A map of a body with three axes

A body that is not an ellipsoid ended by recording what it could not do. It measured what a coordinate means on a body with three unequal axes — that the two latitude conventions differ by 1.40° round Vesta at one parallel and by 6.40° round Phobos, so latitude there is not a function of position — and it stopped before drawing a map of one, because every piece of projection machinery on this site assumes a surface with an axis of revolution.

That assumption is not hidden. It is in the signature: the site’s distortion() takes a latitude and asks the metric for two radii of curvature, M(φ) and N(φ), which are functions of latitude alone. On a body with three unequal axes both principal curvatures depend on longitude as well, and there is nothing to hand the function.

The way through is to stop asking the surface for a latitude at all.

What Tissot’s construction actually needs #

Tissot’s apparatus is usually introduced on a sphere and written in latitude, which makes latitude look load-bearing. It is not. What the construction needs is the first fundamental form — the three numbers E, F and G that say how a coordinate patch measures length — together with the Jacobian of the map.

On an ellipsoid of revolution parameterised by longitude and latitude, E = N²cos²φ, G = M² and F = 0. That is why the site’s distortion() can take two functions of φ and be complete: the metric is diagonal and each entry has a name in the geodesy literature. Written the general way, the same calculation is four lines longer and works on any parameterised surface:

• the pulled-back plane metric is A = JᵀJ, in whatever coordinates the map is written in;
• the principal scale factors are the square roots of the eigenvalues of I⁻¹A, with I the surface’s own first fundamental form;
• the areal factor is |det J| / √(EG − F²);
• and the angular deformation follows from the principal scales exactly as it does everywhere else on this site.

Where F = 0 and the parameters are longitude and latitude, every line of that reduces to the ellipsoidal formulae. That reduction is the check the general route needs and it is asserted rather than assumed: run both on Mars, where both apply, and the areal factors agree to 1.3 × 10⁻¹¹ and the angular deformations to 6 × 10⁻¹⁰ degrees.

What F ≠ 0 means, and why it removes the auxiliary latitudes #

F is the entry that measures whether the coordinate lines cross at a right angle. On any body of revolution it is zero by symmetry, everywhere, for the same reason that a body of revolution looks the same from every longitude.

On a triaxial body it is not, and one figure is the whole of that measurement.

The consequence is larger than the number looks. Every one of the six auxiliary latitudes — conformal, authalic, rectifying, parametric, geocentric, isometric — is an integral along the meridian. Each is defined by walking from the equator to a parallel along a curve, accumulating something, and inverting. Here every meridian is a different curve. There is no the, and so there is no auxiliary latitude to write a formula in.

A formula derived for a sphere, applied to a body that is not one #

The cleanest way to show that the general machinery is measuring something is to hand it a formula whose property is exactly known and watch the property fail.

Lambert’s cylindrical equal-area projection maps a point to (λ, sin φ). On a sphere it is exactly equal-area — asserted here since the collection’s first essays — and it is the projection the projection that shows true size is built on. Applied to a triaxial body’s own parameters and measured against that body’s own first fundamental form, its areal factor is not constant.

Twenty-eight per cent on Vesta. That is not a subtle failure and it is not a rounding artefact of an unfamiliar body: it is the same species of error as Web Mercator is not conformal, which is this site’s headline — a formula carrying a property’s name because of the surface it came from, applied to a surface it did not come from, and never checked.

The check costs one function call, because the machinery does not know or care which body it is pointed at.

The map that does work, and what it gives up #

An equal-area map of this body exists, and it is not hard to build. In the parameterisation (u, v), the area element is √(EG − F²) du dv, so a map with

$$X = u, \qquad Y(u, v) = \int \sqrt{EG - F^2}\; dv$$

has a Jacobian determinant of exactly √(EG − F²) and is exactly equal-area, whatever the body is shaped like. The measurement confirms it at a part in a million, computed from the drawn map’s own derivatives rather than from the construction.

What it gives up is the thing every equal-area map of a spheroid has: that Y is a function of latitude. Here Y depends on u as well, because the strip of surface between two parallels has a different area depending on which way round the body it is taken.

The pole of this map is not a point and not a line. It is a curve, and its shape is a property of the body.

