Sculpting and CAD Surfacing Were Never Supposed to Talk. NeuroCAD Measures Cashman's 2009 Bridge.
Cashman wrote the bridge between subdivision surfaces and NURBS in 2009. Sixteen years later, NeuroCAD is the first CAD platform to wire it as a release-gating regression test.
Cashman wrote the bridge between subdivision surfaces and NURBS in 2009. Sixteen years later, NeuroCAD is the first platform to measure how well it actually works.
Two industries that share a screen and nothing else
A character artist working in ZBrush thinks in clay. They push, pull, smooth, pinch. They do not care which spline interpolates which row of control points; they care whether the shoulder reads as a shoulder. The mesh under their stylus is a subdivision surface — Catmull–Clark, Loop, or one of the OpenSubdiv variants — and the appeal is precisely that the math gets out of the way. The artist never solves a knot insertion problem.
A surfacing engineer in Alias or ICEM lives in the opposite world. Every NURBS patch they lay down has a degree, a knot vector, a control polygon, a continuity contract with its neighbors. They will spend an afternoon arguing whether two patches meet at G1 or G2. They will measure their work with zebra stripes and curvature combs. The clay is gone; the geometry is a contract.
The two worlds share a screen, sometimes — film and game studios stitch them together with re-topology tools, automotive studios push class-A surfaces back into VFX pipelines, mechanical CAD has flirted with subdivision modeling since SolidWorks added “3D sketch” decades ago — but the math has stayed apart. Subdivision surfaces and NURBS are different objects. They evaluate differently. They refine differently. They store differently. Round-tripping between them has been a craft skill, not a guaranteed operation.
Then, in 2009, Thomas Cashman published a Cambridge technical report titled NURBS-compatible subdivision surfaces (UCAM-CL-TR-773, Cashman 2009) that did something quietly revolutionary: he proposed a subdivision rule which, applied to a NURBS control net, produces a limit surface that exactly equals the NURBS surface — except in a small neighbourhood of extraordinary vertices, where the rule degrades gracefully into a generalisation of Catmull–Clark.
Cashman wrote the bridge.
For sixteen years it has sat in the literature, cited politely, implemented in research code, occasionally referenced by OpenSubdiv documentation. It has not, to our knowledge, been published as a measured benchmark inside a production CAD platform. The “NeuroCAD measures it” claim of this post is straightforward: we are wiring Cashman’s algorithm as a workbench-to-workbench bridge between Sculpt (subdivision) and Surface (NURBS Kernel B), and we are publishing the round-trip Hausdorff distance as a contract that every release must pass.
Why the schism existed in the first place
To see why bridging is hard, look at what the two representations actually compute.
A NURBS surface evaluates as
S(u, v) = Σᵢ Σⱼ ( wᵢⱼ Pᵢⱼ Bᵢ(u) Bⱼ(v) ) / Σᵢ Σⱼ ( wᵢⱼ Bᵢ(u) Bⱼ(v) )— a closed-form rational tensor product over a fixed control net with knot vectors that are part of the surface’s identity. Inserting a control point changes the knots; refining the surface changes the patch.
A subdivision surface evaluates as the limit of an iterated refinement scheme. You start with a coarse polygonal control mesh and a refinement rule (Catmull–Clark for quads, Loop for triangles); each refinement step adds vertices and re-positions existing ones; in the limit you get a smooth surface. Stam’s 1998 paper showed you can evaluate Catmull–Clark in closed form, but the representation is fundamentally different from a NURBS patch.
The two surfaces look similar to the eye. A regular Catmull–Clark net produces a surface that is bicubic B-spline almost everywhere — exactly the same family as cubic NURBS. But they are not the same data structure, and naive conversion in either direction loses information:
- NURBS → SubD: a typical CAD conversion samples the NURBS surface and re-fits a control mesh, introducing tessellation error.
- SubD → NURBS: you can extract bicubic B-spline patches from regular regions, but the extraordinary vertices (where the valence isn’t 4) have no NURBS equivalent and require either trim curves or local approximation.
Cashman’s contribution was to propose a single subdivision scheme — non-uniform, with knot intervals carried alongside the control net — such that the limit surface is the NURBS surface in regular regions and gracefully extends NURBS into extraordinary regions where pure NURBS cannot represent the topology.
That is why his thesis is the bridge: the conversion stops being lossy in regular regions, and the irregular regions become explicit, addressable, and (importantly) testable.
What “NeuroCAD measures it” means in practice
NeuroCAD is built around an SDF-native kernel — every feature, every body, every surface ultimately resolves to a signed distance field. NURBS Kernel B is the parametric layer: it lifts NURBS patches to SDFs via a variational reconstruction. The Sculpt workbench, when complete, will run subdivision in the same kernel, expressed as another lift into SDF. The bridge is a third path: a NURBS surface produced in Surface can be loaded into Sculpt as a Cashman-compatible subdivision control net; a subdivision surface produced in Sculpt can be exported back into Surface as a (possibly trimmed) NURBS surface.
