Curves and surfaces: Gems and research review

Review date: 2026-10-04. Repository baseline: 52e29ce. The architecture now has a G1 math kernel implementation; the later phases remain proposed. See the kernel evidence.

The initial architecture was written before reading the chapters discovered through game-dev-gems-toc.md. It selected Bezier kernels, immutable preparation, separate domains/caches and consumer adapters. The review then read relevant chapter contents and updated the final contracts. The preserved baseline and final revision table are in the architecture document.

Evidence and selection

The ignored local references/ tree contains the PDFs linked below. Page numbers are one-based physical PDF pages; printed numbering is recorded separately. Bookmarks and the combined index located articles but were not treated as evidence of algorithms.

Game Programming Gems 2, Graphics Gems 1, GPU Pro 1 and GPU Zen 2 provide extractable text. Game Programming Gems 3 and 4 and Graphics Gems 5 are scans: selected chapters were rendered and read through OCR, with chapter identity and key claims visually checked. Graphics Gems 1's copy emits structural extraction warnings; the cited chapter text is readable, and its page/printed numbers were checked from the extracted page headings. No companion-CD implementations were reviewed or copied. These are original paraphrases and Ludus adaptations.

The index was searched across curve, spline, Bezier, NURBS, arc length, frame, patch, subdivision and tessellation entries. The selected readings cover actual numerical/architectural decisions. Water, implicit fields and general terrain are adjacent systems, not justification for one universal curve/surface module. Supporting modern sources were checked online through primary author/project sites.

Resulting changes

Reading	Useful finding	Concrete architecture consequence
Gravesen, length of Bezier curves	Chord/polygon bounds differ from faster error estimates	Whole-path length interval, prefix uncertainty and strict bounded inversion
de Figueiredo, adaptive curve sampling	Deterministic probing can alias; random probing is heuristic	Full control-hull/interior tests for strict profiles; bounded deterministic refinement
Lowe, nonuniform splines	Segment duration rescales derivatives; C2 constraints form a solve	Domain-aware tangents/derivatives, explicit timed C2 mode, cursor lookup
Dougan, transport frame	Frames depend on earlier frames; Frenet/fixed-up have singularities	Bake history, correct random samples to tangent, explicit restarts and loop policy
Brownlow, patch normals	Interpolated control normals are an approximation	Separate shading normal from analytic geometric normal
Schneider, nearest point/root finder	Cubic distance stationarity is generally degree five; endpoints matter	Global closest search with achieved distance gap; Newton only improves candidates
Bunnell, adaptive subdivision tessellation	Shared edge data avoids rounding seams and topology cracks	Canonical edge/corner positions and synchronized parameter splits
Khoury/Dupuy/Riccio, compute tessellation	Compact keys and GPU pipelines have capacity/precision/seam conditions	GPU as gated backend; bounded buffers and explicit adjacency invariants
Hemingway, GPU NURBS	Rational geometry and trimming introduce distinct data/operations	Preserve rational fidelity; treat trim loops as a separate extension

The closed-loop frame correction, specific transactional owners, numerical certificate requirements, bicubic-to-triangle bound, publication protocol and module boundaries are Ludus decisions. None of these articles specifies the complete modern architecture.

Length of Bezier curves

Source: Jens Gravesen, IV.7, “The Length of Bezier Curves,” Graphics Gems 5. Reviewed PDF pp. 209-215, printed pp. 199-205. Chapter/author and the bound were checked on rendered PDF p. 209.

The chapter bounds length between the endpoint chord and control-polygon length. Subdivision tightens the bounds; a weighted combination converges faster as an estimate. It distinguishes the polygon-minus-chord bound from heuristic estimates and gives cusp/special-curve examples where those estimates underestimate error.

Adopt: Keep lower/upper enclosures and allocate a whole-path absolute error budget across leaves. Store uncertainty in prefixes as well as local lengths. Distance inversion must account for both. For cubic spans the weighted estimate reduces to the midpoint of chord and polygon lengths.

Adapt: Float64 and reviewed conservative arithmetic, explicit precision floors, total leaf/work/depth limits and deterministic work order. These are necessary implementation contracts, not a claim that the chapter proves the behavior of Ludus floating-point code.

Do not adopt: A heuristic convergence test as a guaranteed bound, historical subdivision counts as production budgets, or unconstrained recursion. The counterexamples justify strict versus estimated result profiles.

Adaptive sampling of parametric curves

Source: Luiz Henrique de Figueiredo, IV.4, “Adaptive Sampling of Parametric Curves,” Graphics Gems 5. Reviewed PDF pp. 183-188, printed pp. 173-178. Author/title were confirmed in the rendered contents on PDF p. 3.

