runs on every build.

Vincenty’s method for the distance between two points on an ellipsoid begins by converting both latitudes to parametric latitude, $\tan\beta = (1-f)\tan\varphi$, and then does spherical trigonometry on them with a series correction. That is not an approximation step; it is exact. The parametric latitude is the angle that the point’s projection onto a circumscribing sphere subtends, and an ellipsoid’s geodesics become tractable in it because Clairaut’s relation on an ellipsoid is $\cos\beta\sin\alpha = \text{constant}$ — the same form as the spherical relation, with $\beta$ in place of $\varphi$.

So the auxiliary latitudes are not bookkeeping. Each one is the change of variable that makes a specific hard problem look like an easy one, and choosing the wrong member of the family does not make a calculation slightly worse — it makes it answer a different question.

The parametric gap is the smallest of the five: a quarter of $e^2\sin2\varphi$, which is 5.77 arcminutes at 45°, about 10.7 km. Small compared with the geocentric gap, and still far too large to ignore.

The check that catches it #

Every conversion here is required to round-trip.

Geodetic to geocentric and back must return the starting angle; so must geodetic to conformal and back, which involves an iteration rather than a formula. Across three ellipsoids and every degree from −89° to 89°, the worst error is 6.7×10⁻¹⁶ radians.

That is a weak check on its own — an identity function round-trips perfectly too — which is why it sits alongside the ordering assertion and the independent conversion through Cartesian coordinates. The three together pin the conversions from different directions: one says the transformations are invertible, one says they are in the right relationship to each other, and one says they agree with a route that goes through three-dimensional space and back.

The confusion also has a place in the site’s own audit, and it is a conspicuous one. Two Mercators sit at the far left, each conformal against its own body; the third — the same spherical formula fed ellipsoidal latitudes — is four orders of magnitude to the right of them.

What a coordinate does not say #

The general lesson is about what is missing from a pair of numbers rather than about ellipsoids.

A latitude and longitude, written down, do not record which of six angles the latitude is, which ellipsoid it refers to, or which datum places that ellipsoid relative to the Earth. All three are needed to turn the pair into a position, and none of them is in the numbers.

In practice the first is nearly always geodetic and can be assumed. The second and third cannot: the same numbers on two datums are two places, up to a hundred and sixty metres apart, and there is nothing in the digits that says which.

That is the same structure as the distinction between the quantities that survive a change of coordinates and the ones that do not. A number is a measurement plus a frame, and dropping the frame does not make the number frame-free; it makes it ambiguous.

How the confusion shows in a dataset #

Worth knowing what the failure looks like from the outside, since that is how it is usually met.

A dataset built with the wrong latitude is displaced along the meridian, by an amount proportional to $\sin 2\varphi$. So the signature is a north–south shift that is zero on the equator, zero at the poles, largest in the mid-latitudes, and the same at every longitude.

That pattern is distinctive. A datum error is a roughly constant offset over a region, in an arbitrary direction. A projection error grows away from a central line. A latitude confusion is purely north–south, symmetric about the equator, and its magnitude is fixed at 21.4 km by the flattening rather than by anything about the data.

The awkward part is that a shift of 21 km is large enough to be noticed and small enough to be attributed to the wrong cause. On a continental map it looks like a datum problem, and applying a datum correction to it removes some of the error and leaves a residual that varies with latitude in a way no datum transformation produces.

Why eleven arcminutes rather than three #

A closing note on the arithmetic, because the size of the effect surprises people who know the flattening.

The flattening is $f = 1/298 = 0.00335$, and the latitude gap peaks at $\tfrac12 e^2 = f - \tfrac12 f^2 \approx 0.00334$ radians, which is 11.5 arcminutes. So the gap in radians is essentially the flattening itself.

Put that way it is not surprising at all. What makes it feel surprising is the unit: one part in three hundred sounds like a correction, and eleven arcminutes sounds like an angle. They are the same number, and the second is the one that matters, because a latitude is an angle and a third of a per cent of a right angle is a fifth of a degree.

