From Panel to Cylinder

Wrapping the heightfield around a cylinder: what survives the wrap, what does not, and the two objects it gives you
A carved column read in reflected light, and a thin lamp shell read in transmitted light. The column is built and unprinted. The shell has been printed twice — and the second print overturned this page's ranking of its own two variants, which is §7.

0. The move

§7.2 of the build guide states the assumption everything rests on: the surface is z = f(x, y), one height per pixel. Branch B reads that as a limitation to be escaped. There is a second reading, and it is the cheaper one.

The assumption is not really “height along +Z”. It is “one height per point of a base surface, measured along that surface’s normal”. The plane is a choice of base surface, not the only one.

Choose a cylinder instead and the height becomes a radius. Every property that made branch A tractable follows from the graph structure, not from the plane, so every one of them survives: the mesh is still watertight by construction, the surface still has no undercuts, and every conditioning stage in heightfield.py — guided filter, range compression, background suppression, slope limiting, forms, engraving, round_trunks — runs unchanged on the same array.

base surface      height measured along     what the picture's axes become
------------      ---------------------     ------------------------------
plane (branch A)  +Z, everywhere            x -> bed X,  y -> bed Y
cylinder          radially out from axis    x -> theta,  y -> bed Z
sphere            radially out from centre  no distortion-free choice exists

What does not survive is anything that was really a statement about the plane, and all of it follows from one fact: the picture’s horizontal axis stops being a spatial axis and becomes an angle, and the picture’s vertical axis stops being a bed axis and becomes the print’s build direction. Two consequences change the field itself — the seam (§2) and the overhang limit (§3). A third changes nothing in the field but inverts how you judge and slice it (§4).

§§1–5 build the first object: the v6 carved relief as a solid column, judged by the light that falls on it. §6 builds the second, which uses every word of §§2–4 unchanged and then inverts the design problem — a thin shell whose wall thickness carries the picture, judged by the light that gets through it. §7 is what happened when both shells were printed, and it corrects §6 on the point §6 was most confident about.

1. The geometry

One decision fixes everything else: the picture wraps a full turn, so the circumference carries the painting’s long axis. Keeping Bruegel’s aspect ratio then determines the diameter — it is not free.

That last row is the one worth pausing on. On a flat panel the sample pitch is constant. Here the angular step is constant, so the arc between samples grows with radius: the field is sampled 42 % more coarsely on the peaks than in the valleys. At 16 mm of relief the peak arc is 0.355 mm, still inside the bead, so nothing is lost. The crossing point is at about 26 mm of relief — past that the peaks are sampled more coarsely than the printer can resolve, which is a quiet way to waste detail.

Relief depth means something different here

On the panel, RELIEF_MM = 16 is 9.4 % of the 170 mm width — a shallow carving. Wrapped, the same 16 mm is 42 % of the base radius. The comparison that governs how the object reads is against R₀, not against the picture’s width, and 42 % is a lot: the column visibly swells and narrows as you turn it.

2. The seam

Column 0 and column W−1 of the field are now neighbours. Three separate things follow, and only the first is obvious.

2.1 The mesh must not duplicate a column

The angular step is 2*pi/W, not 2*pi/(W-1): W samples around a closed loop leave W gaps, not W−1. Use the wrong one and the last column lands on top of the first — a duplicated 0.25 mm stripe of picture and a ring of degenerate quads exactly where the join should be.

2.2 Every filter must see across it

This is the one that bites. heightfield.py is a stack of neighbourhood operations — a 13-px guided filter, Gaussians, a 112-px rolling-ball erosion for the background envelope, a 35-px horizontal opening in round_trunks. Every one of them uses SciPy’s mode="nearest" at the array edge, which invents a boundary by repeating the last column. On a flat panel that boundary is real. On a wrapped panel it is a fiction, and the fiction shows up as a vertical band of wrong conditioning down both sides of the join.

The fix is upstream of everything and costs nothing: pad the field circularly before conditioning, crop after. CYL_PAD_MM = 30 is 120 px, comfortably wider than the largest neighbourhood in the stack.

def circular_pad(a, pad_px):
    return np.concatenate([a[:, -pad_px:], a, a[:, :pad_px]], axis=1)

Note what this asserts: that the painting’s right edge really is the left edge’s neighbour. It is — on the finished object. Circular padding is not a convenience here, it is the physically correct boundary condition.