Without a contract, “round-trip” is a marketing word. With a contract, it is a regression test. Specifically:
- Take a NURBS surface
Sfrom Kernel B. - Build a Cashman-compatible subdivision control net
CfromS. - Evaluate the limit subdivision surface
S′fromC. - Measure the Hausdorff distance
H(S, S′)over a high-density sample on the parametric domain. - Demand
H(S, S′) ≤ εfor a publishedε(we target 10⁻⁵ in object-space units for surfaces of unit characteristic length, away from extraordinary vertices; the value relaxes by a documented factor near valence-3, valence-5, valence-6 vertices).
The contract becomes a published benchmark suite. Surface modelers can ship a NURBS patch through Sculpt and prove it survived. Character artists can hand off a sculpted form and prove the surfacer received an exact bicubic B-spline in the regular regions and a measured-quality approximation around extraordinary vertices.
That is the whole pitch: turn a sixteen-year-old academic bridge into a release-gating regression test inside a CAD platform.
What this unlocks for the people who use the tool
Three concrete workflows that are awkward in the current CAD/VFX hybrid landscape and become natural under the Cashman bridge:
1. Class-A surfacing on top of sculpt. A clay model lands in Sculpt. The surfacer takes the Cashman control net, identifies the regular regions, exports them to Surface as bicubic NURBS patches, and finishes them with the usual G2/G3 continuity tools. The artist’s intent survives the math.
2. Mechanical hard-points injected into organic forms. A part has a sculpted handle and a precise mounting flange. The flange is authored in Surface as NURBS, the handle in Sculpt as subdivision. The bridge makes the hand-off lossless in the regular interior of each region; the boundary is a measured-tolerance stitch.
3. Re-topology with a published guarantee. Sculpt’s Dyntopo strokes can be projected back onto a Cashman control net at a chosen subdivision level; the round-trip distance is logged. “Re-topology” stops being a black-box operation and becomes a quantified one.
None of these are speculative. Each is a workflow that exists in industry — Pixar’s OpenSubdiv runs in production at ILM, MPC, Weta; Autodesk Alias couples to Maya; Catia integrates ICEM Surf. What is missing is a published, measured, benchmark-driven round-trip — a contract you can cite when a model fails.
Why we are publishing the test, not just the result
The bridge is an algorithm. The benchmark is a public contract. NeuroCAD ships the second.
Our forthcoming preprint lays out the methodology. The test suite — cashman_bridge_roundtrip and friends, in REV11.5 — runs in CI on every push. The benchmark corpus is a curated set of standard surfaces (Stanford bunny patch, Utah teapot, automotive class-A samples we are licensing) plus synthetic extraordinary-vertex stress cases (valence 3, 5, 6, 7, 8). The Hausdorff distances are reported in the release notes.
If a future change regresses the round-trip, CI catches it before it ships. If a competing implementation publishes their own numbers, they can be compared on the same corpus. If an academic wants to extend Cashman’s scheme to higher degrees or non-tensor-product domains, they have a baseline to beat.
This is the same strategy we have applied to every other publishable contract in NeuroCAD: variable-topology never-fails, feature-tree DCG re-eval latency, replay reproducibility. The pattern is “take a thing the literature already knows, write the contract, ship it as CI, publish the numbers.”
Where this fits in the bigger picture
Hybrid CAD/VFX pipelines are a $multi-billion industry segment that has lived without an interoperable surface representation since the 1990s. Subdivision won the VFX market on the strength of OpenSubdiv. NURBS won the CAD market on the strength of class-A surfacing tools. The two have coexisted by means of expensive re-topology services, import/export filters with documented losses, and the patience of artists.
Cashman’s 2009 paper is the closest thing we have to a peace treaty between the two camps. We are not the first to implement it — the algorithm has appeared in research code and in the OpenSubdiv ecosystem — but, to the best of our literature search, we are the first to turn it into a measured, published round-trip benchmark inside a production CAD platform. The benchmark suite is public, the corpus is public, and the bridge is documented in detail — even though the NeuroCAD platform itself is commercial — so the result is independently reviewable.
The bridge is finally the kind of thing two engineers from different industries can argue about over a single shared number.
What ships, and when
NURBS Kernel B exists today in the kernel; it is the production NURBS layer powering the Surface workbench. The Sculpt workbench is currently a placeholder — the Catmull–Clark refinement primitives are implemented and tested, but the workbench UI and the Cashman bridge itself are queued in REV11.5. Until the bridge lands, this post is advance documentation: the contract we have committed to ship.
When the bridge ships, the round-trip benchmark numbers replace the “we will measure” language in the preprint. Until then we publish the methodology and the corpus, so that the moment the implementation is green, the world can read the result on the same page where we promised it.
Status
| Component | State today |
|---|---|
| NURBS Kernel B | Implemented. Production NURBS evaluation, knot insertion, degree elevation, SDF lift. |
| Catmull–Clark subdivision primitives | Implemented as math primitives; not yet exposed as a workbench. |
| Sculpt workbench (UI, brushes, dyntopo) | Placeholder. Spec exists, code is stub. |
| Cashman NURBS-compatible subdivision bridge | Proposed. Algorithm specified; implementation queued in REV11.5. |
| Round-trip Hausdorff benchmark | Proposed. Test contract drafted; CI gate to be added with bridge implementation. |
No round-trip Hausdorff numbers are quoted in this post that are not labeled as targets. When the bridge lands and CI runs the corpus, the numbers will appear in the release notes and replace target language in the whitepaper and preprint.