What a pipeline does with the parameters, if nobody stops it #

There is a third map in the machinery and it is the one most likely to be drawn by accident: take the two parameters, call them longitude and latitude, and plot them straight. That is the plate carrée written in the body’s own parameters, and it is what any general-purpose tool produces when handed a table of (u, v) and asked for a picture.

Measured on Vesta, its areal factor spreads by a factor of 18.4 across the surface.

That number is not interesting because it is large — the plate carrée is not an equal-area projection anywhere and nobody claims it is. It is interesting because it is the baseline against which the other two have to be read. Lambert’s formula takes 18.4 down to 1.28, which is most of the way; the body’s own map takes it to 1.000. So the spherical formula is not useless on a triaxial body. It is a good approximation carrying an exact name, which is a worse position to be in than being obviously wrong, and it is exactly the position equal-area on the wrong body describes for the ellipsoidal case on Earth.

The comparison also gives the flattening argument a second body to stand on. On the Earth the same substitution — spherical formula, ellipsoidal surface — costs a fifth of a per cent; on Mars six tenths; on Vesta twenty-eight per cent. Those three numbers are one function of one parameter, and the parameter is how far the body is from a sphere.

The spread is the ratio of the longest axis to the shortest #

The four numbers in the spread figure — 1.003 on Earth, 1.006 on Mars, 1.281 on Vesta, 1.424 on Phobos — are asserted above to be one function of one parameter. They are, and the function is simpler than the assertion suggests: it is

$$\text{spread} \;=\; \frac{a}{c},$$

the ratio of the body’s longest semi-axis to its shortest, and nothing else.

Against the four published figures: the Earth’s 6,378.137 over 6,356.752 is 1.00336 against a measured 1.003; Mars’s 3,396.19 over 3,376.20 is 1.00592 against 1.006; Vesta’s 286.3 over 223.2 is 1.2827 against 1.281; Phobos’s 13.0 over 9.1 is 1.4286 against 1.424. Four bodies spanning a factor of a hundred and thirty in the departure from a sphere, and the identity holds to the last digit the measurement prints.

That collapses the whole figure into a division a reader can do in their head, and it makes the claim about the flattening exact rather than qualitative. For an oblate body a/c is 1/(1 − f), so the spread is the flattening to first order — which is why the Earth’s 0.3 per cent and Mars’s 0.6 per cent look like their flattenings, because they are. For a triaxial body the middle axis does not enter at all: only the longest and the shortest decide how badly a spherical equal-area formula fails, and a body squashed in one direction and stretched in another is penalised for the whole range between them rather than for either separately.

It also says what the identity’s zero is. A sphere has a = c and a spread of exactly one, so the formula is exactly equal-area there and the failure is entirely the departure from that equality — which is the content of the whole section, arriving as an equation rather than as four measurements.

The practical use is small and real. Given any body’s published triaxial radii — which is the first thing the IAU working group prints — the cost of drawing it with a spherical equal-area formula is available before any code runs, and so is the answer to whether it matters. A body whose axes agree to a per cent can be mapped with the spherical formula and a stated one-per-cent caveat; one whose longest is a quarter longer than its shortest cannot, and the threshold is wherever the reader’s tolerance falls between them.

What was computed, and how #

Every number above comes from four quantities and one eigenvalue problem.

The surface. A triaxial ellipsoid parameterised as (a cos v cos u, b cos v sin u, c sin v). The parameter v is deliberately not called a latitude anywhere in the code, because it is neither of the two conventions a planetary coordinate is published under and treating it as one is precisely the mistake the ladder’s third rung is about.

The metric. E, F and G from the two tangent vectors, in closed form. The area element √(EG − F²) follows, and the obliquity of the coordinate lines is arccos(F/√(EG)).

The map. A forward function from (u, v) to the plane. Three are implemented: the parametric plate carrée, which is what a pipeline produces when it treats the parameters as longitude and latitude; Lambert’s formula; and the body’s own strip-area map.

The distortion. The Jacobian by central difference — the same second-order-accurate scheme every other figure on this site uses — then the generalised eigenvalues of I⁻¹JᵀJ, whose square roots are the principal scale factors. Nothing in that sequence mentions latitude.