2.3 The rim has only two sides now

hf.apply_rim fades all four edges down to the plate. Left and right are no longer edges; fading them would cut a groove down the join and, worse, would make the two halves of every object that straddles it disagree about its height. cyl.apply_band_rim is the same smoothstep on the vertical axis only.

2.4 What is left over

None of that makes the join invisible, because the painting genuinely has a left edge and a right edge and they are different pictures. Measured on the finished field, the residual step is 5.43 mm at worst, 1.75 mm on average — a cliff running the full height of the column, where the mountains on the right of the picture meet the trees on the left.

It prints without complaint: a radius discontinuity lies within a layer, so it is a sharp corner in the toolpath and nothing more. Whether it reads as a flaw or as the join of a wrapped panel is a judgement to make in the hand. If it reads as a flaw, the honest fix is not to blend the picture into itself — that manufactures a ghost — but to widen the step into a deliberate vertical stile, a few millimetres of flat, and let it be a piece of joinery.

2.5 Giving the seam somewhere to live

None of the above addresses the seam the slicer makes. Every closed extrusion loop has to start and stop somewhere, and a cylinder is the worst case in printing: there is no corner to hide it in, so it lands in the middle of the picture and reads as a scar. On the lamp shell it is worse than cosmetic, because a start-stop is a local thickness anomaly and thickness is the image.

Slicers choose the spot by preference, and the preference is documented: Bambu’s seam logic ranks candidates “concave non-overhang vertex > convex non-overhang vertex > …”. So the fix is not to fight the algorithm but to feed it.

Measure the sharpest concave corner the picture contains, then cut one sharper. The slicer will park the seam in it, on every layer, unprompted.

On the shell’s field the sharpest concave vertex anywhere is 88.7°, and essentially nothing falls below 90°. A V-groove 1.2 mm deep and 1.2 mm wide gives an apex of 60.9° — a 28° margin, on every layer, at a fixed angle. And it is put at the picture’s own join, so the one vertical line the object has coincides with the one discontinuity it already had.

3. The overhang — the wrap’s own constraint

On a flat panel nothing overhangs. Every relief surface faces up; the no-supports guarantee of §7.2 is free. Standing the picture on end spends that guarantee, and it is worth being precise about how, because the two picture axes behave completely differently.

Measured on the real field: 8.4 % of the mesh’s surface area prints into air without a limiter. That is not a supports-and-live-with-it number on a surface whose whole point is its finish.

The fix is one sweep. Walk up the rows from the bottom, and never let a row sit further out than the row below it plus one row’s worth of allowed lean:

step = np.tan(np.radians(max_deg)) * pitch_mm / relief_mm
for i in range(h.shape[0] - 2, -1, -1):          # row 0 is the TOP
    np.minimum(h[i], h[i + 1] + step, out=h[i])

Three properties make this the right shape of fix rather than a hack. It only ever removes material, so it can never introduce a new problem elsewhere. It leaves every inward lean untouched, because a shrinking radius is self-supporting at any angle. And it is exact: after the sweep no row can violate the limit, so there is nothing to iterate and nothing to check afterwards except that the census agrees.

4. The two resolutions swap

An FDM printer has two very different resolutions: the layer height, fine (0.08–0.2 mm), and the bead width, coarse (0.42 mm for a 0.4 nozzle). The build guide’s whole carving strategy — engrave rather than emboss, cap the raised detail, judge 07_print_sim.png and never the raw hillshade — is built on which axis gets which. Wrapping swaps them.

The depth row is the surprising one. On the panel, height is built by stacking layers, so it comes in 0.08 mm steps. On the column, radius is set by where the slicer puts the perimeter within its layer — the bead moves in or out as a whole, and its outer face goes with it. There is no staircase in the relief direction, because the relief direction is no longer the build direction.

The vertical row is the one with a practical consequence. Detail up the picture is now limited by the layer height rather than the bead, which is better — but only down to the field’s own 0.25 mm sample pitch. At a 0.2 mm layer the printer is already sampling the picture more finely than the field describes it.

So print this at 0.2 mm, not at 0.08 mm. On the panel the fine layer bought depth precision; here it buys nothing the field can supply, and costs 2.5× the layers.

