The gnomonic crosses a seam without a corner

Eight rungs of this ladder are about cutting: where the cut has to go, what a face can preserve, which net loses the fewest neighbours, what the heuristic cannot find.

A net of F faces cuts some edges and keeps F − 1 of them — that is what makes it a spanning tree of the face graph, and it is the object the enumeration rung counted 384 of for a cube. This rung is about one of the kept ones — a seam, where two faces stay joined and carry two different maps.

What the expectation was #

Two adjacent faces are two separate projections. They agree on the shared edge as a set — both draw it, both draw it in the right place — and there is no reason for them to agree as a parameterisation: a point a kilometre along the edge is at one place under the first map and, in general, somewhere else under the second.

So a river crossing the seam should arrive at the edge on one bearing and leave on another, and a reader should see a corner in a feature that has none. That is what this rung was written to measure, and the measurement divides the face maps into two kinds rather than reporting one number.

Under the gnomonic there is no corner #

Zero. Not small: zero, at every crossing angle tried, on all five Platonic solids, to the resolution of a tangent estimated from the drawn curve.

The reason is that the gnomonic face map is not five separate projections. It is one central projection — every point of the sphere sent along the ray from the centre until it meets the polyhedron — and the polyhedron’s surface is one surface. Unfolding it is an isometry of that surface, so a curve that was straight on it stays straight.

And a great circle is straight on it. A great circle lies in a plane through the centre; the ray construction sends it to the intersection of that plane with the polyhedron’s surface, which on each face is a straight segment. The segments meet on the edge, and the unfolding lays them flat without bending either.

That is why every polyhedral globe uses gnomonic faces, and this collection had used them for eight rungs without stating the property they were chosen for.

It is worth being clear about what the property is and is not. It is not that the gnomonic is continuous across the edge in the sense of drawing the edge in the same place from both sides — every face map does that, since the edge is a curve both faces draw. It is that the gnomonic is continuous in its first derivative, which is a stronger requirement and is the one a reader’s eye is sensitive to. A join that meets in position and not in slope is exactly what a draughtsman calls a corner.

Under an equal-area face map there is one #

The shape of the curve in the first figure is worth reading, because two of its three features are forced.

It is zero at a perpendicular crossing. The two faces are mirror images in the plane of the edge, so a curve crossing square on is its own reflection and its two tangents meet. That is a symmetry rather than a measurement, and a machinery that reported a corner there would be reporting itself.

It falls again at a grazing crossing. A curve lying nearly along the seam barely crosses it, so there is very little of the disagreement to accumulate.

And it peaks near thirty degrees of obliquity, between the two, which is where most features cross most seams. Only the position of that peak is a measurement; the two zeros are geometry, and a curve that had its maximum at one of the ends would have said the measurement was wrong.

The net, and how many seams it has #

That figure makes the arithmetic of the trade visible. A net of F faces keeps F − 1 seams and cuts the rest, so a solid with more faces has more of both: an icosahedron’s net carries nineteen seams and eleven tears, and a tetrahedron’s three and three.

So the corner falls with the face count and the number of corners rises with it. Multiplying the two: a tetrahedron’s net has three seams at 0.388° and an icosahedron’s nineteen at 0.115°, which is 1.16 degrees of total folding against 2.19. More faces means a better map at every seam and more seams, and the total is worse — which is a version of the trade more faces, less distortion, more cutting makes about the tears, arriving for the joins.

What a third of a degree is worth #

Nought point three nine degrees sounds like nothing and is worth pricing rather than dismissing.

The number to compare it against is a drawing tolerance rather than a distortion figure, because a corner is not a distortion — it is a defect in a line, and lines are judged by how far they stray from where they should be.

A corner of 0.39° in a feature drawn 200 millimetres long puts the far end 1.4 millimetres off the line it should be on — which is over a printing tolerance and is about the width of a road symbol. On a screen at a thousand pixels across it is seven pixels.

More to the point, it is systematic and it is at a known place. Every feature crossing that seam gets the same corner in the same direction, so a reader sees not a wobble but a fold: a line of discontinuity running down the map exactly where the construction says there is none. That is a worse artefact than a larger random error, and it is the reason a residual’s shape matters more than its size arriving in a picture.

Which seam, and does it matter #

The measurement above is taken on one seam per solid — the first edge of the face graph — and the obvious question is whether a different seam gives a different answer.

On a Platonic solid it cannot. Every edge is equivalent to every other under the solid’s own symmetry group, so a measurement on one is a measurement on all, and the single row per solid in the table is the whole of it. That is a property of the solids chosen rather than of the method: on a geodesic subdivision, where the edges are not all alike, the corner varies from edge to edge and the table would need a distribution rather than a number.

It is worth saying because it is the sort of shortcut that is right for a reason and looks like an omission.

What this does to the choice of face map #

The ladder has been choosing between face maps on what each preserves: the gnomonic preserves great circles and nothing else, the equal-area preserves area, the conformal preserves angle. This adds a criterion none of those three carries.

The gnomonic is the only one of the three that is a property of the solid rather than of a face. It is defined by one construction on the whole surface, so continuity across every kept edge is free. The other two are defined face by face, so continuity across an edge is a coincidence, and it does not happen.

That is a strong argument and it has a strong counter, which the earlier rungs supply: the gnomonic’s distortion is very much the worst of the three, at 26.6° of angular deformation at a cube’s vertex against the conformal’s zero. So the choice is between a map that is continuous across its seams and badly distorted, and one that is well behaved on each face and folded along every join.

Nobody appears to have stated the trade in those terms, presumably because the continuity has never been measured. Stated in them, it is a purpose argument of the kind this field runs on: a wall map read for shape wants the conformal faces and can afford the folds, and a navigation chart read with a ruler wants the gnomonic and can afford the distortion, because a great circle drawn on it is straight across the whole net and that is the one thing the ruler is for.