The two latitudes differ because the normal to the surface does not pass through the centre. The same fact has a second consequence, at the level of curvature rather than of angle.

What was computed here #

Every latitude on this page is computed from the ellipsoid’s defining constants. The geodetic-to-geocentric conversion is the one-line tangent relation; the conformal, authalic and rectifying ones are their standard closed forms, with the rectifying latitude obtained from the meridian arc rather than from a series expansion of it.

Three assertions guard the set. Every conversion must round-trip, to 6.7×10⁻¹⁶ radians across three ellipsoids. The six latitudes must arrive in the order the first-order coefficients require, at every latitude from 5° to 85°. And the conformal latitude must be strictly greater than the geocentric one at 60°, which is the pair that agree to first order and would be indistinguishable to any check less specific than this.

The geodetic-to-Cartesian conversion is checked separately, by round-tripping through $x$, $y$, $z$ at three heights and three ellipsoids; the worst loss is 1.9×10⁻⁹ metres. Bowring’s iteration is the piece most likely to be silently wrong there, because it converges to something from almost any starting point.

What the pictures cannot show #

The flattening, again. The cross-sections exaggerate it twelve-fold and say so; at true scale the ellipse and the reference circle behind it would be the same line.

The hero figure also cannot show what any of these angles is for. It shows five gaps, all small, all similar in shape. Which of them belongs in a given formula is a question about the formula, and the figure would look identical if the answer were different.

A third direction called up #

There is a third direction called up, and it is a second departure on top of this one: the plumb line does not follow the ellipsoid normal either. Ten arcseconds of deflection is 309 metres of ground.

The two gaps are of very different kinds and the difference is worth holding onto. Geodetic against geocentric is a construction — it follows exactly from a and f, reaches 11.5 arcminutes, and can be computed anywhere. Geodetic against astronomic is a measurement: it depends on where the rock is, no formula supplies it, and it is what the plumb line is not the normal is about.

The same eleven arcminutes, in a buffer #

The two latitudes differ by up to 11 arcminutes and the essay’s subject is what that does to a coordinate. The applied field finds the same flattening deciding the shape of something much more ordinary: a tolerance written in degrees.

A buffer of a hundredth of a degree is an ellipse on the ground, because the meridian and transverse radii of curvature differ. At 60° north it is 1,114 metres by 558 — a ratio of 1.9966 rather than the sec φ everyone quotes, short by 1,684 parts per million — and on the equator, where a sphere gives exactly one, it is 0.9933, because M is smaller than N by the flattening’s own factor there.

The same two radii decide something a long way downstream, where a stored coordinate’s precision stops mattering against a screen’s. North-south the file’s grid and the screen’s cross at 62.19°; east-west they never cross at all, and the 0.333 per cent by which their ratio is not exactly constant is $N(\varphi)/a$ — the flattening again, in the last place anybody would look for it.

Who found it, and when #

The distinction is as old as the ellipsoidal Earth. Once Newton’s oblate figure was accepted in the 1730s, every geodetic survey had to decide which angle its instruments were measuring, and the answer — the direction of the normal, because that is what a level defines — was clear immediately.

The auxiliary latitudes accumulated through the nineteenth century as the ellipsoidal versions of the projections were worked out: the conformal latitude with the ellipsoidal Mercator, the authalic with the ellipsoidal equal-area projections, the rectifying with the meridian arc. Each was invented by whoever needed one particular spherical formula to keep working.

They are collected and named systematically in Adams’ 1921 treatise for the US Coast and Geodetic Survey, which is where the modern vocabulary comes from, and they appear in that form in Snyder’s manual sixty years later.

Where this goes next #

The body that makes two latitudes necessary is the Earth is a sphere, and when it is not. The projection built on the conformal latitude is transverse Mercator and the series that computes it. And the case where the confusion is shipped to a billion people is Web Mercator is not conformal.