Which means the print simulation has to change too

hf.nozzle_preview encodes the flat panel’s arrangement: an isotropic disc morphology in the picture plane, then round(h / layer). Both halves are wrong here — the morphology should be anisotropic, and the height quantisation should not exist. cyl.nozzle_preview_cyl does bead morphology across theta, layer quantisation down z, and nothing at all across r. The two simulations differ by 0.072 mm RMS over the field, which sounds small until you notice it is concentrated exactly on the fine drawing the guide spent §13 learning how to keep.

5. Printing the column

Euler characteristic 0 rather than 2 is not a defect: a tube open at both ends is a closed surface of genus 1. It is watertight, winding-consistent and positive-volume, which is what a slicer needs.

• Brim, not optional. 170 mm of height on a 905 mm² ring, with material swelling to 108 mm across further up. The mass stays on the axis so it will not tip, but there is nothing else holding that ring down. Brim 4 mm, gap 0, and glue.
• The bore stays empty. The inner wall is a plain 68 mm cylinder; only the outside carries relief. Sparse infill fills the wedge under the peaks, so the 420 cm³ figure is enclosed volume, not filament — slice it for the real number before committing an evening to it.
• One outer wall, slow. The entire deliverable is the outer perimeter of every layer. The build guide’s §10 outer-wall speed and acceleration advice matters more here than anywhere in branch A, because on a column there is no top surface to hide behind.
• Seam placement. The slicer’s Z-seam is a second seam, unrelated to §2’s. Park it on the picture’s own join so there is one visible vertical line instead of two.

6. The lamp shell

The second build off the same wrap, and the opposite reading of it. The column is a solid object that you look at: light falls on it, its surface catches the light, and the relief is the whole point. The shell is a thin wall that you look through: the light is inside, and what reaches you is whatever survived the wall.

On the shell the picture is not the surface. The picture is the thickness.

6.1 Why v1, and why not v6

The column runs the v6 recipe. The shell runs v1 — the canonical four stages of §§1–11 of the build guide, unchanged: snap the depth onto the painted edges, flatten the histogram, compress the range in the gradient domain, add the brushwork back as fine texture. Everything v2 through v6 added is left out, and not for cost.

Every stage in that later machinery is a statement about how a surface catches light. Background suppression exists so objects stand proud of the ground they sit on. Form raising and round_trunks exist so a trunk reads as a cylinder rather than a levee at a grazing angle. The slope limiter rounds columns so the bead can articulate them. A wall read by transmission has no ground to stand on, no grazing angle, and no highlight — there is only how thick it is, everywhere.

One of those stages is not merely unnecessary but actively inverted:

v1 also compresses harder than v6 — COMPRESS_ALPHA 4.0 against 2.0 — which is exactly what a shell wants: a gentle field with no cliffs, spending its whole range on the picture rather than on separating figure from ground.

6.2 The geometry

The 0.8–3.2 mm window is not chosen for structure. It is the range over which white PLA actually does anything optically: below about 1 mm it is properly translucent, above about 2 mm it is nearly opaque, and lithophane practice puts the working range at roughly 0.6–0.8 mm for the highlights and 2.8–3.2 mm for the darks. 0.8 mm is also exactly two perimeters at a 0.4 nozzle, which matters in §6.7.

6.3 Two shells, one picture

Where the picture hangs on the shell is a free choice, and both are exported from one run:

outward   r_out = R0 + RELIEF*h      bore a plain cylinder at R0 - FLOOR
inward    r_out = R0  (a plain cylinder)   bore at R0 - t

Both have thickness FLOOR + RELIEF×h. The two shells transmit exactly the same picture — not approximately, identically.

So the choice is not about the image. It is about three other things.

• Unlit. The outward shell is an object when the lamp is off. The inward one is a plain cylinder.
• Refraction. Surface slope steers light. The outward shell puts its slopes on the exit face, where the ray goes from n ≈ 1.5 to 1.0 — the larger deflection of the two interfaces, and the only one where total internal reflection is available at steep facets. If anything in this object is going to throw a caustic on a wall, that is where it comes from. The honest caveat: an FDM surface is bead-textured and scatters diffusely, and diffuse scatter is a much larger effect than clean refraction, so this is a second-order term riding on an identical first-order one.
• Overhang. Nothing on the inward shell's visible face can overhang, because it is a cylinder.

