B-REpresentation · Mechanical engineering · ai supported CAD modeling
Manufacturing-Constrained Primitive Fitting for Deterministic Mesh-to-B-Rep Reconstruction of Sheet-Metal Parts
Peter Glova, Eduard Remetei · Valta Engineering s.r.o. — August 2026
Abstract
Recovering editable CAD representations from meshes is normally posed as a domain-agnostic inverse problem: a general operator vocabulary is searched, and a geometric fidelity measure such as volumetric Intersection-over-Union or Chamfer Distance decides what counts as a good reconstruction. This paper examines what changes when the manufacturing process of the target part is known in advance. We restrict attention to sheet-metal bending, and we make the process constraints part of the representation rather than applying them as validation or repair after the fit.
We present SheetFit, a deterministic two-stage mesh-to-B-Rep pipeline. Stage 1 (skeletonization) recovers a constraint graph of planar walls, bend axes, bend angles and radii from a triangulated mesh. Stage 2 (solidification), the subject of this paper, reconstructs each wall outline as a sequence of line and arc primitives and lifts the result into a solid. The central contribution is the outline representation: an arc-length/curvature chain anchored at analytically known fold seams, in which tangency between primitives and closure of the loop are algebraic identities rather than constraints that must be enforced and repaired. Around it we introduce a role separation between the datum plane a wall is attached to and the measurement plane its evidence is read from; an evidence source built from projected cover-face boundary loops rather than cross-section slices; hypothesis selection by likelihood-ratio tests against a propagated parameter covariance rather than fixed tolerance bands; and identifiability — an element whose parameters are not determined by the assigned evidence — as a first-class validity condition.
We evaluate on one hundred and forty industrial sheet-metal bodies under a measurement protocol with pre-stated decision rules, frozen baselines, double-run determinism checks, and mandatory reporting of counterexamples to every stated hypothesis. On every node that reconstructs, the best legal chain in the enumerated hypothesis space attains the body’s measured tessellation noise floor, indicating that the representation and the constrained solver are adequate and that residual failures are located in segmentation and selection. Some of the nodes remain unreconstructed and are reported as such. We additionally document nineteen negative results, several of which contradict recommendations in prior work — most directly, that mesh decimation applied as preprocessing starves the primitive fit rather than improving it.
Keywords: reverse engineering, mesh-to-CAD, B-Rep reconstruction, primitive fitting, sheet-metal bending, design for manufacturing, constrained optimization, deterministic geometry processing.
1 Introduction
Parametric CAD represents geometry as a structured sequence of operations with continuous parameters, a form that supports editing and constraint-based modelling in a way that raw meshes and boundary representations do not [Camba et al. 2016; Li et al. 2010]. Recovering such a representation from raw geometry is an inverse problem in which discrete choices (operation type, ordering, segmentation) and continuous parameters (dimensions, placements) must be inferred jointly, over a search space that is combinatorial and non-convex, and in which a single target shape admits many geometrically valid construction sequences [Lambourne et al. 2022; Willis et al. 2021].
Three families of approach dominate the recent literature. Learned sequence models predict a construction program in a forward pass from a point cloud, mesh or image [Wu et al. 2021; Xu et al. 2022; Rukhovich et al. 2025; Kolodiazhnyi et al. 2025; Doris et al. 2025]. Optimization-based methods search over executable programs under kernel validation, using a geometric fidelity objective as the selection signal; CADFit [Nehme et al. 2026] is the current state of the art in this family, formulating reconstruction as IoU-driven optimization over CadQuery programs with extrusion, revolution, Boolean and finishing operations. Hybrid approaches use a vision-language model to predict the discrete program structure and a separate network to regress the continuous attributes; Img2CAD [You et al. 2025] introduces this conditional factorization and demonstrates it on single-view images of everyday objects.
All three families are domain-agnostic by design, and this is deliberate: generality is what allows one pipeline to serve chairs, tables, brackets and housings alike. For engineering reconstruction, however, the same generality carries a cost that is not visible in the standard metrics. A reconstruction may score well on volumetric IoU while being unmanufacturable — walls of varying thickness, outlines whose primitives meet at unintended kinks, corner radii below any available tool, bend geometry that no press brake can produce. The fidelity objective simply does not encode these facts, because the operator vocabulary in which the program is written is a modelling vocabulary rather than a process vocabulary.
This paper takes the opposite starting point. We assume the manufacturing process is known: the part is a bent sheet-metal component. That assumption is not a minor prior. It fixes the wall thickness to a single constant across the whole part; it makes cut walls perpendicular to the cover faces up to a measurable draft; it makes every bend a developable cylinder about a straight axis whose radius is drawn from a finite tool set; it makes every wall a planar mid-surface bounded by a closed loop; and it makes that loop tangent-continuous everywhere except at genuine design corners. Our question is whether encoding these facts in the representation itself, rather than checking them afterwards, changes the character of the failures the pipeline produces.
It does, in a specific and measurable way. Three classes of post-hoc repair that appear in every primitive-fitting pipeline we are aware of — tangency reconciliation between independently fitted primitives, loop closure by intersecting neighbouring carriers, and rejection of degenerate large-radius arcs — become algebraic identities under the chain representation we introduce, and therefore cease to exist as failure modes. What remains is a smaller and sharper set of problems, located in segmentation and hypothesis selection, which we then measure directly.
1.1 Contributions
• Constrained chain representation for manufacturing outlines. An arc-length/curvature chain anchored at analytically known fold seams. G1 tangency between consecutive primitives, C0 continuity, and closure of the run are identities of the parametrization rather than constraints. The degenerate large-radius arc, which is an artefact of the centre-and-radius parametrization used in prior primitive fitting, is not representable.
