The kink at a seam

Six flat pictures of everything establishes what a cube map is and why it exists: six ordinary flat pictures at 90° apiece, covering the whole sphere with nothing worse than a factor of 5.196 in area at a face’s corner. A pole is a line says what it buys — it is the only surface here with no singular direction at all, because it never claims to hold the sphere in one chart.

The price is paid at the joins, and this essay is what the price actually is.

Each face is a gnomonic chart — a plane through the centre of the sphere, which is to say an ordinary pinhole picture — so within one face every great circle is drawn exactly straight. That is the strongest straightness property any surface here has. Cross a seam and the line is still made of straight pieces, and the pieces do not line up.

A kink is not a bend, and the ordinary measure cannot tell them apart #

The distinction in the title is the point of the essay, and it is easy to lose because the usual instrument reports the same thing for both.

Every other surface in this collection bends a straight line: the lines a surface leaves alone measures how far a drawn great circle departs from the chord between its ends, and on a cylinder or a fisheye that departure is a smooth curvature spread along the whole arc. The cube map’s departure from the chord is comparable in size — and it is not a curvature at all. It is two perfectly straight segments meeting at a point.

Three floors and one large number is the shape of that figure, and the large number is what makes it a measurement. The three zeros say: the two computations of the kink agree, each side really is straight, and a midpoint crossing really is unbroken. Any one of those alone would be an arithmetic check. Together with a 44.42° that the same machinery produces, they are a description of the failure — the line is exactly straight, twice, and the two straights disagree about direction.

That distinction has a consequence for anybody sampling such a picture. A curvature can be approximated away by drawing a curve; a kink cannot, because there is no curve there. Interpolating across a seam produces something that is wrong in a way no amount of resolution improves, since the true image genuinely has a corner in it.

Zero at the midpoint, whatever the slant #

The kink is not the same everywhere along a seam, and where it vanishes is worth more than the maximum.

Every line through the midpoint of a seam crosses with no kink at all, at any slant. That is a stronger statement than it looks, and it is the control the essay’s whole claim rests on: a figure showing kinks everywhere would be showing that a cube map breaks lines, which is nearly true and not the useful version. What is true is that a cube map breaks lines except along a particular curve, and the exception is exact rather than small.

The reason is symmetry. The midpoint of a seam is equidistant from the two face centres, and the two gnomonic charts meeting there are mirror images across the seam plane. A great circle through that point has slopes on the two sides that are reflections of each other and therefore equal, so nothing breaks.

Move away from the midpoint and the symmetry goes, and the two arctangents stop agreeing. Increase the slant and the same thing happens faster.

The bound, and where it comes from #

The worst kink is not unbounded, and the number it is bounded by is a familiar one.

Fifty-three point one three degrees is the three-four-five angle, and it turns up here for a reason that can be stated in one line.

It is worth noticing that the bound does not depend on the slant. Both sweeps above — one at 30° to the seam and one at 60° — approach the same 53.130° at the corner, and only the rate differs. That is because the bound is a property of the two chart normals rather than of the line being drawn: at the corner the disagreement between the charts is maximal, and a line arriving there merely samples it. A family drawn at any slant will find the same ceiling if it reaches far enough along the seam. At a cube’s corner three faces meet, and the two charts on either side of a seam disagree most sharply about a direction that lies along the corner’s own diagonal. The half-angle between the two chart normals there has tangent one half, so the total disagreement is twice its arctangent.

That the bound is attained only at the corner — a single point, where three seams meet — is worth saying, because it means the typical kink is much smaller. Along most of a seam a line crossing at a moderate slant breaks by a few degrees, and the 53.130° is a worst case reached in the one place where the cube map is already at its worst on every other measure too, with an area scale of 5.196 and an anisotropy of √3.

Everything bad happens at the corner, and that is not a coincidence #

Three separate measurements of the cube map reach their worst at the same place, and it is worth noticing that they are one fact rather than three.

The kink is bounded by 2·atan(½) = 53.130° and attains it at a corner. The area scale runs from 1 at a face’s centre to 5.196 at a corner. The anisotropy — how much a small circle is stretched into an ellipse — is 1 at a face’s centre and √3 = 1.7321 at a corner.

All three are consequences of one quantity: how far off its own axis a face has to reach to hold a given direction. A face covers 90°, so its extreme directions are 45° off-axis along an edge and about 54.7° off-axis at a corner, and a gnomonic chart’s distortions all grow as the secant of that angle. The corner is simply the furthest any direction gets from any face centre, so it maximises everything at once.

That gives the cube map a clean description as a compromise. Its distortions are those of an ordinary flat picture — the very ones where a surface spends its pixels measures running to ×25 and beyond on a wide plane — truncated at 90° and repeated six times. A plane misbehaves without bound because it is asked to reach arbitrarily far off-axis; a cube face is never asked to reach past its corner, and every one of its numbers is the plane’s number evaluated there.

So the six faces are not a different kind of surface. They are the worst-behaved surface in the collection, used only over the part of its range where it behaves — which is a design worth naming, because it is the opposite of how the other five were arrived at. Those were each derived by asking what property to preserve and accepting whatever behaviour followed. This one starts from a surface whose behaviour is known to be bad and fixes the range instead.

How many seams one line has to cross #

The kink is a per-crossing quantity, and a reader wanting to know what a long line costs needs the count as well.

A great circle on a cube crosses from face to face a bounded number of times, and the bound is small: it meets the cube’s twelve edges in at most four places for a generic circle, and exactly four for one that does not pass through a corner or lie in a face plane. So a straight world line, drawn all the way round, is at most four straight pieces joined by at most four kinks — never a smooth curve, and never an arbitrarily broken one.