And on this build it costs essentially nothing. The steepest upward lean anywhere in the v1 field is 70.7°, but only 0.022 % of cells exceed 45° at all: 9.9 mm² of airborne surface, 0.017 % of the mesh, removed for 0.005 % of the height budget. Compare the column's 8.4 %. The overhang problem is a deep-relief problem — 2.4 mm of relief over 145 mm simply cannot lean far.

6.4 The exponential, which is the real problem

Light through the wall follows Beer–Lambert:

At the placeholder μ = 1.25 /mm, a 0.8 mm wall passes 0.368 of the light and a 3.2 mm wall passes 0.018 — a 20:1 range, which is a good tonal range. The trouble is where it lives:

Half of that entire brightness range is spent between 0.8 and 1.32 mm — the first 22 % of the wall budget. Everything past about 2.4 mm is the same near-black.

Send the heightfield straight to thickness and most of the picture lands in the crushed end. Measured on this field: median normalised brightness 0.35, with 15.6 % of it below 10 % brightness. The fix is the one lithophanes have always used, written out as a stage — choose the brightness you want, then solve for the wall that delivers it:

b_hi = np.exp(-mu * floor_mm)                       # the thin end's brightness
b_lo = np.exp(-mu * (floor_mm + relief_mm))        # the thick end's
b    = b_hi + (b_lo - b_hi) * h              # LIGHT linear in the field
t    = -np.log(b) / mu                        # the wall that delivers it

The ends are unchanged by construction, so the remap stays inside the same wall budget, and it is monotone, so the foot, the overhang limiter and the mesher are all untouched. Median brightness goes 0.35 → 0.68 and the crushed fraction 15.6 % → 1.0 %. It also uses less plastic — 35.6 cm³ against 50.0 — because the midtones no longer sit at 2 mm.

6.5 μ is not a constant of nature

1.25 /mm is a placeholder chosen to span the 0.8–3.2 mm window that practice uses. The real value depends on the filament, its pigment loading, and — because a printed wall is not a solid one — on how solidly it prints. Published work on translucent additive manufacturing applies the Beer–Lambert form and confirms transmittance falls monotonically with thickness, but does not hand you a coefficient for white PLA on an FDM machine. So measure it:

Then set CYL_LAMP_MU_PER_MM and re-run. Nothing else in the build changes: μ enters only through tone_for_transmission and the preview.

6.6 The ends: no rim, a foot and a collar

apply_band_rim fades the field to zero at the top and bottom. On the column that gives a clean recessed border. On the shell it would put the thinnest wall exactly at the two rims — where the object is handled, and where it meets the plate. A 0.8 mm ring at this diameter is 161 mm² of bed contact holding up 145 mm of tube.

So the shell gets the opposite treatment. add_foot raises the bottom 6 mm to at least 2.4 mm of wall — 471 mm² of first layer, and a plinth — and the top 3 mm to 1.6 mm so the rim is not a knife edge. Both are raises, never cuts. The foot thickens going down, so it is self-supporting by definition; the collar thickens going up, but its steepest growth is 21.6°, well inside the 45° limit.

6.7 Printing it

• 0.12 mm, against §4's own advice. §4 argues that a 0.2 mm layer already samples the picture more finely than the field's 0.25 mm pitch, so nothing geometric is gained below it. That argument is about geometry and it still holds. It says nothing about optics: on a wall you look through, every layer boundary is a scattering interface, and coarser layers band the transmitted image whether or not they resolve it. Lithophane practice is unanimous that layer height is the single most important setting. This is the one place where the convention beats my own argument, and the wedge — printed at both heights — settles it in an hour.
• 100 % infill, wall loops to the maximum. A wall that runs 0.8 to 3.2 mm is 2 to 8 beads thick. Any void the slicer leaves inside it is a scattering cavity, and it will show as a blotch that is in no painting. Set walls to 99 and infill to 100 and let the slicer make the whole wall solid.
• White PLA. It transmits evenly across the spectrum; coloured filament tints the image and translucent filament costs you control of the dark end.
• Brim, still. 471 mm² is three times the bare rim but it is still a narrow ring under a tall part.
• No supports, either shell. §6.3.
• Vase mode is not available — the wall thickness is the picture, and vase mode prints one bead of constant width.