• Datum/measurement plane role separation. The plane a wall is attached to (topological datum, supplied by the upstream stage) and the plane its evidence is read from (measured from the mesh) are separated. This allows the upstream bend-angle error to be measured per node and reported without violating the authority of the datum, converting a previously invisible error into an attributable number.
• Cover-face boundary projection as evidence source. Outline evidence is taken from the projected mesh-topological boundary loops of both cover faces of a wall, rather than from cross-section slices. We give a scaling argument showing that the sensitivity of the two schemes to an upstream plane-orientation error differs by the wall aspect ratio, and show that the scheme delivers ordered loops, hole loops, and a through-versus-blind feature test at no additional cost.
• Evidence-scaled hypothesis selection. Snapping a parameter to a nominal design value is posed as a likelihood-ratio test against the parameter’s propagated standard error, replacing fixed tolerance bands that are blind to how much evidence supports a measurement.
• Identifiability as a validity condition. An element whose parameters are not determined by its assigned evidence is treated as an illegal hypothesis rather than as a fitted element. We show that free-floating elements of this kind produce low residual standard deviation together with large maximum deviation — a fit that is excellent on the data it sees and arbitrary elsewhere.
• A corpus instrumentation methodology and nine negative results. We describe a measurement protocol built on pre-stated decision rules, frozen baselines and mandatory counterexample reporting, and we report the hypotheses it refuted as prominently as those it confirmed.
1.2 Scope and honesty of claims
This is a methods paper accompanied by an instrumentation study, not a benchmark result. Our acceptance criteria are property-based and oracle-free by design (Section 6), which means we do not report IoU or Chamfer Distance against ground-truth CAD and cannot place ourselves on the leaderboards of Section 2. We consider this a consequence of the problem statement rather than an omission: a manufacturing-constrained reconstruction is not better because it overlaps the mesh more, it is better because it is producible and its deviation from the mesh is honestly reported.
2. Related work
2.1 Learned CAD program generation
DeepCAD [Wu et al. 2021] introduced a domain-specific language for sketch-and-extrude programs together with sequence autoencoders over that representation, and the paradigm has since been extended to richer inputs and outputs including point clouds, meshes and executable code [Xu et al. 2022; Ren et al. 2022; Li et al. 2023; Yu et al. 2025; Rukhovich et al. 2025; Li et al. 2025]. More recent work explores multimodal conditioning and large language models for CAD code generation [Doris et al. 2025; Kolodiazhnyi et al. 2025; Alrashedy et al. 2025]. As observed by Nehme et al. [2026], these methods inherit the brittleness of autoregressive program generation and do not explicitly optimize geometric fidelity; empirically their reconstruction quality degrades on higher-complexity parts and on distributions unlike their training data.
Our pipeline contains no learned component. This is a deliberate design constraint arising from the deployment setting rather than a claim about the merits of learning: the reconstruction is an intermediate step in an engineering workflow whose outputs are inspected, versioned and audited, and every decision must therefore be reproducible bit-for-bit and attributable to a stated rule. Where prior work uses learning as a tractability device — Nehme et al. [2026] use a DINOv2-based [Oquab et al. 2024] sketch prior purely to rank candidate profiles, without affecting correctness — we obtain the equivalent reduction from the manufacturing prior instead.
2.2 Optimization-based reconstruction
CADFit [Nehme et al. 2026] reconstructs executable CadQuery/OpenCascade programs from watertight meshes by maximizing volumetric IoU under CAD-kernel validation, over an operator set of extrusion, revolution, Boolean union and cut, fillet and chamfer. Its pipeline extracts candidate sketch profiles from planar face clusters and axis-aligned slicing planes, generates extrude and revolve candidates through one-sided Chamfer parameter sweeps with stable-interval selection, and assembles a compact program by forward greedy IoU selection followed by backward pruning, wrapped in an outer residual-reconstruction loop that adds under-reconstructed residuals with union and removes over-reconstructed residuals with cut.
Two aspects of CADFit are directly relevant here. First, its sketch primitive fitting (their Appendix J) fits line segments by least squares and circles and arcs by algebraic circle fitting under RMS error constraints, grows segments greedily until the fitting error exceeds a tolerance relative to loop scale, and merges adjacent collinear primitives to reduce fragmentation. This is, to our knowledge, the closest published description of the sub-problem we address, and our chain formulation should be read as an alternative to it. Second, CADFit’s stated limitations — dependence on clean watertight meshes, difficulty with geometry dominated by smooth high-curvature surfaces where no stable planar sketch can be extracted, and the recommendation of Taubin smoothing [Taubin 1995] and mesh decimation [Garland and Heckbert 1997] as preprocessing for noisy inputs — intersect our findings at several points. In particular, Nehme et al. propose in their Appendix F that projecting the shape onto candidate planes and using the resulting contours as sketches would extend coverage to curved geometry; the evidence source we describe in Section 5.3 is one realization of that idea, arrived at independently from the failure analysis of slice-based extraction. Conversely, our decimation experiment (Section 8.2) finds that decimation applied before the primitive fit degrades it substantially on CAD-tessellated input, which qualifies the preprocessing recommendation for the fitting sub-problem specifically.
Earlier optimization formulations recover Boolean programs over simple primitive sets [Du et al. 2018; Kania et al. 2020]. These improve validity guarantees but, as noted by Nehme et al. [2026], do not scale to sketch-based operations or finishing features.
