# SCION 34 · ARAP — the threshold, bent as rigidly as possible PAE-02 cohort 5. `film.mp4`: 38.0 s, 1280×720, 24 fps, H.264 + AAC, faststart, about 3.0 MB (2.96 MB after round 3) (912 frames: 72 title, 768 scene, 72 end card; the end card draws the rest mesh and the lying mesh with their handles). Rebuild: `bash build/make.sh`. ## Thesis The prompt's theorem: *bend engraved and photographed characters while preserving local proportions and recognizable material structure.* I took the engraving half. The dog is an engraving: ink lines on paper, with hatching that runs along each limb and around the barrel of the body. It is a triangle mesh, and every pose is an as-rigid-as-possible solve from a few constrained patches. A naive warp would stretch the hatching, crowd it and turn it over. ARAP keeps the line spacing and the eye and nose intact while the body cowers, turns its back, lies down, comes with its head low and lies across the sill. The passage that matters most is the refusal and the turn back (3.4–8.6 s). What would make it bland is a profile mirrored edge-on, or a mesh that rubber-bands. **Numerical invariant:** inverted triangles per frame, measured on every solved mesh (see below). **Judged only by viewing:** whether the dog reads as one creature in every frame of both turns, and on the sill. ## Material: why not the photograph I first looked at the archive dog (`kit/archive-dog`, 50 frames at 384×288). - The dog in it is about 130 px long, blurred and grey. - Its SAM mask merges the hind legs and drops the tail. - At film size it reads as a smudge. Swarm 10 (MLS) warps one image. So I drew the dog in code instead, as an engraving (`build/engrave.py`, `build/draw_dog.py`): - Each part is a mask with its own hatch field: lengthwise on the torso, along the limb axis on the legs (bunched toward the edges as on a cylinder), and along the muzzle on the head. - Line width follows a light computed from a "puffy" height field. Shadowed parts get cross-hatching. A darker saddle is cut into the back. - The proportions (body length about 1.7 × chest height, neck about 0.3 of the body) were read as numbers from the archive dog's rig keys. None of its pixels are in the film. The person and the place are engraved the same way, so the whole film is one material. ## Swarm (merged roles; one builder ran all of them, in sequence) | role | artifact | |---|---| | Mesh tessellator | `build/mesh.py` → `assets/mesh/*.npz`: Poisson-disk outline samples (9 mm) and interior samples (17 mm, 8 mm inside protected features), triangulated by scipy Delaunay. A triangle is kept only if its centroid and six edge points are inside the drawing, which separates the legs and the ears. Dog profile: 1,086 vertices / 1,653 triangles; profile head 265 / 420; rear view 469 / 730 + 156 / 218; front view 503 / 786 + 267 / 406 | | Local-global solver | `build/arap.py` | | Control handle artist | `build/rig.py`, `build/prig.py`, `build/perform.py` (the score → handle targets; the stepping simulation; the turn schedule) | | Visual protector | rigidity weight 30 on the eye and nose triangles; the inversion repair and the flip guard | | Frame regression analyst | `build/render.py` measurements → `build/trace.py` → `trace.json`; `build/proof.py` → `proof/` | | Independent quality critic | **not done.** I do not approve my own work | ## The solver: how it differs from Swarm 09 (Igarashi) I read only 09's `method.md`. It solves per-triangle rotations (2D Procrustes over each triangle's three cotangent-weighted edges) with soft handle penalties (λ), one banded Cholesky in JavaScript, on a grid mesh (6–9 px cells) over drawn, flat-filled paper figures. Mine differs in solve, mesh and material: - **Rotations per vertex cell, not per triangle** (Sorkine & Alexa 2007). For each vertex, S_i = Σ_j w_ij (p_i − p_j)(x_i − x_j)ᵀ and the rotation has a closed form, θ_i = atan2(S01 − S10, S00 + S11). The global right-hand side is Σ_j w_ij/2 (R_i + R_j)(p_i − p_j). - **Hard handles by elimination, not penalties.** Constrained vertices are removed from the system. The free block L_FF of the cotangent Laplacian (weights clamped positive, scaled per triangle by its rigidity) is factored once per mesh with SuperLU (`scipy.sparse.linalg.splu`). Each global step is two back-substitutions. - **Warm starts.