6.8 What the lamp still needs

This section builds the shell and stops there, which was the point. What is left is not geometry:

• A source and a diffuser. The bore is 63 mm clear, open at both ends. A bare point source inside a 32 mm radius will hot-spot; the transmission previews on this page all assume even illumination from within, which is a lie that a diffuser has to make true.
• Rotation. Mechanical, and outside this pipeline.
• The projection. The genuinely open question, and it does not follow from anything here. A shell has no imaging optic, so what lands on the wall is not the picture but a badly defocused version of the transmission map, brightest where the wall is thinnest, its sharpness set by how small the source is. The outward shell's surface slopes steer some light on top of that. Whether the result is a shadow of Hunters in the Snow or an interesting smear is a question for a dark room and a bulb, not for this page.

7. In the hand — L1, and the answer to it

Everything above this line is a prediction. Two shells have now been printed, and this section is the first thing on this page that was measured off an object rather than off an array. It is written in the order it happened: a shell that failed, a diagnosis with two separate parts, a second shell that fixed both, and then a result none of the modelling on this page could have produced — because the model is right, and the ranking built on top of it was wrong.

7.1 L1: what a depth-driven wall looks like in plastic

L1 is the shell of §6.2 exactly as specified: v1 recipe, depth-led, 145 mm tall, wrapped once, relief facing out.

The picture is faintly there if you know where to look, and it is not there otherwise. This is not a print defect; the mesh built and sliced clean. It is the field. Local contrast on L1's transmitted range measures 0.0197 rms overall and 0.0146 across the hunters — under two percent of the range, sitting behind white PLA's own 57–63 % haze. There was never anything to see.

The torch photograph closes the other question in the same afternoon. §6.8 lists the projection as genuinely open; with a source small enough to cast anything, nothing survives the wall. That is consistent with the published haze figures for white PLA — 57.6 / 61.3 / 62.9 % at 0.8 / 1.6 / 2.4 mm — and it means the projection needs a different object (clear filament, or a mask cut through), not a better lamp.

7.2 Two complaints, two independent changes

Standing L1 on the table produces two separate objections, and it is worth keeping them apart because they have nothing to do with each other.

• Nothing reads. The wall is driven by the wrong quantity. → drive it from luminance.
• It does not read as a round thing. A picture wrapped once around a cylinder is a flat picture bent into a tube. It has a front, a back, a join you are meant not to notice, and at Ø64.9 you can only ever see a sliver of it. Walking around it is not rewarded. → wrap the picture with its own reflection.

The second change is the one that turns a rolled-up poster into an object made in the round: follow the picture with its mirror image, so that both joins are mirror lines rather than seams. A kaleidoscope, not a panorama.

It is nearly free. mirror_wrap runs the samples 0…W−1 and then W−2…1, so the reflection shares its end columns with the original and no column is duplicated. Both joins are then continuous exactly, to 0.0 — there is no step to hide, because a mirror line is not a discontinuity. The circumference doubles, and with it the diameter, which is the whole gain: twice as much picture is in front of you at any angle, and the object acquires two vertical axes of symmetry that read as deliberate rather than as a joint. The cost is that the painting appears twice, which on a cylinder nobody sees whole is a smaller price than it sounds.

Circumference is samples × pitch throughout: 815 × 0.25 and 1628 × 0.25. §6.2's table rounds L1's to 203.74 and its diameter to 64.85, one hundredth low in each; the values here are the arithmetic.

The luma recipe also deletes most of the pipeline. The guided filter existed to snap a depth map onto the painting's edges, which is pointless when the field is the painting; add_detail existed to put luminance back into a depth field, where it already is; and compress_range is actively harmful here, 0.1172 → 0.0987, because it attenuates large gradients — which on a depth map are the global recession and on a luminance map are the figures themselves. The whole v1 chain was scaffolding for getting luminance into a depth-driven field, and it evaporates when luminance is the field. The lamp build never loads the depth model at all.

7.3 What the pipeline predicted about the two shells

§6.3 says the two shells transmit the same picture, and says it in the strongest possible form: not approximately, identically. That claim is correct, and it is worth putting a number on how correct. The two transmission maps this build exported differ on 2.59 % of their pixels, by a mean of 0.14 % of the range — and the whole of that difference is the overhang limiter, which the build now runs at a different angle for each shell, 60° outward against 70° inward. Nothing else in the two fields differs at all.

