A conformal map onto a face

What a face can preserve built two maps onto a polyhedral face — the gnomonic, which draws every great circle straight, and an area-preserving one built by integrating a differential equation — and ended by recording that the third member of the family was owed. A conformal face map exists, it is the one Lee’s projection uses, and the note said it needs the Dixon elliptic functions and would be a complex-analysis library living inside a mapping one.

That was the wrong reason to defer it, and the right one is more interesting.

A conformal map of the sphere is an analytic function of a conformal coordinate. That is what conformal means, once a coordinate is in hand in which the sphere’s own metric is a scalar multiple of the flat one. So a conformal face map is a power series, and choosing which power series is a linear problem — because what is asked of it on the face’s boundary is that one Cartesian coordinate be constant there, which is linear in the coefficients.

No elliptic functions. A least-squares fit, and a singularity at each corner that no series can remove.

The chart, and why it is not Mercator’s #

The obvious conformal coordinate is the isometric one, λ + i·ln tan(π/4 + φ/2), which is Mercator’s, and it is the wrong choice here. The reason is worth recording because it was tried first.

The whole economy of this construction is its symmetry. A face has n-fold rotational symmetry about its centre and the map must commute with it, which removes all but every nth coefficient and turns a fit with forty unknowns into one with six. In the isometric plane a rotation of the sphere about the face’s own axis is not a rotation of the plane — it is not even close, away from the origin — so the symmetry is not expressible and the saving is lost.

In the stereographic chart centred on the face, it is. A rotation about the centre becomes ζ ↦ e^{2πi/n}ζ exactly, so the basis is ζ, ζ^{n+1}, ζ^{2n+1}, …, the mirror symmetry makes the coefficients real, and the chart is bounded on a region, which the isometric one is not near a pole. The sphere’s metric in it is ds = 2|dζ|/(1 + |ζ|²), which is one line and is checked against arc length rather than trusted.

What is asked of the series, and what is not #

The fit has one condition and one free quantity.

On the face’s edge, the image must lie on a straight line — for the edge chosen to sit on the +x axis of the chart, that is Re f(ζ) = s for a half-width s. The position along that edge is left free, which it must be: a conformal map is not allowed to be told where each boundary point goes, and a fit that specified it would have no solution.

The half-width s is carried as an unknown of the same fit, because fixing both the leading coefficient and the width would over-determine a map whose overall size is arbitrary. One edge is fitted rather than all n, because the symmetry is already in the basis and fitting every edge would weight each corner n times over.

What is not asked for is conformality, and this distinction governs how every number in this essay should be read.

The conformality is not an approximation. A polynomial in a conformal chart is analytic wherever it is evaluated, so the angular deformation is zero to the arithmetic’s floor whatever the fit did, at four terms or at forty. Reporting that as a success of the fit would be a caption that cannot fail.

The image is the approximation. A short series gives a map whose face edge is not quite straight, and that residual is what more terms buy. At eight terms the cube’s mapped edge departs from a straight line by 0.74 per cent of its own half-width; at sixteen, by 0.38 per cent.

The corner, where the map stops being analytic #

A cube’s spherical face is a spherical square, and its corners are not right angles. Three faces meet at a cube vertex, so the interior angle of the spherical face there is 2π/3 = 120°, while the flat square it is carried onto has a corner of 90°.

A conformal map that takes a 120° corner to a 90° one behaves like ζ^{3/4} there. Its derivative behaves like ζ^{−1/4}, which is unbounded — so the scale factor of a conformal polyhedral map is infinite at the solid’s corners.

That is not a failure of the fit. It is the angle deficit, arriving as a singularity of the map instead of as a tear. The deficit is where all of the polyhedron’s curvature is concentrated, as the globe on a solid computes; a conformal map must respect angles everywhere it is analytic, so it cannot be analytic where the angles do not add up.

The exponent is a closed form for every solid, and it is different for each:

The five exponents in that column are not five separate facts. Where q faces meet at a vertex the spherical corner is 360°/q, and a regular p-gon’s flat corner is 180°(p − 2)/p, so the exponent is q(p − 2)/2p — which reproduces every row. Written that way it can be compared against the vertex’s angle deficit, and the two turn out to be the same quantity.