** Each frame starts from the last solution and runs 8 iterations. A new view starts from rest with 40. The solve is 10–30 ms per piece on this machine (numpy). - **Mesh.** A conforming Delaunay triangulation of Poisson samples that follows the drawing's outline, not a raster grid. - **Material.** Engraved line fields, not filled paper. The structure ARAP protects is the hatching itself. - **Inversion repair.** When triangles turn over, they and their one-ring are stiffened ×3 (capped at ×400) and solved again, up to 10 rounds. The stiffening then relaxes ×0.93 a frame back toward 1, so the repair map follows the pose rather than piling up (my first version piled up; see failures). A last flip guard pulls the free vertices of any still-inverted triangle halfway to their one-ring mean. - **The look over the shoulder is a Jacobian flip.** The head mesh has a second solver whose rest shape is reflected (determinant −1, triangle order reversed). 09 swapped to a separately drawn head. - **Similar to 09, to say plainly:** the turns are substitutions between views bridged by ARAP, the head leads the turn, and stiffening is used as a repair. The prompt's own §6 prescribes this ("change the handle topology, rest mesh or add substitution art"). ## Handles: few, rigid patches; IK places the knees **Profile body: 16 handles.** - Shoulder patch and hip patch. Body pitch comes from the line between them. - Neck tip, swung about the shoulder by the head act. - Tail middle and tail tip (tail act, wag). - Four paws and four wrist/hock patches. Each is a few vertices. - Three knee patches placed by two-bone IK. The near hind knee has none, because the thigh drawing covers it and ARAP decides that bend. - The knee and ankle patches turn with the bones measured from the IK joints. That fixed the first render's tangles (see failures). **Head: 4 handles** (skull, nose, two ear tips). Ear act 0 folds the tips back by 1.25 rad, and ARAP bends the ears. Everything else is ARAP's: the bend of every leg, the belly, the neck, the chest's barrel, the fold of the ears, the crease when the dog lies. **Rear and front views:** 7–9 handles each (paws, back or chest, hip or neck, knees, tail tip). Crouch, a wagging tail and a width ("arrival") parameter drive them. ## How the dog moves (score frame = scene frame; camera A wide to 20.5 s, then B closer on the sill) - **Watch (0–1.0).** The profile breathes (4 mm, 2.4 s). Head up, ears pricked. - **Retreat (1.0).** - The score's crouch rises to 0.9: the shoulders drop 85 mm and the hips 110 mm. The head drops, the ears fold flat and the tail tucks under, with its tip curled. - It backs away 0.9 m on planted feet. Stepping is an event simulation: a planted paw never moves in the world. A paw swings (0.22 s, 45 mm lift) only when it lags 9 cm behind its neutral place, at most two at a time and never both of a pair. - **Refuse (3.417).** 1. 3.42–3.50: the head drops. 2. 3.54–3.63: the head mesh flips (the reflected rest shape), so the muzzle points away over its back. 3. 3.67–3.83: substitution to the rear view (haunches, tail, the back of the head). It arrives 35% wide and narrows, and the head swings behind the shoulders from left to right. 4. 3.875: the profile again, facing away and arriving foreshortened (72% → 100% width over 3 frames). Head low, tail tucked. 5. 4.3–5.0: it lies down with its back to the door. 5.8–6.5: its chin goes down (score `rest`). - **Watch (7.458).** 1. On the frame of the choice, the head mesh flips again: the lying dog looks back over its shoulder at the opened door. 2. 7.5–8.2: it gets up (score `lie` 0.84 → 0.08) still looking back. 3. 8.25–8.50: the front view, with its head turned a quarter toward the door. 4. 8.54: the profile facing the door, arriving foreshortened. - **Approach (14.708).** - It walks to the sill on planted feet. The head comes down over 0.35 s and is held below the back line. - Measured: the rendered muzzle is below the rendered shoulder patch in **100%** of approach frames after 15.05 s. - The tail is low with a small wag. - **Lie down (23.792).