A conformal map of a body with three axes

A map of a body with three axes built the equal-area map of a triaxial body from the body’s own strip areas, held its areal factor at 1 to a part in a million, and recorded plainly what it could not do:

No conformal map of a triaxial body. The equal-area case is built and measured; the conformal one needs an isothermal coordinate, which on a surface with no axis of revolution has no closed form.

That is wrong, and the correction is older than the difficulty. A triaxial ellipsoid has an isothermal coordinate obtainable by two one-dimensional quadratures, in coordinates Jacobi published in 1839 in order to integrate geodesics.

Why the site’s own coordinates could not do it #

The parameterisation everything here has used for a triaxial body is (a cos v cos u, b cos v sin u, c sin v) — longitude and a parametric latitude — and it has one defect that decides everything.

The coordinate lines do not cross at a right angle. On a body of revolution they always do, which is why every auxiliary latitude on this site exists and why none of them survives on a triaxial body; on Vesta they are 1.56° from square and on Phobos 7.50°. An isothermal coordinate is one in which the metric is a scalar multiple of the flat one, and a metric with a cross term is not a multiple of anything.

So the failure recorded was real: in those coordinates there is no route to a conformal map. What was missing was that the coordinates were a choice.

The coordinates Jacobi found #

Jacobi’s ellipsoidal coordinates are the surface’s own lines of curvature, and for a triaxial ellipsoid with a ≥ b ≥ c they have closed forms:

$$x = a\cos\omega\sqrt{\frac{a^2 - c^2 - (b^2-c^2)\sin^2\beta}{a^2-c^2}}, \quad y = b\cos\beta\sin\omega, \quad z = c\sin\beta\sqrt{\frac{b^2 - c^2 + (a^2-b^2)\sin^2\omega}{a^2-c^2}}$$

with β and ω running over the same ranges a latitude and a longitude do. On a body of revolution they collapse to the ordinary ones.

Two properties matter and both are measured here rather than taken from the pedigree. The lines cross at right angles, to 2 × 10⁻⁸ degrees. And the metric is of Liouville’s form — a factor times du²/A(u) + dv²/B(v), with each function of one variable alone.

The second property is the one that does the work, and it is worth seeing as an equation. If E/G = f(β)/g(ω), then

$$U = \int\sqrt{f}\,d\beta, \qquad V = \int\sqrt{g}\,d\omega$$

turns the metric into ρ(dU² + dV²), and (U, V) is isothermal. Both integrals are one-dimensional. There is no partial differential equation anywhere, no boundary-value problem, and nothing to iterate — which is precisely what the recorded shortfall assumed there would be.

The map, and what it holds #

The isothermal coordinate used directly is the conformal map, exactly as Mercator’s coordinate is Mercator’s map. The measurement is then the ordinary one: hand it to the same Tissot machinery every projection here passes through, generalised to a surface with a metric.

It comes back conformal to 1.1 × 10⁻⁶ degrees on Vesta and 1.1 × 10⁻⁶ on Phobos, which is the noise floor of a numerically differentiated Jacobian and is the same figure the genuinely conformal projections of the sphere reach.

The areal spread of 162 is not a fixed property of the body, and the figure says so: it depends on how close to the singular points the sampling goes, since the scale factor there is unbounded. Sampled no closer than 12° to an umbilic it is 79; at 6° it is 162. What is fixed is that the spread is unbounded, which is the statement that matters.

Four singular points, not two #

Every conformal map of a closed surface has somewhere it cannot describe. On a sphere or a body of revolution those places are the two poles, and Mercator’s northing going to infinity — the reason a square world forces a cut at 85° — is the familiar face of it.

On a triaxial body they are the four umbilics — the points where the two principal curvatures are equal, where the surface is locally spherical, and where the coordinate system degenerates because μ = ν. On Vesta they sit at ±63.10° of latitude, on the prime meridian and the antimeridian; on Phobos at ±37.57°.

They are not at the poles, and there are four of them. As the two larger axes approach one another the four slide together in pairs and become the two poles of an oblate body — so Mercator’s singular points are these four, degenerate, which is a statement about the sphere that only a triaxial body makes visible.

The singularity is a clean reciprocal — the fitted exponent is −1.000 — which is a different shape from the one a conformal map onto a polyhedral face meets at a solid’s corner, where the exponent is the flat corner angle over the spherical one and is 3/4 for a cube. Both are conformal maps failing at a point; the reason differs, and so does the rate.