Set that beside the fraction-straight curve below and the trade sharpens. A short line has no crossings and is perfect. A line long enough to go all the way round has four, which is few — but they are the only defects it has, and each is a corner rather than a gentle departure.

It also explains why the fraction of exactly-straight lines falls so steeply with arc length. The chance of avoiding a seam is the chance of staying inside one 90° face, and that becomes unlikely quickly as the arc grows — not because the surface degrades, but because the faces are only so big.

What the seam costs, as a fraction of lines #

The kink is an angle, and the practically useful question is how often a reader meets one.

That curve is the honest summary of the trade and it says something neither the kink angle nor the area scale does. A cube map has a scale below which it is perfect. A short enough line lies inside one face, is drawn exactly straight, and has no error of any kind — 95.8 per cent of lines at a 5° arc. Every other surface bends every line at every length, by a little.

So the two kinds of surface are not better and worse; they are differently shaped. A curved surface distributes a small error over everything. A cube map has zero error over most things and a large one over the rest, and which is preferable depends entirely on whether the work being done is local.

For sampling a texture, local is exactly right and the cube map wins outright — which is why it is the format graphics hardware uses, and why a pole is a line ends by naming it as the only bounded answer. For drawing a picture somebody looks at whole, a seam running through the middle of a long straight edge is the worst possible artefact, and the smooth surfaces win.

What a chart is, and why this is the general case #

The cube map is usually presented as an engineering trick. It is better understood as the ordinary way of covering a curved thing, and the seam is the ordinary price.

A sphere cannot be covered by one chart without a singular direction — a pole is a line argues that from topology, and the cube map is the alternative it names. Cover it with several charts instead and each can be well behaved, and what appears is a transition between charts: two descriptions of the same neighbourhood that have to be reconciled where they overlap.

Here the transition is trivial to state — the two faces are flat and the change of chart is a projective map — and its visible consequence is the kink. The two charts genuinely disagree about which direction a line is going, and neither is wrong. The disagreement is a property of the pair rather than of either — the same shape of statement as undoing a picture made on a curve, where what a rectification needs is not a property of the surface alone.

That is the same shape as the seam a shadow crosses when it runs from a floor onto a wall, or the join between two bands of a scroll, which a scroll is not a panorama measures: two regions each internally consistent, and a boundary at which the reconciliation is visible. A shadow across an edge measures one of those, and the family resemblance is not an analogy — in each case a single object is being described by two maps and the boundary is where the descriptions meet.

Why the seam is where the reader is, and the pole is not #

There is an asymmetry between this essay’s defect and the one next door that decides which surface to prefer, and it is not about size.

A pole is a line measures a failure that sits at one direction, and a direction is something a rig can be pointed away from. Aim the singular direction at the tripod, or at the sky, and no reader ever meets it. The failure is severe, unbounded, and easy to hide.

A cube map’s failure is spread along twelve edges, which between them run through every part of the sphere. There is nowhere to point them. A long straight edge in a room — a skirting board, a roofline, the top of a wall — will cross a seam somewhere, and the kink will be in the middle of the thing a reader is looking at.

So the two surfaces trade a large defect that can be hidden against a moderate defect that cannot. For a texture sampled by a machine that never sees a whole line, the cube map’s trade is obviously right. For a picture displayed to somebody, it is obviously wrong, and the equirectangular surface’s wasted pole is the cheaper price.

That is the practical content of the whole comparison, and it is why neither surface is the better one. A defect’s severity and a defect’s reachability are separate properties, and the collection’s other measurements only report the first.

What this does not settle #

Three limits.

The kinks here are measured on the unfolded net, which is a way of laying the six faces out on a page and is not how a cube map is used. In a viewer the faces are sampled independently and the seam is never drawn at all; the kink then appears as a discontinuity in what is sampled rather than as a visible corner. The angle is the same number either way, and the artefact it produces is not.

The closed form and the measurement off the net agree to 10⁻¹³ degrees, which is the arithmetic floor, and that agreement is a check on the two implementations rather than evidence about cube maps. Both compute the same geometry from the same face normals; they share the geometry and not the code, which is what makes the check worth running and less than it might appear.

And nothing here measures resampling, which is what a reader actually meets. A cube map is stored as six rasters, and a line crossing a seam is sampled from two of them; how the two are blended is a filtering question with its own literature. The geometry says the true image has a corner. What a filter does with that is somebody else’s subject.

What a draughtsman would do with it #

One consequence is worth drawing out because it is constructive rather than cautionary, and it is the sort of thing the rest of this collection is about.

A cube map’s faces are gnomonic charts, which is to say ordinary perspective pictures — so every straightedge construction in this collection works, unmodified, inside a single face. Vanishing points behave, the measuring point behaves, a harmonic conjugate found with a straightedge is the harmonic conjugate. Every projectivity is two perspectivities applies inside a face and does not apply across a seam, because the two faces are two different picture planes.

That gives a clean rule for working on such a picture. Anything local is ordinary perspective and needs no new machinery at all. Anything crossing a seam has to be carried across by the change of chart, which is a projective map between the two face planes — computable, exact, and not something a straightedge can execute by itself.

So the cube map is the one surface in this collection on which the classical constructions remain available. Every curved surface breaks them, because a construction that joins two points with a straight line is assuming the surface draws lines straight; the cube map does draw them straight, six times over, with an explicit map between the pieces.

Six charts, and one angle at the joins #

The object is a straight line crossing between two flat pictures, and the finding is that the failure has a different kind from every other surface’s in this collection.

Not a curvature spread thinly along an arc, but two exactly straight pieces meeting at an angle — zero along the midpoint of every seam, bounded by the three-four-five angle at the corners, and absent altogether from any line short enough to stay inside one face. A surface that is perfect below a scale and broken above it, which is what covering a sphere in pieces buys and what it costs.