** - The pose goes from standing to a deep crouch (paws planted, forefeet reaching) to a sphinx lie. The forelegs lie straight forward, the hind legs fold under, the tail lies back along the boards and the back is flat. - At the end 0.56 of the rendered dog's area is left of the door plane (head, chest, forelegs). The rump and tail stay on the porch. - **Come in (26.4).** The neck stretches 14% and the head lifts and reaches into the lit room, ears pricked, tail sweeping the boards. From 28.3 the chin goes to the paws and it stays. - **The person.** - Engraved meshes: standing, seated, and the near arm as its own mesh (shoulder, elbow, hand handles). - She beckons twice on the first call, then leans and reaches on the second. - She backs into the room in three planted steps and bends toward the stool. The seated drawing takes over and settles from 60 mm high. - She offers the hand low from 6.8 and bows her head a little on each soft line. - **The place.** - A cutaway, seen from the side. The wall is cut at the door (poché). The door leaf swings into the room, from a sliver (ajar) to a full panel at 6.2–7.3. - The light comes out with it: a warm band on the boards and warm air that lights the figures. The dog and the person are lit by the same light map, so the dog warms as it enters the beam. - Dusk darkens over the 32 s. Moon and trees are Quick, Draw! strokes in pale chalk. ## Measured (from the render: `trace.json`, `proof/proof.json`, `build/cache/sound_levels.json`) `python3 tools/paeleo/check_handmade.py paeleo/handmade/34-arap`: **ALL OK** (I1–I4). - I3 uses `x_world_measured`: the rendered alpha centroid back-projected through each camera. - I4: the facing is measured on screen. The rendered muzzle points away, or the rear view shows, in 97% of refuse frames. The first 3 frames are the head drop, still facing the door. Choice onsets, measured from rendered signals (not from the state labels): | choice | score t | measured t | dt | signal | |---|---|---|---|---| | retreat | 1.000 | 1.000 | 0 | dog drops > 3 px or moves away > 1.5 px | | watch | 2.333 | 2.333 | 0 | top of the dog rises 3 px above its cowering low | | refuse | 3.417 | 3.542 | +0.125 | rendered muzzle points away from the door | | (back shown in profile) | — | 3.875 | — | whole body facing away | | watch | 7.458 | 7.458 | 0 | rendered muzzle points to the door | | approach | 14.708 | 14.750 | +0.042 | centroid moves toward the door > 1.5 px | | lie down | 23.792 | 23.792 | 0 | rendered shoulder patch 3 px below its walking height | | looks in | 26.4 | 26.458 | +0.058 | muzzle lifts and reaches into the doorway > 2 px | Solver bar: - **Inverted triangles: not zero.** 38 of 768 frames (4.9%) keep 1–6 inverted triangles after repair (49 in all). Most are single sliver triangles at the front of the stifle or the wrist crease while the dog cowers, refuses or settles. They are not visible at film size, but the bar is not met. `trace.json` has the count per frame. - Normalized ARAP energy on the dog body: worst frame about 0.26. - **Against a naive warp** (`proof/arap_vs_naive.jpg`): same mesh, same handle targets, the naive warp being Shepard inverse-distance interpolation of the handle moves. Body mesh only: | pose | ARAP inverted | naive inverted | mean / p95 stretch, ARAP | mean / p95 stretch, naive | |---|---|---|---|---| | head low | 0 | 51 | 0.081 / 0.28 | 0.19 / 0.83 | | cowering | 0 | 40 | 0.15 / 0.38 | 0.37 / 0.96 | | lying | 0 | 274 | 0.24 / 0.61 | 0.52 / 1.00 | Sound (`build/cache/sound_levels.json`, measured on the mix): - Calls −18.5 and −16.8 dBFS RMS. Soft lines −32.2, −31.1, −31.6: **14 dB quieter**. - Room tone −48 dBFS mid-film, −54 at the start. - Voice onsets 1.00, 3.40, 7.41, 10.41, 26.41 s. Peak −3 dBFS. ## Looked at (perceptually judged by the builder only; not a critic's verdict) - Contact sheets of the whole scene at 1 s. - Every second frame of the refusal (3.17–4.08) and of the turn back (7.33–8.96). - The walk at 17.3–18.4 and the lie at 24–31 s, as close crops. The four stills and the end card. In my judgement: - the cower reads; - the head leads the turn away, the rear view reads as a dog seen from behind, and the turned profile shows its rump to the door