The deficit at a vertex is 360° less the q flat corners that meet there, which is 360°(1 − α) exactly. So

$$\alpha = 1 - \frac{\delta}{2\pi}, \qquad \text{and the scale diverges as } d^{-\delta/2\pi}.$$

The rate at which a conformal polyhedral map blows up at a corner is the angle deficit there, divided by a full turn, and nothing else. The cube’s 90° of deficit gives an exponent of exactly a quarter; the tetrahedron’s 180° gives a half; the dodecahedron’s 36° gives a tenth. Every number in the table above is one subtraction from a number the globe on a solid already computed for a different purpose.

One step further closes it. Gauss–Bonnet fixes the total deficit of any convex polyhedron at 4π, and on a solid whose vertices are all alike that total is shared equally, so δ = 4π/V and the divergence exponent is 2/V: a quarter for the cube’s eight vertices, a half for the tetrahedron’s four, a tenth for the dodecahedron’s twenty, a sixth for the icosahedron’s twelve. The severity of the singularity is decided by counting the corners, and by nothing about the map at all.

That is what turns the practical advice at the end of this essay from a preference into arithmetic. A conformal polyhedral map wants many vertices, because the curvature it cannot represent is a fixed 4π that has to be put somewhere, and putting it in twenty places rather than four makes each place milder by exactly the ratio of the counts. It is the same conservation the gnomonic version obeys with the opposite sign — fewer faces mean less cutting and worse stretching — and the two preferences pull against each other on the same conserved quantity.

What was computed, and how #

The design matrix has one row per boundary sample and one column per coefficient, plus a column for the half-width. Two hundred samples along one edge, six to thirty-two coefficients, solved by normal equations with the columns scaled to unit norm.

That scaling is not decoration. The columns are Re ζ^{nm+1} on a region where |ζ| ≤ 1, so the last column’s norm is smaller than the first’s by whatever |ζ| typically is raised to the power kept. Solving the unscaled system loses the high-order coefficients entirely, and the residual then stops falling as terms are added — which reads exactly like convergence and is the opposite of it.

The conformality is measured with the site’s own distortion(), the same routine that measures Mercator, applied to the solved map as an ordinary projection object. That is the whole reason the solver returns a projection rather than a pair of functions: every measurement the site owns then applies to it unchanged.

The corner exponent is measured by walking towards the corner along the great circle from it to the face centre, at distances from 8° down to 0.125°, and fitting the slope in logarithms.

A face of a cube, on the sphere it came from #

Before the maps, the object being mapped is worth one figure of its own, because spherical square is a phrase that hides an assumption.

The corner angles are what the whole essay turns on, and they are visible here: the spherical square’s corners are visibly fatter than right angles, which is the 120° the table below records, and it is the amount by which a spherical polygon exceeds its flat namesake — the spherical excess this site computes elsewhere from Gauss–Bonnet.

The three face maps, side by side #

The ladder now has all three members of the family, each exact in the one property it exists for.

There is no fourth column, and the reason is the theorem the whole site is built on: the trade-off is two lines shows that conformal and equal-area together force an isometry, which the curvature forbids. That argument was made about the sphere and it applies verbatim inside a single face, because a face of a spherical polyhedron is a piece of sphere.

Adding the third member changes neither of the first two, and that is worth saying plainly: the gnomonic and the area-preserving face maps are exactly what they were. What changes is that the family is now closed, and that the closing member is the one which cannot be drawn at its corners at all.

How much of a face the corner spoils #

The exponent decides how fast the scale grows, and the more useful question for a drawing is how large the region is in which it has grown by any given amount. That follows from the same law and it shrinks far faster than the exponent does.

Take some reference distance from the corner and ask where the scale has doubled relative to it. Since the scale goes as d^(−2/V), doubling requires the distance to fall by a factor of 2^(V/2). On a cube that is a factor of sixteen; on an octahedron, eight; on an icosahedron, sixty-four; on a dodecahedron, one thousand and twenty-four. So the neighbourhood of a corner in which a dodecahedral conformal map is visibly stretched is three orders of magnitude smaller than the corresponding neighbourhood on a cube, for a difference of only 0.15 in the exponent.

That is the practical form of the whole argument and it inverts the impression the exponent table gives. Read as five numbers between 0.5 and 0.9 the solids look comparable; read as the size of the spoiled region they are not remotely. A tetrahedral conformal map is stretched over a visible fraction of every face, which is why Lee’s map looks the way it does; a dodecahedral one is exact to the eye everywhere except within a thousandth of each vertex, where no reader is looking and no feature is resolved.

