Concept

Seam — where it appears

The edge where two faces of a piecewise-flat picture surface meet, across which a straight line is continuous in position and broken in direction. It is the reason a piecewise-flat surface cannot be treated as one map, and the kink at it is the image of the crease rather than anything about the line.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

8–26 meye at 1.6 m26–70 meye at 4.0 m70–200 meye at 11.0 mthree stations, buttedfar band ×6.9

Three distances in one landscape

A landscape assembled from a low station for the near ground, a level one for the middle and a high one for the far gives its furthest band 6.9 times the picture one camera would allow it — because the image of a fixed depth interval falls as one over depth squared, and its own station gives that back. What it costs is a jump in the rate at which depth runs, 2.50 at the first join and 2.75 at the second, and three views from one height leave no seam at all.

scroll · Threedistances
recovered 1.60 m8–26 mrecovered 4.00 m26–70 mrecovered 11.00 m70–200 melevation recovered from each band's own marksnot read off a ledger

A landscape that changes its rule halfway up

A reader handed only the marks of a three-station landscape recovers each band's own camera height without being told any of them — 1.60 m, 4.00 m, 11.00 m, to 3.6e-15 m. What that same reader cannot recover across a join is a common ground: the next band's own marks read as a ground point 2.40 m away from the true one at the first seam, 7.00 m at the second.

scroll · Threedistances
leftfrontrightbackupdownacross the left/front seam: 1.80°, with each side straight to 7e-16corner area ×5.196anisotropy √3 = 1.7321 there

Six flat pictures of everything

There is one way to photograph the whole sphere and keep every straight line straight, and it is to stop using one surface. Six flat pictures at ninety degrees cover everything, each of them a perfect pinhole, and the price is paid entirely at the seams — where a straight line does not bend but kinks, by an angle that reaches 45 degrees and is exactly zero for the lines lying in the seam's own plane.

curved · Cubemap
planeevery linecylinderone parameter · parallelno meeting pointstereographicone parameter · concurrentthey meet to 1e-12equirect.one parameter · parallelno meeting point3 of 4 keep a curvethe signature has three values, not eight

The lines a surface leaves alone

Only the plane draws every straight line straight, which is easy to measure and which gets read as though a curved surface bent everything. It does not. Every curved picture surface here keeps a one-parameter family of world lines exactly straight, and the family is a curve on the sphere of lines rather than a region of it — so a grid of samples finds none of it, which is what the first version of this measurement reported.

curved · Straightfamily
three eyes, flat ground1.6 m4 m11 mone eye, stepped groundground 9.4 mground 7.0 mground 0.0 mone eye, 11 m99 ground samplesagree to 6e-14 px

The stations are also a staircase

A level eye draws a ground point on a row that depends on the eye's height and the ground's only through their difference. So a landscape drawn from three stations at 1.6, 4 and 11 m over flat ground is, to 6 × 10⁻¹⁴ px across ninety-nine samples, the same picture as one eye at 11 m over ground stepped 9.4 m and 7.0 m above the far plain. The station heights a reader recovers are recovered on an assumption the marks neither supply nor test — and what would decide between the two readings is a cliff 103% of a band's own height, which is exactly where the convention lays its mist.

scroll · Threedistances
the join, 26 mnear band: 21.8°far band: 45.0°215 px wide under either8 m roadturns 23.2°, keeps its width

A seam breaks direction, not size

An eight-metre road crossing the first join of a three-station landscape is drawn 215.4 px wide under either band's rule, identically, and a six-metre post 161.5 px tall under either — the eye's height cancels out of any size taken at one depth. What does not cancel is where those sizes sit. The road's edges are turned 23.2° from each other and the post's foot lands 64.6 px out of place, and a painter butting two bands can absorb the offset and can never absorb the turn.

scroll · Threedistances
42.0°27.3°13.5°13.5°27.3°42.0°leftfrontrightbackupdownmidpoint: 0°7 straight lines, at 60° to the seamworst 42.04° · midpoint 0°

The kink at a seam

A cube map is six flat pictures, so every great circle is drawn exactly straight inside a face — to 3e-15 of its chord — and breaks at the join. The break is a kink and not a bend, it is exactly zero at a seam's midpoint whatever the slant, and it is bounded by 2·atan(½) = 53.130° at the corner.

curved · Cubemap

Named alongside it

The objects these essays reach for when they reach for this one.

ElevationFree parameterMoving viewpointPiecewise mapStation pointAnisotropyCube mapDepth compressionGnomonicPicture surfaceStation-disagreementArea scale

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