The strip areas are Simpson’s rule over the parameter, at 120 intervals, and their accuracy is checked the way the rest of the site checks a quadrature: by requiring the resulting map’s measured areal factor to be one. It is, to 3 × 10⁻⁷.

Two routes to the same areal factor #

The site’s habit is that no quantity is printed until two independent routes agree on it, and the areal factor here has two.

The first is the drawing’s own: differentiate the forward map numerically, take the determinant, divide by √(EG − F²). The second is the construction’s: the strip-area map was built to have a determinant of √(EG − F²), so its areal factor is one by design. The two agreeing to three parts in ten million is a check on the quadrature, on the differencing step, and on the sign conventions in the metric — any one of which could be wrong in a way that produced a plausible map.

That check matters more here than usual because there is no external answer to appeal to. On the Earth a suspicious result can be compared against a published transformation or a national grid; for Vesta there is nothing to compare against except the machinery’s own consistency, which is why the machinery is required to disagree with itself when it is wrong. The refusal is the oblate case: run the same general routine on Mars and it must return exactly what the ellipsoidal formulae return, and it does, to eleven digits.

The angular deformation is enormous, and that is not the body’s fault #

The equal-area map of Vesta shears by up to 169°, which looks alarming beside the 125° a Mollweide map of the Earth reaches. It is not a fact about the body.

A cylindrical equal-area map of a sphere also shears without bound at the pole: holding area while stretching one direction forces the other to shrink by the reciprocal, and near the pole the parallel is short. The measurement here is the same effect on a body whose parallels are shorter still. What the comparison shows is that the trade-off argued in the trade-off is two lines is not a statement about spheres. It is a statement about surfaces with curvature, and every one of these bodies has some.

Where the model stops #

The bodies are ellipsoids and the real ones are not. Vesta has a crater at its south pole large enough to have removed about one per cent of its volume, and Phobos is not smooth at any scale a lander cares about. The triaxial ellipsoid is a fitted figure, exactly as WGS84 is for the Earth, and everything above is a statement about that fitted figure. What it is not is a shape model: those exist, they are polygon meshes with millions of facets, and a map drawn on one is a different object with a different provenance — the same argument that keeps coastline datasets out of computing an area needs a surface.

The map is one construction, not a family. The strip-area map is the obvious equal-area map of this surface and there are others; nothing here searches for the best one, in the sense Chebyshev’s criterion means. On a body with no axis of revolution the regional criteria this site owns would all still apply — they are integrals over a region, and a region is a set of points — but the aspect machinery would not, because rotating a triaxial body about anything other than a principal axis does not leave the parameterisation intact.

Nothing here is conformal. A conformal map of a triaxial body exists — the surface is locally conformally flat, as every smooth surface is — but constructing one needs an isothermal coordinate, which on this surface has no closed form. The site’s own solved-series machinery could in principle be pointed at it, and that is the honest description of a thing not yet done rather than a thing shown to be impossible.

Who found it, and when #

The triaxial ellipsoid is older as a figure of the Earth than as a figure of anything else. Clairaut and later Laplace considered whether the Earth’s equator might be elliptical; nineteenth-century geodesists measured it and reported equatorial semi-axes differing by a few hundred metres, a result that survived into the twentieth century and then dissolved as the measurements improved. Jacobi’s theorem — that a sufficiently fast-spinning fluid body settles into a triaxial ellipsoid rather than a spheroid — was published in 1834 and is why the shape belongs in the discussion of small bodies at all.

The mapping of small bodies is recent and practical. The IAU’s working group on cartographic coordinates has published triaxial radii for asteroids since the first spacecraft encounters, and the projection literature for them is thin: Snyder’s 1985 survey of map projections says nothing about surfaces without an axis of revolution, because in 1985 there were no maps of such a surface to make.

Where the ladder goes next #

The machinery here answers what a coordinate on a triaxial body means and what a map of it costs. It says nothing about where the zero of height on such a body is, and that turns out to be the harder question — not because the geometry is difficult but because there is no sea to settle it.

On Earth the geoid is found rather than chosen: water finds the equipotential surface by itself, and the reference ellipsoid was then defined to be a level surface so that the two agree. Every other body has to choose. Mars has chosen twice this century, and the two choices differ by kilometres.