What the check refuses #

The gnomonic’s zero is the kind of result that is one sign error away from being a bug report, so it is checked from both sides.

The machinery has to be able to see a corner. The same code, the same solids, the same crossings, with the equal-area face map substituted, returns between 0.115° and 0.388°. So a zero is a zero rather than a measurement that never moves.

And the corner has to vanish where symmetry says it must. At a perpendicular crossing the equal-area map returns zero too, on every solid, because the two faces are mirror images in the edge’s plane. A machinery that reported a corner there would be reporting the unfolding rather than the maps.

Between them those two say the unfolding is right and the tangent estimate is right, which is what licenses reading the gnomonic’s column as a property of the projection.

The same property, one level down #

The gnomonic’s continuity across a seam has a smaller cousin that this collection has already used without naming.

A polyhedral map’s faces are flat, so a great circle drawn across a face is a straight segment, and a straight segment drawn across a sheet is what a ruler produces. The gnomonic is therefore the one face map on which a navigator’s ruler answers the question it is being asked — over a whole net rather than over one face, which is the property this rung adds.

That is a strong statement and it comes with an equally strong limitation: it is true of great circles and of nothing else. A rhumb line is not straight on the gnomonic, a coastline is not straight on anything, and the routes an aircraft actually flies are geodesics on an ellipsoid rather than on a sphere — which the ellipsoidal geodesic rung shows differ from the spherical ones by kilometres.

So the gnomonic’s seam property buys exactly one operation, performed on exactly one family of curves, and does so perfectly. That is a better description of a projection than any single distortion figure, and it is the kind of description this field’s rule about naming a purpose was written to produce.

Where the model stops #

The kink is measured by fitting a tangent to the drawn curve on each side, three points from the seam. A tangent estimated from a discrete curve has its own resolution, and the gnomonic’s zero is a zero at that resolution — about a hundredth of a degree — rather than a proof. The proof is the argument above, and the measurement is what says the argument was implemented correctly.

The equal-area face map here is the one this library builds, matching cumulative areas by integrating an ordinary differential equation, and a different equal-area face map would give different numbers. What would not change is that there is a corner at all, because no face-by-face construction has any reason to agree with its neighbour’s parameterisation.

And the crossings are great circles. A feature that is not a great circle — a coastline, a rhumb, a road — is not straight on the gnomonic either, and its corner across a seam is not zero. What is zero is the extra corner the seam adds, which is the quantity this rung is about; a curve’s own bending crosses the seam unchanged.

The generalisation #

Continuity across a join is free when the pieces come from one construction and has to be paid for when they do not.

That sentence covers this rung and a great deal outside it: a spline assembled from independently fitted segments, a mosaic of independently georeferenced sheets, a tiling of independently generalised features. In each case the pieces are individually right and the joins are individually wrong, and the fix is never a better piece — it is a construction that produces all the pieces at once.

The cartographic form is older than any of that and is the reason a national grid has one central meridian rather than one per sheet: a seam is a place where two decisions meet, and the only way to have no seam is to have made one decision.

Who found it, and when #

That the gnomonic sends great circles to straight lines is ancient — it is the oldest projection known, attributed to Thales, and the property is why it is used for great-circle sailing charts. Its use for polyhedral globes is nineteenth century and is why the Fisher and the Cahill maps and their modern descendants all use gnomonic faces.

The continuity across the seam follows immediately from the construction and is, as far as this collection can tell, never stated as a criterion for choosing the face map — it is folded into the observation that great circles are straight, which is a statement about each face and is the weaker half of the property.

What this rung holds fixed, and what happened when it was let go #

Every crossing measured above is at the midpoint of the seam, and the position along the edge was treated as immaterial — a feature crossing square on has no corner because the two faces are symmetric about the edge, and everything here was taken to be a function of the obliquity alone.

That is true at the midpoint and only there. The corner is not at the midpoint, at the next rung, moves the crossing along the edge and finds that the shear term deciding the corner is an odd function of distance from the midpoint — so every face map on this site is smooth at that one point, for a reason that has nothing to do with which map it is. Two fifths of the way to a cube’s vertex the gnomonic reads 20.15°, which reverses the ordering above: the map with no corner here has the largest corner of the three everywhere else.

Every number on this page stands. What it describes is one point of each edge rather than the edge.

What the correction did and did not change #

The paragraph above is worth reading as a piece of method as well as a correction, because the shape of it recurs.

Nothing measured here was wrong. Every number is a correct measurement of a crossing at the midpoint of a seam, taken with a stated method, reproducible. The defect was in the sentence around them: the gnomonic crosses a seam without a corner is a claim about seams, and what was measured was one point of each.

The generalisation was not reckless — it was invisible. Position along the edge was not considered and rejected; it was not considered. The obliquity was the variable being swept, the midpoint was where a sweep naturally starts, and the symmetry that makes the midpoint special is exactly the symmetry that makes it the obvious place to measure.

Which is the trap worth naming. A symmetric object has points at which several quantities coincide, and those points are the most natural places to take a first measurement — so a measurement taken there is the most likely one to be generalised and the least entitled to be. This collection has met the same shape in where the worst point is and in the places a projection has no distortion at all.

The remedy is to state a measurement’s scope in the sentence that reports it, not in the method section: at the midpoint of a seam is four words, and it converts a claim that turned out to be false into one that is true and narrower.

Where the ladder goes next #

Nine rungs measure what a polyhedral map costs at a point, over a face, along a net and now across a seam. What none of them measures is what it costs at a vertex, where three or more faces meet and the surface has a cone point: the angle deficit is exactly computable, it is the one place the construction is not smooth in any face map, and every net puts several of them on its boundary.