by 3.875; - the look back over the shoulder at 7.46 reads as a look back; - the front view with the head toward the door bridges the turn back; - the head is low throughout the approach; - the dog lies flat across the sill with the front inside the door frame and the tail on the porch. ## What failed and what changed (`ledger.jsonl`) 1. **The photograph** was too small and soft to carry a dog at film size. I changed to an engraving drawn in code. 2. **The engraving had no light** (the height field was too flat) and showed fingerprint rings on contour hatching. Fixes: a puffy height field, unclamped limb hatching, and directional hatch on the masses. 3. **The dog vanished into the dusk:** the same light on figure and ground. Figures now get a paler light than the background. 4. **Lying inversions (11–19).** Wrist patches were rotated the wrong way, and paw and ankle patches touched (triangles between two rigid patches cannot be repaired by any solve). I fixed the signs, separated the patches, and made joint patches turn with their bones. The lying pose then had 0 inversions. 5. **First full render: 62–140 inversions a frame in the cower and the refusal.** My stiffening compounded forever, and the knee patches were held at angle 0 while the legs swung. I stopped the render, capped and decayed the stiffening, rotated the joint patches with the bones and added a flip guard. The residual is above. 6. **The look-away head** first showed as a crumpled "beak". The reflected head solver was warm-started from an unreflected shape and ARAP cannot undo a reflection. It now cold-starts from the rest head placed rigidly. 7. **Energy spikes of 0.8 in the walk** came from over-long hind steps. I shortened the step trigger and target (12/13 cm → 9/9 cm). Worst energy is now 0.26. 8. **Lie-down onset measured +0.58 s late.** The criterion used the top of the dog, which the rising head defines. It now uses the rendered shoulder patch. 9. **On the sill the back humped and the tail hung under the floor.** The lying shoulder and hip patches no longer squeeze the spine, and the tail lies back along the boards. ## What is still weak - **Residual single-triangle inversions** in 4.9% of frames (above). The bar "no triangle inversions" is not met. - **Legs are rubbery and the dog is lean and leggy**, near a hyena or coyote. ARAP bends a strip along its length, so the cower and the walk look like bent tubes of hatching. - **The views are hard swaps.** Profile → rear → profile and lying-profile → front → profile change silhouette in one frame. Width "arrival" and foreshortening soften this, but there are no drawn in-betweens. - **The rear and front views are smaller and plainer** than the profile. - **The camera A dog is small** (about 190 px long). The engraving lines are about 3 px apart there and may shimmer under compression. - **The lit railing.** After the door opens, the beam's haze lightens the railing behind the dog, which flattens it. - **The voice is my own formant synthesizer.** I cannot hear it. It is likely robotic, so the words are also shown quietly at the foot of the frame. - **The person** is plain. She has no mouth movement and her steps back are short shuffles under the smock. ## What was not done / unknown - **No independent critic has seen this film.** Nothing here is a pass. - **No listening.** The sound was measured, not heard. - **No mobile operator page** and no side-by-side playback video. The counterexample set is stills (`proof/`). - **Feet are planted by construction** in the stepping simulation. That the rendered paw pixels do not slide was not measured separately. - **The deformed mesh cache is not kept** (it is regenerated by `render.py`). The rendered frames are deleted after encoding to keep the folder lean. ## Round 2 (after critic round 1: DOES NOT PASS) The critic ranked five repair conditions. Each is listed below with what changed and what was measured. Everything was re-rendered (all 768 frames) and re-checked: `check_handmade.py` **ALL OK**; onsets unchanged (retreat 0, watch 0, refuse +0.125, watch 0, approach +0.042, lie down 0, looks in +0.058); head below the back in 100% of the approach. 1. **Looking in, 26.4–27.5 s.