The chapter adaptively samples a general curve using local flatness probes and reuses samples. It explicitly describes aliasing when a deterministic interior probe misses undulations, and proposes randomized probes near the midpoint as a practical mitigation. Its applications include rendering and approximate arc-length parameterization.

Adopt: Spend refinement work where geometry requires it; reuse samples and preserve ordered source parameters.

Adapt: For the known polynomial core, use control-hull and complete subpatch bounds, deterministic subdivision and explicit budgets. A center-point or edge-only surface test cannot see every interior deviation.

Do not adopt: Random sampling as a certificate, wall-clock-seeded cooking, unbounded recursion or treating a flatness probe as a guaranteed length error. A future black-box evaluator can offer estimated sampling; it cannot inherit the polynomial core's strict guarantees without additional derivative/bound information.

Nonuniform splines

Source: Thomas Lowe, §2.4, “Nonuniform Splines,” Game Programming Gems 4. Reviewed PDF pp. 180-190, printed pp. 171-181. Rendered pp. 180 and 189 checked identity, derivative scaling and endpoint/lookup discussion.

The chapter separates rounded, smooth and timed nonuniform paths. Its normalized Hermite spans scale endpoint velocities by segment width; world acceleration needs a second scaling. Smooth acceleration constraints form a tridiagonal system, with an iterative smoothing alternative. It also discusses cached segment lookup, precomputed coefficients and fixed-step forward differences.

Adopt: Differentiate geometry, timing and continuity; rescale derivatives into their declared domain. Prepare domain ends and use a caller-owned cursor for coherent traversal. Add an optional timed C2 solve with explicit endpoint/ periodic conditions, checked residuals and failure.

Adapt: Chord spacing is approximate constant-speed traversal. Use the bounded arc map when true distance control matters. Prefer a bounded direct solve for the stated continuity conditions over assuming a few smoothing passes suffice.

Do not adopt: A timing-driven tangent damping heuristic while still promising exact C2 continuity, or forward differences for random/variable-step sampling. Fixed-grid forward differences remain a benchmark candidate with drift controls.

Parallel transport frame

Source: Carl Dougan, §2.5, “The Parallel Transport Frame,” Game Programming Gems 2. Reviewed PDF pp. 208-212, printed pp. 215-219.

The chapter incrementally rotates a frame with tangent changes, describes camera and lofting uses, and emphasizes dependence on the previous frame. Its comparison shows Frenet failures on straight spans/inflections and fixed-up failures near alignment with the tangent. It suggests coarse frame samples with quaternion interpolation.

Adopt: Prepare frame history once, retain the seed/policy, and give straight and inflected paths a transported frame.

Adapt: Use a modern RMF implementation after numerical tests. Random samples correct interpolated orientation to the analytic tangent. Frame angular error is distinct from distance error. Closed-loop holonomy requires an explicit preserve/compensate/authored-roll policy; that addition is our reasoning beyond the chapter.

Do not adopt: Treating all zero cross-products as a harmless skip, which confuses parallel and opposite tangents, or inferring bank/physics from transport. Cusps, zero tangents and reversals need visible singularity handling.

Fast patch normals

Source: Martin Brownlow, §4.3, “Fast Patch Normals,” Game Programming Gems 3. Reviewed PDF pp. 338-341, printed pp. 349-352. Rendered pp. 338 and 340 checked chapter identity and the accuracy discussion.

This article's proposed optimization interpolates normals attached to the control mesh using the patch's basis. It explicitly acknowledges that those normals are approximate and can conceal shading discontinuities after skinning. It is not an exact analytic-derivative normal algorithm.

Adopt: Keep an optional normal field as an independently interpolated shading attribute, including its authoring/deformation policy.

Adapt: Normalize/check the shading result and keep it distinct from normalized Cross(Su, Sv), which defines the actual geometric normal on a regular patch. UV coordinates are also distinct from patch parameters.

Do not adopt: This approximate field for projection/contact geometry, surface regularity or proof of continuity. Smoothed lighting cannot weld a surface or repair a geometric seam. The chapter therefore improves the design primarily by clarifying a boundary.

Nearest point on a curve

Source: Philip J. Schneider, XI.7, “Solving the Nearest-Point-on-Curve Problem,” Graphics Gems 1. Reviewed PDF pp. 617-621, printed pp. 607-611. Supporting article VIII.2, “A Bezier Curve-Based Root-Finder,” in the same PDF, reviewed pp. 420-427, printed pp. 408-415. The publisher's contents also identify Schneider and the printed pp. 408-415 for the supporting article.