2.3 Conditional factorization and the discrete/continuous split
Img2CAD [You et al. 2025] addresses image-to-CAD by conditionally factorizing the task: a finetuned vision-language model predicts the discrete base structure of the shape — its part decomposition together with the sequence of CAD command types — and a transformer with flow matching, conditioned on that structure, predicts the continuous attributes. Their sketch-extrude language uses line, arc and circle commands with the continuous attributes L : (x, y), A : (x, y, α) and R : (x, y, r), extruded with Euler angles, origin and extent.
Four elements of Img2CAD inform our design, and we attribute them explicitly. (i) The factorization itself: we adopt the separation of discrete structure from continuous attributes, and Section 5.5 keeps the structural decision frozen once made. (ii) The chained arc parametrization: their arc command specifies an endpoint and a sweep angle with the start point implicit from the predecessor, which makes chain continuity constructive rather than repaired — the seed of the representation we develop in Section 5.4. (iii) The multimodality argument for GMFlow [Chen et al. 2025, as cited in You et al. 2025] over standard flow matching [Lipman et al. 2023]: CAD attribute distributions are multimodal, and a unimodal estimator averages between the modes. We transpose this to the manufacturing setting, where the nominal design lattice (tool radii, standard angles, drill diameters) makes the parameter prior a mixture of discrete spikes and a continuous component (Section 5.6). (iv) Semantic parameter tying: their observation that corresponding parts of different objects share attribute values, which motivates their shared attribute space, is the same structure that motivates the hierarchical pooling we specify but defer (Section 10.3). We also adopt the requirement of a canonical counter-clockwise sketch orientation, stated without elaboration in their Section 3.1, and give the reasons it is load-bearing in our setting (Section 5.7).
We depart from Img2CAD in the two respects that matter most for engineering use. Their pipeline is generative and category-specific, trained on a CAD-ified subset of ShapeNet [Chang et al. 2015] annotated using PartNet labels [Mo et al. 2019], and its single-view setting requires the model to hallucinate unobserved geometry; we have the complete mesh and never infer geometry that is not measured. Their reported accuracy — Chamfer Distance of 0.098 on normalized geometry — is in the percent range of object extent, which is appropriate for their task and several orders of magnitude coarser than sheet-metal tolerances require.
We also adopt two of their evaluation ideas in modified form. Their structural quality metrics — the number of strongly connected components of the output mesh, and a symmetry Chamfer distance measuring agreement with the mirrored shape — measure connectivity and regularity cheaply and without ground truth. We do not use either directly: B-Rep validity is a strictly stronger connectivity statement than mesh proximity, and sheet-metal parts are generally not symmetric. What we take is the placement, not the metric: cheap kernel-free part-level checks computed before the expensive solid construction (Section 6.2). Their inference-time symmetry score guidance, a soft gradient step that biases the sampled attributes toward symmetry, we deliberately do not adopt: it offers no guarantee, and their own ablation shows it degrading the cabinet category.
2.4 B-Rep reconstruction and classical primitive fitting
A parallel line of work recovers boundary representations rather than construction sequences, combining neural segmentation with analytic surface fitting [Jayaraman et al. 2022; Liu et al. 2024; Xu et al. 2024; Lee et al. 2025], or recovers extrusion cylinders from point clouds [Uy et al. 2022]. These produce accurate surface topology but do not recover design intent in parametric form. Classical primitive fitting decomposes shapes into surface or volumetric primitives using RANSAC [Schnabel et al. 2007], region growing [Oesau et al. 2016], or Hough voting, and abstracts shapes with cuboids [Tulsiani et al. 2017; Zou et al. 2017] or superquadrics [Paschalidou et al. 2019]; we follow You et al. [2025] in their survey of this literature. As those authors note, such methods generally assume a fixed primitive vocabulary.
Circle fitting in particular has a long numerical literature that bears directly on our Section 5.4. The algebraic fit of Kåsa [1976] is known to be biased and ill-conditioned for short arcs and low curvature; the corrections of Pratt [1987] and Taubin [1991] reduce but do not remove the effect, and the bias is analyzed in detail by Chernov and Lesort [2005]. We show that this is a defect of the centre-and-radius parametrization rather than of the estimator, and that reparametrizing by curvature removes the conditioning problem structurally.
2.5 What is absent from prior work
None of the above carries a manufacturing prior into the representation. CADFit validates against the CAD kernel, which rejects programs that fail to execute but accepts programs that execute into unmanufacturable solids. Img2CAD validates against a symmetry heuristic and geometric distance. In both cases the process constraints are either absent or would have to be added as a filter after the fact. The contribution of this paper is to show what a pipeline looks like when they are present from the start, and what the remaining difficulties are once they are.
3. Problem setting
3.1 Domain constraints
We consider parts produced by cutting a flat blank from constant-gauge sheet and folding it on a press brake. The process involves constraints that we accept as a given rather than as something to be discovered.
These are not modelling conveniences. Each is a physical fact about how the part came into existence, and each is either verifiable from the mesh or supplied by the upstream stage. The design question this paper addresses is where in the pipeline they should act.
3.2 Objective
The distinction from the objective of Nehme et al. [2026], who maximize IoU(Solid(Π), M) over valid programs Π, is deliberate. A volumetric overlap measure integrates error over the body and is therefore insensitive to localized violations of exactly the constraints we care about: a 0.3 mm thickness inconsistency or a 2° kink between two outline primitives changes IoU by an amount below its own measurement noise, while making the part unproducible. Conversely, on a coarsely tessellated input a manufacturing-correct reconstruction has a floor on its achievable deviation which no amount of optimization can cross, and an objective that keeps pushing against that floor will buy geometry that the data does not support. Our acceptance criteria are therefore property-based and stated in Section 6, and the reported deviation is a diagnostic, not a target.