** There is a new head drawing and mesh, `L_head`: the dog's head from three-quarters behind. It shows the back of the skull and the backs of both ears; the muzzle peeks past the far cheek, aimed up and away into the lit room, above the woman's hands. It cross-fades in from the profile head as the score's `look` rises (0.35 → 0.6) and back out as it falls (by 28.3 s), and it is ARAP-solved like the other heads (ear tips fold with the ear act). The dog uses the profile head toward her and this head toward the room. Crops: `proof/look_in_crops.jpg` (26.25, 26.75, 27.50 s at 100%). 2. **Hindquarters, 23.8–32 s.** I stopped asking ARAP to fold the standing mesh into a lie (the prompt's own §6: change the rest mesh). The lying dog is its own engraved drawing and mesh, `P_lie` (857 vertices, 1,392 triangles): forelegs along the boards, the near hind leg already folded (thigh, hock on the ground), the tail laid back. - The standing mesh goes only as far as a deep crouch with the paws planted. Over the score's `lie` 0.42–0.62 it dissolves into `P_lie`, which arrives 70 mm high and settles. - On the lying mesh ARAP does only breathing, the neck and head, the settle and the tail sweep, so nothing has to fold. - **Measured: 0 inverted triangles in all 197 frames from 23.79 to 32 s** (the lie at the sill), and 0 in the refusal lie too. - The end card now shows the lying rest mesh solved, with no fold. Crops: `proof/lying_crops.jpg` (24.5, 27, 30 s at 100%, each labelled with its inversion count). 3. **The turns.** No frame-to-frame cut changes the view any more (`build/perform.py: view_for`): - The outgoing drawing foreshortens over two frames; in the turn away its head mesh has already flipped to lead the turn. - The two drawings then overlap for two frames, weighted 0.67 / 0.33 and then 0.33 / 0.67. In those frames the incoming view arrives ARAP-widened (rear or front view handles at 1.7 → 1.55 × width) and narrows over three more frames, while the rear view's head swings across behind the shoulders. - The profile then arrives foreshortened (55% → 100% width). - Refusal: 3.50–4.08 s; turn back: 8.17–8.67 s. - What it is: deformed drawings plus a short two-frame cross-dissolve, not drawn in-betweens. A faint double image shows in those overlap frames. 4. **The standing silhouette.** The neck is shorter (the head joint 47 mm lower and 23 mm back) and the chest deeper (40–45 mm lower brisket, 30 mm lower belly). The lying and crouching heights were changed to match. 5. **Shimmer.** - Hatching is 1.6× coarser on every figure: dog lines about 4 px apart in the wide shot, 6 px close. - Cross-hatching is off: it made dotted patterns at film scale. - The background's ruled lines are 1.45× wider; the dotted cross-hatch on the porch front, roof, posts, steps and picture is replaced by solid tone or single lines. - The scene wireframe overlay is removed: that was the "dotted fringe". The mesh is shown on the end card instead. - The figures are rasterised at 3× supersampling with a texture pre-filtered to 0.7 px/mm. - Looked at, at 100% crops, around 18 s and 27 s. Still weak / unknown after round 2: - **The standing and walking profile still turns 1–7 single triangles over in 56 of 768 frames** (107 in all). They sit in the cower (1.7–2.3 s) and are recurring singles in the walk (15–21 s), at the hind leg's swing. None are in any lying frame. - **The overlap frames of the turns show a ghosted double image** for two frames per view change. - **`L_head` is small at film size** (about 50 px). The back-of-head reading depends on the two ear backs and the muzzle tip. - Worst normalized ARAP energy is 0.39, at a walk frame. - At the end, 0.54 of the dog's rendered area is inside the door frame. - No critic has seen round 2. ## Round 3 (after critic round 2: DOES NOT PASS, narrowly) 1. **No double images.