Where the four points are, in closed form #

The umbilic latitudes are quoted above as measurements, and they are also a formula in the three semi-axes, which is worth writing down because it explains the degeneration argument rather than merely asserting it.

An umbilic of a triaxial ellipsoid lies in the plane of the longest and shortest axes, with

$$x = \pm a\sqrt{\frac{a^2-b^2}{a^2-c^2}}, \qquad y = 0, \qquad z = \pm c\sqrt{\frac{b^2-c^2}{a^2-c^2}}$$

and its planetocentric latitude is the arctangent of the second over the first. On Vesta’s axes that gives 63.10° and on Phobos’s 37.57°, which are the two figures the measurement returned — a closed form and a numerical search agreeing, which is the check the section needed and did not have.

The formula also says what happens at the two boundaries of the triaxial family, and both limits are informative.

As the two larger axes come together, b → a, the first radical goes to zero and the second to one, so x → 0 and z → c: all four umbilics slide onto the two poles, in coincident pairs. That is the oblate case, and it is the sense in which Mercator’s two singular points are four points that have collided. Nothing about the sphere shows this, because on a sphere every point is an umbilic and the count is not defined.

As the two smaller axes come together, b → c, the first radical goes to one and the second to zero, so the four umbilics collapse in pairs onto the two ends of the long axis, at latitude zero. That is the prolate case, and it is the reason a prolate body’s conformal map has its singularities on the equator rather than at the poles — the same geometry, rotated ninety degrees, and a fact that a treatment which only ever considers oblate bodies has no way to state.

So the four points are the general case and the familiar two are a degenerate one, twice over and in two different places. Vesta at 63.10° and Phobos at 37.57° sit between the two limits, and their positions say how far each body is from being a body of revolution: Vesta’s equatorial section is 2.7 per cent out of round and its umbilics are two thirds of the way to the poles, while Phobos’s is 12.3 per cent and its umbilics are less than half way. The umbilic latitude is a shape parameter of the body, readable off a conformal map, and it is the second such invariant this essay has found after the modulus.

The one number a conformal map preserves #

The conformal map carries the body onto a rectangle, and the rectangle’s proportions are not free. Any further conformal map of the rectangle can move it about and rescale it, and cannot change the ratio of its width to its height. That ratio is the modulus, and it is a conformal invariant of the body: 1.4618 for Vesta, 2.2397 for Phobos.

This is the one quantity in the essay that has no analogue in the site’s ordinary business. A conformal map of the sphere has no modulus, because the sphere is conformally a plane and every conformal map of it is a Möbius transformation of every other. A body with a rectangle has a number attached to it that is neither a length, an area nor an angle, and that no conformal map can touch.

What was computed, and how #

The point on the body from the two coordinates, in closed form, checked against the ellipsoid’s own equation at 625 points: worst residual 10⁻¹⁶.

The first fundamental form by central difference, from which the orthogonality and the separation both follow. The two one-dimensional integrals by Simpson at 720 steps, tabulated with the integrand at every node — because the integrand is the answer’s derivative, and interpolating without it is the trap this site has now met twice. Linear interpolation between nodes made the map report 38.95 degrees of angular deformation, on a map conformal by construction, because the distortion machinery differentiates at a step eight times finer than the table’s spacing. A Hermite cubic through two values and two known slopes is fourth order and costs nothing.

There was a second and smaller trap in the same measurement. The map has a seam, as every cylindrical map does — the isothermal table stops at ω = ±π — and a derivative taken across it measures the cut rather than the map. Sampled on the edge, 37 of 1,556 points reported 38.94 degrees and the rest reported none, which is what a seam looks like when it is mistaken for a result.

Gauss–Bonnet, which was also owed #

One more recorded shortfall is paid in passing. This site exercises Gauss’s theorem locally on a triaxial body by two independent routes — the curvature from the embedding and from the metric alone — and noted that the global statement, that the total curvature of a closed surface is 2πχ, had not been run.

It is one quadrature in whatever coordinates are to hand, and Jacobi’s are the ones with no obliquity in them. Integrating the Gaussian curvature over Vesta gives 12.56637 against 4π = 12.56637, a relative error of 1.6 × 10⁻¹¹; over Phobos, 1.3 × 10⁻¹¹.