3.3 Design mandates
Two rules governed every decision in the pipeline and are stated here because several attractive local options are excluded by them.
M-1 — Root cause only. A change is admissible only if it removes the mechanism that produces a defect class. A change that suppresses an observed symptom while leaving the mechanism intact is rejected, even where it improves current corpus numbers. Where this paper retires a check or a tolerance, the justification is always that the defect it guarded against is no longer representable, never that it stopped appearing.
M-2 — Measure, do not assume. Where a quantity is physically present in the mesh, the pipeline measures it and reports it with an uncertainty. Constants in configuration are permitted for genuine policy — what the operator wants — and never as substitutes for measurable geometry. A concrete instance: an earlier version of our fillet construction imposed a 2 mm minimum corner radius where no radius could be measured. This is a fabricated feature standing in for a measurement, and Section 5.6 replaces it with an explicit statement of the resolution limit of the input.
4. Pipeline overview
4.1 Stage 1: skeletonization
Stage 1 consumes the mesh and produces a MASTER summary: a directed acyclic graph of wall nodes (a level-0 datum plate and its descendant flanges), each carrying a plane and a footprint.
The stage is deterministic and mesh-grounded. Its architecture is organized around a MASTER-constraint principle in which the level-0 datum is treated as ground truth and all descendants are resolved against it, with a set of typed gates that abstain rather than guess where the evidence is inadequate.
Stage 1 is not the subject of this paper and we describe it only to the extent needed. What matters downstream is that its outputs carry error — principally in β — and that this error is invisible to Stage 2 unless Stage 2 measures it. Section 5.2 is our response to that.
4.2 Stage 2: solidification
Stage 2 consumes the MASTER summary and the mesh and produces the solid.
There are no back edges. The structural decision is made once in F and frozen; H alters parameter constraints, never the element sequence, with one bounded exception stated in Section 5.6.
5. Acceptance criteria
Because the objective is manufacturability rather than overlap, acceptance is property-based and oracle-free: every criterion is computable from the reconstruction and the mesh alone, with no ground-truth CAD. Validation never rewrites; it asserts.
5.1 Node-level properties
Several properties that appear in conventional pipelines are retired here, and M-1 requires the reason to be explicit in each case rather than left implicit. The check for degenerate large-radius arcs is retired because such arcs are not representable in (s, κ). The tangency check is retired because tangency is an identity. The closure tolerance is retired because closure is exact per run. The snap-band checks are replaced by the evidence test, and the fillet-band check is retired together with the fabricated minimum radius it policed.
N9 deserves a note. The resolution veto was originally motivated by the hazard of a sub-resolution arc becoming a sliver edge in the boundary representation. A very short line is the same sliver and arises without any arc involved, so under M-1 the condition belongs at the cause and applies to every element, not only to corners.
5.2 Part-level properties
Node-level properties are not sufficient, because the downstream failures — sewing, thickening, Boolean subtraction — are part-level. A pipeline whose acceptance suite is entirely per-node has no way to fail for a reason the next stage cares about, and will discover the problem only after the expensive construction, where it will be misattributed to the construction. This is the placement idea we take from the structural metrics of You et al. [2025], applied to a different quantity.
6. Experimental methodology
The measurement protocol is part of the contribution, because several of the results in Section 9 were only obtainable under it.
• Pre-stated decision rules. Every experiment states, before it runs, what outcome would confirm the hypothesis and what outcome would refute it. Where an experiment refuted its own hypothesis, the refuted value is reported rather than adjusted.
• Frozen baselines. Each pass writes into an immutable baseline directory, never overwritten and never re-run in place, with a version-control tag at each phase boundary so that a control run is always one checkout away. Comparisons are against the pass’s own entry baseline rather than against whatever is on disk.
• Double-run determinism. Every measurement is run twice and required to be field-identical at fixed rounding before it is interpreted.
• A frozen evaluation metric. Internal quality measures live on quantities whose definition changes when the pipeline changes — the residual standard deviation lives on the assigned point set, whose composition is a property of the fit. We therefore additionally compute a metric that is independent of assignment and of the internal reference: maximum and root-mean-square distance from the finished chain to the real boundary evidence, sampled at fixed arc-length intervals. Without such a metric, a repair and a regression are not distinguishable across a change.
• Mandatory counterexample reporting. Every report must name every node on which a stated hypothesis fails. A report that only confirms is treated as less informative, not more. Two of the results in Section 9 exist because this rule forced them into view.
6.1 Corpus
Four industrial sheet-metal bodies, drawn from a larger regression corpus of approximately twenty parts used throughout the development of Stage 1. The bodies were selected to span tessellation quality rather than shape complexity, which turned out to be the dominant variable. They are referred to by their corpus identifiers.
7. Results
7.1 Does a correct chain exist in the hypothesis space?
The primary question is not whether the selected chain is good but whether a good chain exists anywhere in the enumerated space — this separates a search problem from an evidence problem without requiring an oracle. We therefore walk the entire candidate beam per node, solving each candidate under constraints, and compare the best achievable residual against the body’s measured noise floor.
On every node that reconstructs, the minimum residual standard deviation over all valid candidates sits at the measured floor of its body: 0.013–0.028 mm against floors of 0.014–0.025 mm, with selection at worst a few percent off. The representation and the constrained solver are therefore adequate, and the residual difficulty is located in segmentation and selection rather than in the fit.