Beer–Lambert is a function of path length and nothing else. There is no term in it for where in the wall the thickness variation sits, and no term for light moving sideways. Both shells have the same t(θ, z) by construction, so the model is obliged to predict the same image — and it did.

Having established the tie, §6.3 ranked the shells on what was left over: unlit appearance, a second-order refraction argument, and overhang. A seam analysis done after this page was written then decided it, and decided it on printability. Redo §2.5's corner measurement on a luma field and the numbers invert: hard tonal edges become near-full-relief jumps across one or two samples, so the field's own sharpest concave corner is 27.0°, not 88.7°, and on the outward shell the channel loses to the picture on 99.8 % of layers. Beating 27° would need a groove about 6.7 mm deep — deeper than the wall. On the inward shell there are no rivals at all, because the outer contour is a circle with two notches in it: margin +127°, every layer, unprompted.

So the inward shell was picked, and L2 was printed inward first — on a printability argument, taken while the optical question was believed to be a tie.

7.4 The hand disagrees, and by a lot

They are not the same picture. On the outward shell the birch trunks have edges, the hunters have spears, the dogs are countable and the village has roofs. On the inward shell the same content is a grey weather system: the trees survive as soft columns, the figures as smudges, and everything below about two millimetres is gone.

“Looks blurrier” is not a measurement, so here is one. Each photograph is converted to log luminance and band-passed at a series of spatial scales; the scale axis is put in millimetres on the object using the drum's known 129.55 mm diameter as the ruler. Working in the log domain is what makes the photographs comparable: a JPEG value is roughly radiance to the power 1/2.2, so its logarithm is (1/2.2) × log radiance plus a constant, and a band-pass throws the constant away. Exposure and white balance drop out; the same constant of proportionality is left in every frame.

At eight-millimetre features the two objects are comparable. At two millimetres the inward shell holds 40 % of the outward shell's relative contrast, at one millimetre 23 %, at half a millimetre 18 %. The deficit is not a level shift; it is a slope, and it grows exactly as the features get smaller. Something is low-passing the inward shell.

7.5 Which channel is doing the work

The obvious explanation is that the wall diffuses, and that a thickness pattern written on the far side of a scattering wall has to diffuse across the whole wall before it can leave, while a pattern written on the exit face does not. That story is plausible, it is what lithophane practice has always assumed, and it is not what these photographs show. The control that settles it is free: turn the lamp off.

Read the top row. Turning the lamp on changes the outward shell's half-millimetre band by four percent — 0.0193 to 0.0201. At one millimetre the lamp buys ten percent, at two millimetres twelve, at four seven. Whatever is drawing the trunks and the dogs, it is not the light coming through the wall; it is room light landing on a relief. Above four millimetres the lamp does not merely fail to help, it actively costs: 0.0531 → 0.0385 at eight millimetres, because flooding the shell from inside fills in the relief's own shadows and substitutes its own broad tonality.

Now read the bottom row. With the lamp off the inward shell measures 0.0025 at half a millimetre — on a plain cylinder with no relief at all, which is the noise floor of this measurement. With the lamp on it measures 0.0028. The transmitted image contributes essentially nothing below a millimetre. What the lamp does supply is the coarse end, where it lifts the same object from 0.0110 to 0.0302 — nearly threefold.

Put the modulation on the face you look at. Not because transmission through it is sharper — that is still unproven — but because a surface you can see is a second, free, exactly-registered picture, and a surface you have buried is not.

7.6 What these photographs do not establish

The house rule is that a headline claim carries its own controls, so here are the ones that were run and the one thing that is still open.

Two smaller caveats belong on the record. The room light was daylight through large windows and was neither controlled nor measured, so the lamp-off and lamp-on frames of a pair are minutes apart rather than simultaneous. And every number above is a ratio taken within one photograph or between band-passes of the same log image, which is why none of them depends on μ — still a placeholder, still unmeasured, and still waiting on the wedge of §6.5.

7.7 The sky, the birds, and the two joins

This is also the register where the inward shell is at its best. Where the painting is broad tone — the sky, the frozen ponds, the far snowfield — the transmitted image is doing all the work on both objects and they converge. It is the drawing that is missing, not the picture.