The nearest-point article derives the stationary dot-product equation, explains that a cubic produces a generally fifth-degree polynomial, and evaluates candidate roots plus endpoints. The root-finder article uses Bernstein form, control-polygon signs and subdivision rather than depending on one initial Newton guess, and notes accumulated floating-point error.

Adopt: Global candidate coverage, explicit endpoints and bounded search. V1 uses control-hull distance branch-and-bound with a global achieved gap; safeguarded Newton only improves the candidate upper bound.

Adapt: The stationary equation extends to 3D, but clipping, repeated/endpoint roots, coincident stationary functions and numerical certification need new tests. A nonzero degree-five polynomial has at most five distinct real roots; do not size a general isolator around the nearest-point text's narrower sample root-count wording.

Defer: A shared Bernstein root-isolation subsystem until another query needs it. Recursion depth or a small parameter step is not by itself a strict world-distance guarantee.

Adaptive subdivision tessellation and displacement

Source: Michael Bunnell, Chapter 7, “Adaptive Tessellation of Subdivision Surfaces with Displacement Mapping,” GPU Gems 2. The index locates the chapter; the publisher-hosted author chapter was read, particularly §§7.1.4-7.1.6 and §7.2.1.

The chapter preprocesses patch organization, evaluates limit geometry, chooses one patch's data for shared edges/corners to avoid floating-point disagreement, and handles T-junctions. Displacement changes its refinement test.

Adopt: Canonical shared position samples, matching boundary subdivision, topology-aware tessellation and displacement-aware error bounds.

Adapt: V1 uses synchronized dyadic rectangular patch grids; a later local adaptive backend must preserve the same seam invariants. Amplitude and derivative variation affect different error requirements.

Do not adopt: Its pixel-shader/render-to-texture architecture, CPU readback loop, dated performance comparison or an edge-only test as a general bicubic interior guarantee. GPU/CPU choice is a Ludus measurement decision.

Compute tessellation

Source: Jad Khoury, Jonathan Dupuy and Christophe Riccio, Chapter 1, “Adaptive GPU Tessellation with Compute Shaders,” GPU Zen 2. Reviewed relevant text in PDF pp. 15-28, printed pp. 3-16: implicit key representation, pipeline overview, T-junction discussion, results and limitations.

The chapter encodes binary triangle subdivision paths in compact keys, updates double-buffered GPU arrays, and uses compute/indirect commands for rendering. It discusses key-width precision, buffer capacities and a T-junction avoidance rule observed under a stated screen-density configuration.

Adopt conditionally: Compact private refinement descriptors and reuse of static topology are candidates for a later GPU backend.

Require before integration: Explicit output/append capacity, overflow behavior, synchronization and capability tests. A constant-sized key per node does not make the total number of nodes or output triangles constant.

Do not adopt: The empirical distance/edge-size seam condition as proof for arbitrary views, deformations or displacements. Shared boundaries remain explicit. The chapter's results are for its hardware, mesh and shader workload, not Ludus.

GPU NURBS and trimming

Source: Graham Hemingway, geometry manipulation Chapter 3, “GPU-Based NURBS Geometry Evaluation and Rendering,” GPU Pro 1. Reviewed background and implementation/trim excerpts from PDF pp. 82-100, printed pp. 67-85, especially §3.1, §3.3's storage/evaluation description and §3.4.1-3.4.2. This is a selective reading, not a validation of its full code.

The chapter defines weighted rational curve/surface evaluation and separates the underlying surface from a trimming profile. Trim preparation includes inverse projection into surface parameters and a rendering mask.

Adopt: Rational source metadata cannot disappear during an allegedly exact conversion. Trimming is an independent domain/topology problem with boundary/ hole semantics. Position evaluation alone does not complete trimmed geometry.

Adapt: V1 can bake explicit approximations; exact retained NURBS needs a separate homogeneous representation, knot/weight/denominator validation and fidelity tests.

Do not adopt: Texture masking as robust collision/trim topology, unrestricted Newton inverse projection, old GPU storage limits, or an unqualified precision/ speed claim. Define knot counts and endpoint multiplicities under one tested index convention rather than translating notation without checking it.

Current primary sources and SOTA assessment

The primary-source checks below establish the availability/properties of individual techniques. Their selection and integration are our judgment. No source makes one architecture universally optimal.