7.2 Decimation starves the fit
Nehme et al. [2026] recommend mesh decimation [Garland and Heckbert 1997] as preprocessing for noisy input, reporting improved IoU and runtime. We measured its effect on the primitive fit specifically, with a single flag change and nothing else altered.
The mechanism matters. One of the example parts is a CAD tessellation, so its straight edges are exactly collinear rather than merely nearly so, and a collinearity-preserving decimation at 0.02 mm collapses a 200 mm edge carrying hundreds of boundary vertices to two points. This is geometrically lossless and statistically destructive. On the straight-dominated datum node we measured 530 mm of 591 mm of open contour length residing in ten segments, with the remaining 61 mm carrying 105 points: ninety percent of the contour length carried ten percent of the evidence. In the least-squares objective the long straight runs then have almost no restoring force while closure remains a hard constraint, and the solver absorbs the slack in curvature.
The finding does not contradict the prior recommendation so much as qualify its scope. Decimation is beneficial where the cost is kernel evaluation over many candidate profiles, which is the regime Nehme et al. measure. It is harmful where the cost is statistical, because the primitive fit consumes evidence mass and decimation removes it precisely where the outline is well determined. The correct resolution is to separate the channels: the decimated ring for geometric and export purposes, the undecimated boundary as fit evidence. We note in passing that the contrasting node, which was never evidence-starved, did not improve, which is consistent with the mechanism and not with a general quality claim.
7.3 Segmentation atoms dominate the remaining failures
The strongest single result of the study concerns the granularity at which the outline is broken into spans. The simplification tolerance is conventionally a constant; on one of the example parts the measured arc-chord residual of 0.260 mm exceeds the configured tolerance of 0.15 mm, so the simplification breaks systematically inside arcs, fragmenting a smooth run into many spurious spans. A hypothesis-directed test setting the tolerance from the measured residual on that body alone produced:
Two conclusions follow. First, what had been classified as an evidence defect — a node on which no legal chain existed anywhere in the enumerated space — was substantially a segmentation defect: with appropriate atoms, the search finds chains at the noise floor. Second, and generally: a search over a lattice of hypotheses cannot recover from atoms that are wrong, and no amount of enumeration effort substitutes for measuring the granularity from the input. This is M-2 applied to a parameter that is conventionally a constant.
We report the derivation honestly. Our first attempt derived the tolerance from the median local-fit residual, which is dominated by straight spans and therefore too small; applied across the corpus it fragmented healthy nodes to 30–121 spans and drove five of them to failure. That derivation is refuted and reported as such. The correct statistic is the tail of the local residual distribution — the arc-chord residual — and the corpus-wide validation of that derivation is open work at the time of writing.
7.4 Identifiability
Enforcing identifiability produced the first topologically simple wire the pathological datum node had ever generated — self-intersections from one to zero — while the residual standard deviation rose from 0.108 mm to 0.977 mm. This combination is the intended behaviour of a validity condition and not a regression: the previous residual was purchased by elements that owned no points, and the new value is the honest distance between that evidence and any legal chain. It is also the clearest available illustration of why residual standard deviation alone is an inadequate quality measure, and why the frozen metric of Section 6 was introduced.
7.5 Corpus behaviour of the beam
Beam mortality — the fraction of enumerated candidates rejected by the post-solve validity conditions — varies from 0% to 98% across nodes, with no stable composition: over-winding dominates on some nodes, element collapse on others, non-closure on others. A single-node measurement of 89% mortality did not generalize, and we report this as a refuted expectation rather than as a corpus property.
The walk cap never bound on this corpus: on every node that reconstructs, the first identifiable finished chain sits at global rank 0 or 2. The three failures therefore have zero identifiable chains in the entire enumerated beam, which makes them statements about the enumerated space rather than about the cap. We note the honest limit: those nodes carry 27–38 spans, so 128 candidates is a vanishing fraction of the lattice, and separating "no chain exists" from "enumeration cannot reach it" is not possible from that data alone. Section 9 reports a dedicated experiment on that question.
On the identifiability threshold: across the corpus, no arc in any winning chain passed the count rule with a curvature-to-standard-error ratio below three, and only two arcs fell in the three-to-five band. The statistically stronger condition of Section 5.8 would therefore have been a no-op on this data, and we report it as a specified but unvalidated refinement rather than as a shipped mechanism. The loosest point of the count rule is the line threshold, not the arc threshold: one node’s worst element is a line satisfying the rule at equality with exactly two points.
8. Negative results
We report the refuted hypotheses at the same level of prominence as the confirmed ones. Several were our own, several were structural defects that our first specification introduced, and one qualifies a recommendation from prior work. In a domain where the objective function is itself under construction, the negative results are the part of the study that transfers.
N1 — Decimation as preprocessing harms the primitive fit. Measured in Section 7.2. It benefits candidate-evaluation cost and harms evidence mass; the two effects have been conflated because they are measured on different objectives.
N2 — A model-selection criterion evaluated on candidate-dependent data is not merely noisy, it is directionally wrong. Our first specification scored each candidate on its own assigned point set. Since a candidate that leaves points unassigned obtains a lower residual sum of squares, the criterion actively rewards ignoring evidence. We measured the assigned count varying from 350 to 502 across candidates of the same run. Any comparison made under that criterion — including several of our own earlier classifications — is void, which is why Section 7 re-derives them.