Which is what mirror_wrap promised and, unusually for this page, delivered without a correction: the joins are continuous to 0.0 because a reflection cannot have a step. The seam channel itself stays invisible when the lamp is on, exactly as §2.5 intended — it is subtracted from both surfaces, so the wall thickness never changes and the transmission genuinely cannot see it. With the lamp off it shows as a faint vertical crease catching the room, which is the only place it has ever been visible.

7.8 What changes in the recipe

• CYL_LAMP_DRIVER = "luma", permanently. The depth lineage has no business on a wall read by transmission. This is settled by L1 and not open.
• The outward shell is the build. CYL_LAMP_MODES should still export both — the run is one pass and the inward mesh costs nothing to keep — but outward is what goes on the plate.
• Pay the seam by hand, and read §2.5 as history. Its 1.2 × 1.2 groove is what severed the wall; the shipped profile is 0.8 deep by 3.2 wide, whose flanks tilt 26.6° instead of 63° and leave a narrowest band of 0.744 mm, 1.77 beads. Even so the outward shell's channel loses on 99.8 % of layers against a luma field's 27° corners, and beating that would need a 6.7 mm groove, which is deeper than the wall. So paint the seam in the slicer: two minutes, free, robust, and the correct price for the better object. The measured alternatives were both worse — de-notching buys 110.5° for −7 % contrast and still loses on 28.6 % of layers, and a 0.5 mm circumferential blur fixes it completely at a cost of −24 % of the hunters' local contrast, which is the entire point of the object.
• Do not inherit the inward shell's 70°. Its justification does not survive §7.6. If an inward shell is ever built again it wants 60° like the other one, or a fresh argument.
• μ is still a placeholder and the wedge is still unprinted. Nothing in this section waits on it; the tone remap does.

8. The code

Two new files, one config block. Nothing in heightfield.py, depth.py or mesh.py changes.

CYL_HEIGHT_MM = 170.0         # print height; A1 mini Z is 180 mm
CYL_RELIEF_MM = 16.0          # peak-to-valley, radial. Compare against R0, not width
CYL_WALL_MM = 4.0             # shell thickness at zero relief (was BASE_MM)
CYL_MAX_OVERHANG_DEG = 45.0   # radius growth per unit rise. None = off, supports on
CYL_PAD_MM = 30.0             # circular pad so filters see across the seam
CYL_REUSE_FLAT_DEPTH = False  # True = reuse a depth cache inferred at another size
CYL_LAYER_MM = 0.20           # the layer height you will actually slice at

CYL_BUILD = "lamp"             # or "column"
CYL_LAMP_HEIGHT_MM = 145.0    # circumference follows from the aspect ratio
CYL_LAMP_FLOOR_MM = 0.8       # thinnest wall = 2 perimeters at 0.4 mm
CYL_LAMP_RELIEF_MM = 2.4      # so the wall runs 0.8 -> 3.2 mm
CYL_LAMP_MU_PER_MM = 1.25     # PLACEHOLDER. Print the wedge and measure it.
CYL_LAMP_TONE = "linear-light"  # solve thickness for the brightness you want

9. What it opens

The reason this branch is worth its evening is not either object. It is that heightfield_to_shell is an instance of a general move, and several other ideas on the list are the same move with a different base surface — or, in the lamp's case, the same surface read by a different kind of light.

References

• The build guide — §7.2 for the heightfield assumption this page generalises, §12–13 for the carving stages that run unchanged, §10 for the slicer settings that matter more here.
• painting2lamp — the lamp path of §6 as a single browser page: image in, STL out, offline, verified against cylinder.py to 1.1e−5 mm.
• From Relief to Sculpture — the other branch out of the same heightfield, which spends the assumption rather than moving it.
• The Poisson solve — still the engine under compress_range and limit_slope, both of which run here unmodified.
• Weyrich et al. (2007), Digital Bas-Relief from 3D Scenes — the compression this whole family rests on says nothing about the base surface, which is why the wrap is cheap.
• Yuan, Bourell et al., Additive Manufacturing of Optically Translucent Parts (SFF Symposium 2014) — applies Beer–Lambert to printed polymer plates from 0.38 to 4.74 mm and finds transmittance falling monotonically with thickness. It reports no coefficient you can borrow, which is §6.5's whole point.
• Lithophane 3D printing guide — the 0.6–0.8 mm / 2.8–3.2 mm working window, white PLA, 0.12 mm layers, 100 % infill and maximum wall loops. Conventions, not measurements; checked 2026-08-05.