Source checked on 2026-10-04	Design implication and limit
Yuksel, Schaefer and Keyser: Catmull-Rom parameterization	Centripetal regular segments avoid cusps/self-intersections within the analyzed family; not a global nonintersection guarantee
Wang, Jüttler, Zheng and Liu: RMF paper record/abstract	Double reflection is a strong prepared-frame candidate; the published convergence statement does not validate a singular Ludus path
Yuksel: A Class of C2 Interpolating Curves, 2020	Local C2 non-polynomial curves are a meaningful extension; exact cubic conversion is unavailable in general
OpenSubdiv Far overview	Separate topology refinement, value refinement and stencil/patch evaluation; reuse topology without assuming numerical error stays valid
OpenSubdiv surface/data semantics	Scheme, boundary, crease and face-varying data require dedicated metadata/contracts
SciPy PCHIP documentation	Shape-preserving scalar interpolation is C1 with possible second-derivative jumps; optional cooker technique without a runtime Python dependency

The RMF paper's full text was not available from the institutional record. That source supports selection for investigation, not implementation equations. The Yuksel sources were reviewed through author summaries/formulations; this task does not claim a new proof or exhaustive comparison of every curve family.

“SOTA” here means preserving modern extension paths and rigorous ownership/ numerical contracts while choosing a small kernel that can be tested and optimized. A newer formulation with desirable authoring properties is not automatically the lowest-cost runtime format. The C2 family and established OpenSubdiv integrations show why extensions deserve explicit fidelity and dependency gates rather than forced conversion.

Index-only leads and rejected scope expansion

The index also lists “Subdivision Surfaces for Character Animation” in Game Programming Gems 3, “High-Performance Subdivision Surfaces” in Game Programming Gems 7, “Trigonometric Splines” in the same volume, GPU Pro Phong tessellation, and Eberly's curve/surface chapters. These remain reading leads; this review does not attribute implementation recommendations to their unread contents.

The scanned “Quick and Simple Bezier Curve Drawing” title/introduction was encountered while locating the length chapter. Its general-degree factorial/power approach was not reviewed as an implementation candidate. The polynomial kernel already has a fixed degree and a bounded reference path.

Water and terrain sources do not warrant adding simulation, arbitrary field meshing, trimming, robust boolean geometry and subdivision to v1. They are different product requirements. No third-party implementation or copyrighted code is added by this design task.

Evidence still needed

Implement and independently test conservative arithmetic before labeling results certified. Compare Horner/de Casteljau, cache footprints, direct/BVH projection, whole-candidate/incremental edits and CPU/GPU preparation on actual native/ browser fixtures. Record source/derived memory and active-plus-candidate peaks. Historical chapter timings and subdivision counts are not acceptance numbers.

The architecture's implementation phases require warning-clean builds, unit/ sanitizer tests, pinned format/tidy, header/build budgets, SDK and Wasm validation. This review supplies an implementation handoff and evidence trail, not proof that those future implementations already pass.

Conference and journal research backlog

Recorded 2026-10-06. This section preserves venues and an initial reading queue for improvements after the initial curves/surfaces implementation. The architecture owns representation, ownership, numerical contracts and G2-G6 gates; the G1 evidence ledger records only the fixed-size kernel's completed validation.

Implement and validate G2-G5 first, then compare promising ideas against that recorded baseline. Source readings needed to implement already selected algorithms and strict numerical guarantees still belong to the initial work. The broader research pass and alternative backends follow the baseline; optional G6 capabilities and conditional extensions keep their existing acceptance gates.

Venue scopes, source metadata, abstracts and selected introductory passages were screened when recommending these resources. The three full papers below remain queued for detailed review. The questions and experiments are Ludus proposals, not evidence of a measured improvement or an adopted implementation. Preserve the dated Gems review above when adding later findings.

Venues to search

Venue	Reading priority after baseline	Topics to screen for	Ludus work it could inform
Computer Aided Geometric Design	First	Bezier/B-spline algorithms, approximation, continuity, subdivision, projection and geometric bounds	G2 distance/source preparation; G3 queries; G4 patches
Reliable Computing	First	Interval arithmetic, guaranteed accuracy, error-free transformations and floating-point error analysis	Conservative arithmetic and precision floors across G2-G4
SIGGRAPH technical papers / ACM Transactions on Graphics	First	Curve authoring, transported frames, sweeps, geometric modeling and interactive techniques	G3 frames/sweeps; G5 editing; alternative curve families
Symposium on Geometry Processing / Computer Graphics Forum	Next	Meshing/remeshing, differential geometry, topology, parameterization and deformation	G4 patch networks and mesh quality; G5 editing
SIAM Conference on Computational Geometric Design	Next	Spline mathematics, approximation and geometric algorithms	G2-G4 numerical and representation alternatives
Solid and Physical Modeling / Computer-Aided Design	Selective	Tolerancing, geometric validity, constraint solving, robust computation and CAD representations	G4 adjacency; conditional NURBS, trimming and intersection work
High-Performance Graphics	When G6 has measured demand	GPU evaluation/tessellation, graphics architecture, bounded output and end-to-end performance	G6 renderer-private acceleration after RHI prerequisites