** Every frame now shows exactly one drawing (`perform.py: view_for` returns one layer; the stand/lie and profile/turned-in head changes are single-frame swaps in `render.py: expand`, not dissolves). - Turn away: the profile narrows over 4 frames (100% → 52%, 3.46–3.58 s) as the head drops toward the door. A hard swap at the narrowest frame (3.625) brings the rear view, arriving 12% wide and narrowing while its head swings across. A hard swap at 3.83 brings the profile turned away, arriving at 50% width and widening over 4 frames. - Turn back: the standing dog narrows (8.17–8.29 s), swaps to the front view (8.33–8.46), then swaps to the profile facing the door at 50% width (8.50). - Lie: the deep crouch was reshaped so its hip height (285 mm) matches the lying mesh's arrival height (286 mm). The swap happens on the frame the score's `lie` passes 0.55, the beat when the dog drops. - Look-in: the head swaps on the frame `look` passes 0.45. - I extracted frames from the encoded `film.mp4` at 3.50, 3.58, 3.62, 3.67, 3.71, 3.83, 3.88, 3.92, 4.58, 4.62, 7.50, 7.62, 8.17, 8.25, 8.33, 8.46, 8.50, 26.50, 26.75 and 27.50 s and looked at them: one body and one head in each. 2. **The over-the-shoulder head.** The turn away no longer uses a mirrored head: the profile head drops toward the door as the body narrows. Looking back from 7.46, the mirrored head is set at an absolute angle of −0.12 rad, so the muzzle points left to the door, level or a little down, at its back line. It is always solved from a rigid placement, so it has no beak. In the encoded film at 7.50 and 7.62 the eye, the dark muzzle and nose, and both ears read. 3. **Looking in.** `L_head` has a foreshortened muzzle with mass: wider at the base, rounded at the end, with a dark nose tip and highlight. The head is tilted 0.42 rad so the muzzle aims up and left into the lit room, above and past the woman's hand. 4. **Hind legs.** - The cower lowers the hips 75 mm (was 125) and drops the hocks (the metatarsus slants back 0.4 rad × crouch). - The deep crouch lowers the hips 170 mm (was 215), with the hocks at 0.7 rad. - The two-bone IK keeps thigh and shank at their rest lengths. - Looked at on the encoded film at 1.50, 1.83, 2.08 and 4.25 s: four distinct feet, no leg looping over itself. - Measured inverted triangles: 3 in 1.5–2.1 s (single slivers), 0 in 4.0–4.5 s, 0 in all lying frames (23.79–32 s). Measured after round 3: - `check_handmade.py` ALL OK. - Onsets: retreat 0, watch 0, **refuse +0.208** (first rendered frame whose view faces away, 3.625; the narrowing and head drop start at 3.458), watch 0, approach +0.042, lie down 0, looks in +0.058. - Head below the back in 100% of the approach. - Inverted triangles over the film: 79, in 53 standing or walking frames. - Worst normalized energy 0.61, at the cower's end (2.0–2.6 s). The hip is lowered less there, so the spine bends more. ## Round 4 (after critic round 3: DOES NOT PASS, very narrowly) 1. **"Two rumps, a doubled tail, an extra thigh" at 3.85–4.5 s.** There was no second mesh: `perform.json` has exactly one layer per frame (checked, max 1), and the renderer draws one mesh per frame. The doubling was inside the one drawing: - The near thigh and shoulder masses carried their own outline stroke inside the body, which read as a second rump. - The tucked tail swung down along the back of the thigh, which read as a second tail. - Fix: those two masses no longer draw an inner outline (only hatch direction changes there). A tucked tail now rotates 1.6× further, so it passes under the belly behind the thigh. Crops of the encoded film at 3.88, 4.00, 4.25 and 4.50 s show one rump, one tail and four legs. 2. **The hind legs at 7.5–8.3 s.** The standing mesh is now never driven past the deep crouch (score `lie` clipped to 0.5): the old path had started the lying fold on it before the swap to the lying mesh. The deep crouch is shallower: hips −115 mm, shoulders −100 mm, hocks slanted 0.45 rad. The swap to and from the lying mesh is a ~45 mm drop on the frame `lie` passes 0.55. Crops at 7.62 and 7.75 s on the encoded film show four distinct hind and fore feet, nothing crossed. 3. **Optional, done.** The figure outline is heavier (7 mm stroke, about 1.4 px wide in the wide shot). The far railing is lighter (rail tone 0.55 → 0.38, balusters 0.45 → 0.30). Re-measured after a full re-render: `check_handmade.py` ALL OK. Onsets unchanged (0, 0, +0.208, 0, +0.042, 0, +0.058). 0 inverted triangles in the lying frames. 79 single triangles over the film, in 53 standing or walking frames. The film is 2.92 MB.