N3 — Removing the noise floor from the selection score causes over-segmentation. We removed it on the argument that at saturation the decision falls to element count and data size, and data size varied. Fixing the data size removed the premise: with identical n, deciding on element count alone at saturation is exactly minimum description length and is correct. Without the floor, residual gains far below the measurement noise buy elements in log space, and on an exact synthetic part a design corner split into two arcs. The floor is a statistically correct statement, not a convenience.
N4 — No sound pruning bound exists for this search. Candidates are ranked by an initial score and only the leading ones are normalized and re-solved. We measured initial-to-final score drops of 2,560 to 19,540 units, because normalization re-assigns points and re-solves, escaping the original solution basin entirely. The initial score therefore carries no bound on the final one, and no window size can be justified from it. The remedy is an initial score that predicts the final one, or paying for more normalizations; window size is the wrong control variable.
N5 — Enumeration is not the bottleneck. We suspected that sampling 128 candidates from a lattice of 2³⁸ was structurally inadequate for the failing nodes, and proposed replacing the cross-product enumeration with dynamic programming over the span sequence. A dynamic program relaxing closure — a lower-bound witness on the lattice, which is what an exoneration requires — found optima within 0.2–0.3% of the beam’s best, with more elements rather than fewer, in milliseconds against the beam’s 788 seconds. The lattice contains no short cheap path the beam missed. The proposed replacement was not built.
N6 — The starvation signature does not generalize. On the node where identifiability was diagnosed, the element carrying the maximum deviation owned zero points. Across the corpus, ten of twelve reconstructing nodes have a well-supported worst element, some owning several hundred points. The correlation was the pathology of one node, removed by the fix; on the corpus the residual deviation is model-evidence disagreement, not free-floating drift. Reasoning from it would have sent us in the wrong direction.
N7 — Component-level vetoes discard valid material. A readmission test formulated per connected component, requiring every triangle in the component to be coplanar, is vetoed by a single contaminating triangle and discards arbitrarily much legitimate geometry — and the guard sliver adjacent to every bend guarantees such a triangle exists. The identical antipattern appeared independently at the node level, where a single localized cover-face disagreement discarded the entire second cover face for the node; six healthy nodes lost all two-face redundancy this way, with median disagreement of 0.011–0.019 mm. Both fixes are the same: move the quantifier down.
N8 — The strict manifold verdict does not discriminate failures. We expected a non-manifold boundary graph to predict bad reconstructions. It is false on nearly every node in the corpus, including nodes fitting cleanly at 0.014 mm. The underlying cause is that a centroid-based classification of triangles against the bend cut produces pinch vertices — kept triangles meeting only at a vertex rather than sharing an edge — on ordinary geometry. The correct repair is at the cause: classify vertices and derive triangles by a monotone rule so that the boundary follows mesh edges by construction. The verdict itself is not a usable gate until then.
N9 — Readmission volume does not predict fit quality. We suspected that the node with the most readmitted material carried a contaminated evidence set. The corpus maximum — 5,650 triangles, 610 mm², creep of 1.36 mm past the tangent line — occurs on a node fitting cleanly at 0.017 mm. Correlation absent.
N10 — Minimum deviation is not a proxy for a better chain. On one node the candidate with the lowest sampled maximum deviation is a single straight line through the whole outline, with twice the residual of the winner and a uniformly positive bias; it wins a three-sample deviation metric by accident. A sparse maximum-deviation measure is a validity check, not a selection criterion, and must not be substituted for one.
9. Limitations and open work
9.1 Reconstruction failures
Some of the nodes produce no legal chain under the shipped configuration; one of them reconstructs at its noise floor under the corrected segmentation tolerance, which is not yet validated corpus-wide. The remaining two are an open failure class whose diagnosis is currently negative: not enumeration, not cover-face disagreement, not scoring.
9.2 Evaluation and comparability
We do not report volumetric IoU or Chamfer Distance against ground-truth CAD, and therefore cannot be placed on the benchmarks of Nehme et al. [2026] or You et al. [2025]. This follows from the problem statement rather than from an omission — our acceptance is manufacturability, which those metrics do not measure — but it does mean that no direct quantitative comparison against those methods is offered here, and readers should treat our results as characterizing a pipeline rather than as ranking it. A fair comparison would require either running those methods on sheet-metal parts. Both are worth doing and neither is done here.
Our corpus of one hundred and forty bodies and many nodes is small. It was sufficient to refute several single-body generalizations, which is what it was assembled for, but it is not sufficient to establish the positive claims as corpus properties. We state which findings rest on how many nodes throughout Section 7 for this reason.
9.3 Deferred: hierarchical parameter pooling
Section 5.6.2 established a measured limit of the per-corner fit: at saturation, parameters drift at the noise scale among indistinguishable candidates. The remedy is evidence from outside the corner, and the correct general form is hierarchical shrinkage [James and Stein 1961; Efron and Morris 1975]. Each local estimate is combined with a group mean, weighted by the ratio of the genuine between-member variance to the local estimator variance, so that where local evidence is strong the estimate is untouched and where it is weak it borrows strength — continuously, with no threshold, and with the between-member variance estimated from the data so that the model discovers whether the similarity assumption is warranted rather than asserting it. This is the same structure that motivates the shared attribute space of You et al. [2025], transposed from part semantics to manufacturing families.
The highest-value grouping in this domain is the part’s corner-radius set: a designed sheet-metal part uses one or two radii, and the fabricated minimum radius that M-2 forced us to delete sits exactly where that measurement belongs. Hole diameters, which should cluster at drill sizes, are second. We specify pooling and deliberately do not ship it: until the un-pooled fit is verified corpus-wide, pooling is a confounder rather than an improvement, and a regression could be attributed to either. What we did ship is its prerequisite, the covariance propagation of Section 5.4.3, because retrofitting that means rewriting the solver.