That is not a new result about either body — it is a theorem, and a body shaped like a sphere has to give 4π — which is exactly why it is worth running. A quadrature over a surface, with an area element from one route and a curvature from another, agreeing with a topological invariant to eleven digits, is a check on the whole apparatus rather than a measurement of the asteroid.

Where the model stops #

The bodies are ellipsoids. Vesta and Phobos are not, and a body that is not an ellipsoid is about exactly that gap: the triaxial figure is a fitted shape, and the real surface departs from it by kilometres. Everything here is about the fitted shape.

The construction is for a triaxial ellipsoid specifically. The Liouville property is not a general fact about surfaces — it is what makes the ellipsoid’s geodesic flow integrable, and it is why Jacobi cared. A general convex body has no such coordinate and would need a numerically solved isothermal coordinate, which is the difficulty the original shortfall described and which remains real for anything that is not a quadric.

And the map is not useful for planetary cartography as it stands. A rectangle with four singularities on its edges is not what a data product wants; what a working scheme would do with this coordinate is what the sphere’s conformal maps do with Mercator’s — use it as the chart in which some other map is an analytic function.

What this makes available #

Two things are now possible on a triaxial body that were not, and both are worth naming because they are the reason a conformal map is more than a curiosity.

A conformal chart is where analytic functions live. Once a surface has an isothermal coordinate, every conformal map of it is an analytic function of that coordinate — which is the fact this collection uses to solve for conformal maps of regions of the sphere by least squares. The same machinery could in principle be pointed at a region of an asteroid, and solving for the map instead of choosing it is the construction that would do it.

And a conformal coordinate is what a grid needs. Every national grid on Earth is conformal, for the reason a surveyor needs one scale factor per point rather than two — so a body with a conformal coordinate is a body on which a survey grid could be defined at all. That is not an idle observation about asteroids: the same argument applies to any body whose figure is fitted with three axes rather than two, and the number of such bodies with spacecraft data is growing.

Neither is done here. What is done is the coordinate, and the coordinate is the thing that was recorded as missing.

The generalisation #

The lesson is not about asteroids and is worth stating in the form the shortfall’s own wording invites. The obstruction that was recorded was a property of a coordinate system, not of a surface. No conformal map existed in the coordinates the site had; one existed in coordinates that had been in the literature since 1839, for a problem that had nothing to do with conformality.

The general form: a construction that requires a metric to have a particular shape is a question about the chart, and the answer is often available in a chart somebody built for a different reason. Separability, orthogonality and integrability are properties of coordinates, and the surface is entitled to have several sets.

The shortfall, restated rather than struck out #

One line of the recorded note survives the correction and is worth keeping in its narrower form.

A general convex body still has no closed-form isothermal coordinate. What has been shown here is that a triaxial ellipsoid has one, and the reason is specific: its lines of curvature separate the metric into Liouville’s form, which is the same property that makes its geodesic flow integrable and which is not shared by surfaces in general. A body shaped like Vesta but not fitted by an ellipsoid — the real Vesta, with its craters — has a metric that separates nowhere, and a conformal map of it would need an isothermal coordinate computed numerically by solving an elliptic equation over the surface.

So the note was right about the difficulty and wrong about where it starts. It starts one step further out than it said: not at a body with no axis of revolution, but at a body that is not a quadric.

Who found it, and when #

Jacobi introduced the ellipsoidal coordinates in 1839 and used them to integrate the geodesic equations on the triaxial ellipsoid — the first non-trivial integrable geodesic flow, and the reason Liouville’s form has his name attached to the metric that admits it. Liouville himself set out the class of surfaces whose metric separates that way, and the connection to isothermal coordinates is immediate once the form is written down.

What is recent is nothing in the mathematics. It is that a planetary body’s conformal map is now a thing somebody might want, since the bodies with three visibly different axes are exactly the small ones a spacecraft has been sent to.

Where the ladder goes next #

Two maps of a triaxial body now exist here and each destroys what the other keeps. The obvious next question is the one this site asks for the sphere and has not asked for a body with three axes: what is the least distortion a compromise map of such a body can reach, and does the criterion that answers it for a region of the sphere — Chebyshev’s — have a form here at all? That would need the boundary machinery of solving for the map instead of choosing it pointed at a surface whose conformal chart has just become available.