SGP publishes its proceedings in Computer Graphics Forum; SPM full papers are published in Computer-Aided Design. The linked 2026 conference pages are archive and scope entry points, not a future attendance schedule. Priorities and the mapping to Ludus are our selection criteria. Screen relevant sessions and bibliographies rather than treating every article at a venue as applicable.

Initial reading queue

ID	Source and attribution	Full-review status	Question and possible experiment after review
CSR-01	Jens Gravesen, Adaptive subdivision and the length and energy of Bézier curves, Computational Geometry 8(1), pp. 13-31, 1997. DOI; publisher abstract	Queued; metadata/abstract screened. The related Graphics Gems 5 chapter was reviewed above.	Which refinement/work allocation gives the best G2 cost at the same whole-path length and inversion residual budgets? Compare deterministic split priorities on lines, cusps, uneven spans and nearly stationary paths; retain prefix uncertainty and strict failure semantics.
CSR-02	Wenping Wang, Bert Jüttler, Dayue Zheng and Yang Liu, Computation of rotation minimizing frames, ACM Transactions on Graphics 27(1), article 2, 2008. DOI; coauthor-hosted manuscript	Queued; abstract/introduction screened; full manuscript located.	Review the double-reflection equations, regularity assumptions and error analysis before use. After G3, compare frame angular error and work against the validated baseline, including inflections, near-zero tangents, random-access sampling and closed-loop residual twist.
CSR-03	H. Jiang, H. S. Li, L. Z. Cheng, R. Barrio, C. B. Hu and X. K. Liao, Accurate, Validated and Fast Evaluation of Bézier Tensor Product Surfaces, Reliable Computing 18, pp. 55-72, 2013. Journal paper	Queued; abstract, introduction and arithmetic assumptions screened	Can compensated de Casteljau evaluation improve difficult patch samples at acceptable cost? Compare the current float64-intermediate reference, compensated evaluation and an independent high-precision oracle. Test cancellation, extreme/subnormal values, derivatives and output narrowing; establish actual enclosures separately from improved point accuracy.

CSR-02's manuscript is newly located research context. The earlier statement about full text being unavailable describes the institutional record checked on 2026-10-04; it does not assert that no author copy exists. Finding a manuscript does not constitute a full review or validate a singular Ludus path.

CSR-03 assumes double-precision round-to-nearest arithmetic with no underflow. Any trial must check that assumption against native/Wasm subnormal behavior, the exact operations used and final float32 rounding. A compensated point estimate alone does not certify subdivision, norms, length accumulation or transformed bounds.

Review and trial records

After baseline validation, screen these archives for concrete G2-G6 problems and extend this queue with stable IDs, exact author/title/year, DOI or primary URL, access locator, priority and status. Record the date and scope of each search. Read original sources before crediting an implementation to them.

Track Queued, Reading, Reviewed, Trial planned, Trial complete, Adopted, Deferred or Rejected. A completed record includes:

1. Source ID, review date and exact sections/pages consulted; the relevant idea, assumptions, limitations and departures from Ludus's current design.
2. A testable improvement hypothesis, affected phase/module and bounded trial.
3. The baseline revision, reproduction commands, fixtures, seed, toolchain and backend; links to the test, benchmark or PR. Generated results stay in out/.
4. Before/after achieved errors and failure behavior, build/query/sample time distributions, allocations, resident/cache bytes and active-plus-candidate peak memory. Mesh trials also record output counts and shared-edge agreement.
5. An adopt/defer/reject decision and reason, with links to changed contracts, code attribution and independent validation where adoption is justified.

Use the architecture's existing workload and numerical gates. Compare on native and Wasm where supported, and include difficult/invalid inputs rather than only a favorable throughput case. Preserve transactionality, bounded work, explicit approximation and the no-exceptions/dependency boundaries. A performance gain cannot justify weakening a strict guarantee or silently widening tolerances.

No source in this new queue has a completed trial or adoption record yet.