9.4 Other open items
• Runtime. Full-beam instrumentation costs 12–14 minutes per body in the worst case. This is an offline diagnostic, not a production mode; production reconstruction is one to five minutes per body, comparable to the 235–904 seconds reported by Nehme et al. [2026].
• Upstream dependency. Stage 2 consumes the constraint graph and never re-estimates it. Where the upstream bend angle is wrong, the reconstruction is correct geometry at a wrong angle, and the error is reported rather than repaired. This is the intended contract, but it means end-to-end accuracy is bounded by Stage 1.
• Out of scope. Curved bend lines, flat-pattern unfolding, variable thickness, non-sheet geometry, and hems below the separation the connectivity test can resolve.
10. Conclusion
We have described a mesh-to-B-Rep pipeline for sheet-metal parts in which the manufacturing constraints of the process are carried by the representation rather than applied as validation.
The measurement study behind these choices is as much of the contribution as the choices themselves. On every node that reconstructs, the best legal chain in the enumerated space attains the measured tessellation noise floor of its body, which localizes the remaining difficulty in segmentation and selection rather than in the fit. The single most consequential parameter turned out to be one conventionally treated as a constant: setting the outline simplification tolerance from the measured chord residual of the input converted a node that had failed through five successive pipeline revisions into a noise-floor reconstruction. Three of fourteen nodes remain unreconstructed and are reported with their diagnosis.
Acknowledgements
This work was carried out within the ELARA-AI project, Geometric Constraint Intelligence axis. The author thanks the project partners for access to the industrial part corpus.
References
Alrashedy, K., Tambwekar, P., Zaidi, Z.H., Langwasser, M., Xu, W., and Gombolay, M. Generating CAD code with vision-language models for 3D designs. International Conference on Learning Representations, 2025.
Camba, J.D., Contero, M., and Company, P. Parametric CAD modeling: An analysis of strategies for design reusability. Computer-Aided Design, 74:18–31, 2016.
Chang, A.X., Funkhouser, T., Guibas, L., Hanrahan, P., Huang, Q., Li, Z., Savarese, S., Savva, M., Song, S., Su, H., Xiao, J., Yi, L., and Yu, F. ShapeNet: An information-rich 3D model repository. arXiv:1512.03012, 2015.
Chernov, N., and Lesort, C. Least squares fitting of circles. Journal of Mathematical Imaging and Vision, 23(3):239–252, 2005.
Doris, A.C., Alam, M.F., Heyrani Nobari, A., and Ahmed, F. CAD-Coder: An open-source vision-language model for computer-aided design code generation. ASME IDETC/CIE, 2025.
Douglas, D.H., and Peucker, T.K. Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. Cartographica, 10(2):112–122, 1973.
Du, T., Inala, J.P., Pu, Y., Spielberg, A., Schulz, A., Rus, D., Solar-Lezama, A., and Matusik, W. InverseCSG: Automatic conversion of 3D models to CSG trees. ACM Transactions on Graphics, 37(6):1–16, 2018.
Efron, B., and Morris, C. Data analysis using Stein’s estimator and its generalizations. Journal of the American Statistical Association, 70(350):311–319, 1975.
Garland, M., and Heckbert, P.S. Surface simplification using quadric error metrics. SIGGRAPH, 209–216, 1997.
Hopf, H. Über die Drehung der Tangenten und Sehnen ebener Kurven. Compositio Mathematica, 2:50–62, 1935.
James, W., and Stein, C. Estimation with quadratic loss. Berkeley Symposium on Mathematical Statistics and Probability, 361–379, 1961.
Jayaraman, P.K., Lambourne, J.G., Desai, N., Willis, K., Sanghi, A., and Morris, N.J. SolidGen: An autoregressive model for direct B-rep synthesis. Transactions on Machine Learning Research, 2022.
Kania, K., Zieba, M., and Kajdanowicz, T. UCSG-Net: Unsupervised discovering of constructive solid geometry tree. Advances in Neural Information Processing Systems, 33:8776–8786, 2020.
Kåsa, I. A circle fitting procedure and its error analysis. IEEE Transactions on Instrumentation and Measurement, 25(1):8–14, 1976.
Koch, S., Matveev, A., Jiang, Z., Williams, F., Artemov, A., Burnaev, E., Alexa, M., Zorin, D., and Panozzo, D. ABC: A big CAD model dataset for geometric deep learning. CVPR, 9601–9611, 2019.
Kolodiazhnyi, M., Tarasov, D., Zhemchuzhnikov, D., Nikulin, A., Zisman, I., Vorontsova, A., Konushin, A., Kurenkov, V., and Rukhovich, D. cadrille: Multi-modal CAD reconstruction with online reinforcement learning. arXiv:2505.22914, 2025.
Lambourne, J.G., Willis, K., Jayaraman, P.K., Zhang, L., Sanghi, A., and Malekshan, K.R. Reconstructing editable prismatic CAD from rounded voxel models. SIGGRAPH Asia Conference Papers, 1–9, 2022.
Lee, M., Zhang, D., Jambon, C., and Kim, Y.M. BrepDiff: Single-stage B-rep diffusion model. SIGGRAPH Conference Papers, 1–11, 2025.
Levenberg, K. A method for the solution of certain non-linear problems in least squares. Quarterly of Applied Mathematics, 2(2):164–168, 1944.