The singularity is therefore not the reason to avoid a conformal polyhedral map. It is a reason to choose the solid, and the choice it argues for is the same one the cutting argues against.

What the singularity costs a real map #

Lee’s conformal projection of the world onto a tetrahedron is a published, printed map, so the singularity is evidently survivable. What it costs is worth stating.

At a corner the scale is infinite, so a neighbourhood of every vertex is drawn arbitrarily large. The map is still usable because the divergence is slow: the scale grows like d^{−1/4} for a cube, so at a tenth of a degree from the corner it is only about three times its value at a degree. On a tetrahedron, where α = 0.5, the divergence is d^{−1/2} and the same tenfold approach costs a factor of three as well — but the tetrahedron’s corners are fewer and its faces vastly larger, so more of the world sits near one.

The practical consequence is that a conformal polyhedral map is best on a solid with many faces and mild deficits, which is the opposite of the direction the gnomonic version prefers, where fewer faces mean less cutting.

How fast the series converges, and why it is not fast #

A boundary condition on a smooth curve is fitted by a power series geometrically: each term buys a constant factor. A boundary with corners is fitted algebraically, because the function being represented has a branch point at each corner and a polynomial has none.

For the face map the fitted rate is about −1.1 in the number of terms, so halving the edge residual costs roughly doubling the series. That is slow by the standards of a series and fast by the standards of a map: sixteen coefficients gives an edge straight to a part in three hundred, which is finer than the width of a drawn line.

What a net of these would look like #

The conformal face map has an odd property when the faces are put back together: the map is conformal on each face and the faces meet at angles, so the net is conformal everywhere except along the cuts and at the corners — which is exactly the description of Lee’s published map.

That separation is worth stating because it is what makes the polyhedral family modular. The choice of solid fixes the cutting; the choice of face map fixes the distortion; and the two decisions are independent, which is unusual — on every other projection family in this site the analogous choices are entangled.

Where the model stops #

This is not Lee’s map. Lee’s projection is the exact conformal map, expressible with Dixon’s elliptic functions, and it has the singularity built in rather than approached. What is computed here is a sequence of exactly conformal maps whose images converge on the right polygon. For a drawing the difference is invisible at sixteen terms; for a definition it is the whole difference, and a national mapping agency wanting a reproducible projection would want the closed form.

Only regular faces are handled. The basis is built from the n-fold symmetry, so an irregular face — which is what a subdivided geodesic solid has — has no symmetry to exploit and needs the general Schwarz–Christoffel machinery with an unknown boundary correspondence. That is a genuinely harder problem and it is not attempted.

The corner exponent is measured on the interior of the face and never at the corner itself, because the map’s derivative there is infinite and any measurement of it would be a measurement of the sampling. The essay’s claim is about the approach, which is what a singularity permits anybody to say.

Who found it, and when #

Oscar Adams published the conformal map of the world onto a square in 1925, and onto other polygons in the years around it, using elliptic integrals — a line of work that goes back to Peirce’s quincuncial projection of 1879, which is the hemisphere-onto-a-square case and is the earliest conformal map with a polygonal boundary in the literature.

L. P. Lee gave the conformal projection onto a tetrahedron in 1965 and onto the other Platonic solids afterwards, using Dixon’s elliptic functions — the functions Alfred Dixon introduced in 1890 for the curve x³ + y³ = 1, which have three-fold symmetry where the Jacobi functions have four-fold, and which nobody had had a use for.

The construction here is not theirs. It is the same object obtained by the numerical route the Schwarz–Christoffel literature calls the parameter problem, made easy by the symmetry — and it is worth noticing that the numerical route is the one that shows the corner behaviour as a measurement rather than as a term in a formula.

Where the ladder goes next #

The family of face maps is now closed, and every one of them is a map onto a face. What has not been asked is how many faces there should be.

The five Platonic solids are where polyhedral mapping usually stops, and they stop for a reason with nothing to do with mapping: there are only five. Subdividing an icosahedron gives a convex solid with any number of faces, every one of them still flat and still unrollable — so the family does not stop, and the trade it offers has two exponents that can be fitted.