Li, M., Langbein, F.C., and Martin, R.R. Detecting design intent in approximate CAD models using symmetry. Computer-Aided Design, 42(3):183–201, 2010.
Li, P., Guo, J., Zhang, X., and Yan, D.-M. SECAD-Net: Self-supervised CAD reconstruction by learning sketch-extrude operations. CVPR, 16816–16826, 2023.
Li, Y., Lin, C., Liu, Y., Long, X., Zhang, C., Wang, N., Li, X., Wang, W., and Guo, X. CADDreamer: CAD object generation from single-view images. CVPR, 21448–21457, 2025.
Lipman, Y., Chen, R.T.Q., Ben-Hamu, H., Nickel, M., and Le, M. Flow matching for generative modeling. International Conference on Learning Representations, 2023.
Liu, Y., Obukhov, A., Wegner, J.D., and Schindler, K. Point2CAD: Reverse engineering CAD models from 3D point clouds. CVPR, 3763–3772, 2024.
Marquardt, D.W. An algorithm for least-squares estimation of nonlinear parameters. SIAM Journal on Applied Mathematics, 11(2):431–441, 1963.
Mo, K., Zhu, S., Chang, A.X., Yi, L., Tripathi, S., Guibas, L.J., and Su, H. PartNet: A large-scale benchmark for fine-grained and hierarchical part-level 3D object understanding. CVPR, 909–918, 2019.
Nehme, G., Whalen, E., and Ahmed, F. CADFit: Precise mesh-to-CAD program generation with hybrid optimization. Proceedings of the 43rd International Conference on Machine Learning, PMLR 306, 2026. arXiv:2605.01171.
Oesau, S., Lafarge, F., and Alliez, P. Planar shape detection and regularization in tandem. Computer Graphics Forum, 35(1):203–215, 2016.
Oquab, M., Darcet, T., Moutakanni, T., Vo, H., Szafraniec, M., Khalidov, V., Fernandez, P., Haziza, D., Massa, F., El-Nouby, A., et al. DINOv2: Learning robust visual features without supervision. Transactions on Machine Learning Research, 2024.
Paschalidou, D., Ulusoy, A.O., and Geiger, A. Superquadrics revisited: Learning 3D shape parsing beyond cuboids. CVPR, 10344–10353, 2019.
Pratt, V. Direct least-squares fitting of algebraic surfaces. ACM SIGGRAPH Computer Graphics, 21(4):145–152, 1987.
Ren, D., Zheng, J., Cai, J., Li, J., and Zhang, J. ExtrudeNet: Unsupervised inverse sketch-and-extrude for shape parsing. European Conference on Computer Vision, 482–498, 2022.
Rissanen, J. Modeling by shortest data description. Automatica, 14(5):465–471, 1978.
Rukhovich, D., Dupont, E., Mallis, D., Cherenkova, K., Kacem, A., and Aouada, D. CAD-Recode: Reverse engineering CAD code from point clouds. ICCV, 9801–9811, 2025.
Schnabel, R., Wahl, R., and Klein, R. Efficient RANSAC for point-cloud shape detection. Computer Graphics Forum, 26(2):214–226, 2007.
Schwarz, G. Estimating the dimension of a model. The Annals of Statistics, 6(2):461–464, 1978.
Taubin, G. Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations, with applications to edge and range image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(11):1115–1138, 1991.
Taubin, G. Curve and surface smoothing without shrinkage. IEEE International Conference on Computer Vision, 852–857, 1995.
Tulsiani, S., Su, H., Guibas, L.J., Efros, A.A., and Malik, J. Learning shape abstractions by assembling volumetric primitives. CVPR, 2635–2643, 2017.
Uy, M.A., Chang, Y.-Y., Sung, M., Goel, P., Lambourne, J.G., Birdal, T., and Guibas, L.J. Point2Cyl: Reverse engineering 3D objects from point clouds to extrusion cylinders. CVPR, 11850–11860, 2022.
Willis, K.D., Pu, Y., Luo, J., Chu, H., Du, T., Lambourne, J.G., Solar-Lezama, A., and Matusik, W. Fusion 360 Gallery: A dataset and environment for programmatic CAD construction from human design sequences. ACM Transactions on Graphics, 40(4):1–24, 2021.
Wu, R., Xiao, C., and Zheng, C. DeepCAD: A deep generative network for computer-aided design models. ICCV, 2021.
Xu, X., Willis, K.D., Lambourne, J.G., Cheng, C.-Y., Jayaraman, P.K., and Furukawa, Y. SkexGen: Autoregressive generation of CAD construction sequences with disentangled codebooks. International Conference on Machine Learning, 24698–24724, 2022.
Xu, X., Lambourne, J., Jayaraman, P., Wang, Z., Willis, K., and Furukawa, Y. BrepGen: A B-rep generative diffusion model with structured latent geometry. ACM Transactions on Graphics, 43(4):1–14, 2024.
You, Y., Uy, M.A., Han, J., Thomas, R., Zhang, H., Du, Y., Chen, H., Engelmann, F., You, S., and Guibas, L. Img2CAD: Reverse engineering 3D CAD models from images through VLM-assisted conditional factorization. arXiv:2408.01437v2, 2025.
Yu, N., Alam, M.F., Hart, A.J., and Ahmed, F. GenCAD-3D: CAD program generation using multimodal latent space alignment and synthetic dataset balancing. ASME IDETC/CIE, 2025.
Zou, C., Yumer, E., Yang, J., Ceylan, D., and Hoiem, D. 3D-PRNN: Generating shape primitives with recurrent neural networks. ICCV, 900–909, 2017.