VoxelForge/Standalone/Research/WorldAssemblerM3/Research13/REPORT.md

Key recommendations

1. Add one shared curve sweep evaluator for passage bores, the L45 ramp and R12 graph edges. Evaluate a smooth spline over global arc length, rather than taking a minimum over piecewise-linear capsules or discrete ramp stamps.
2. Accelerate the global search with a conservative BVH. Seed each surviving spline span from its chord projection, refine with a safeguarded solver, and compare all candidates whose bounds could beat the current best. One Newton solve from one seed is not a global closest-point solve.
3. Keep the floor cap separate from tube ownership. Use an explicit floor profile and R3’s floor-near candidate rule for overlapping bends and coils; keep relief and smooth unions away from the floor zero set.
4. Use a sphere sweep as the fast, bounded field. A general framed cross-section sweep is not guaranteed to be a true distance field on tight curves. Use rotation-minimising frames for its orientation, but validate curvature and retain conservative bounds for culling.

Math and exactness

Represent the spine as a C1 or preferably C2 cubic spline (C(s)), parameterised by arc length (s). Interpolate radius (r(s)) and floor height (h(s)) as explicit route attributes. The round sweep is:

[ \phi_{\text{tube}}(p)=\min_{s\in[0,L]} \left(|p-C(s)|-r(s)\right) ]

This gives a continuous field with the correct zero set for the union of swept balls. It is exact distance outside that union; inside overlapping spheres it is only a distance bound, as the swept-volume literature cautions. The exact nearest-point equation for a cubic Bezier has degree five; Iñigo Quílez’s compact Bezier distance function is for a quadratic curve, so it is a useful small-case reference, not a complete cubic-tube solution. IQ’s 2D distance functions, point-to-NURBS distance paper, swept-volume SDF analysis

For the floor, define a separate cap from the selected route branch. For a single-valued floor graph (h(x,y)), a useful field is ((h(x,y)-z)/\sqrt{1+\lambda^2}), where (\lambda) bounds the floor slope. Its zero set is the authored floor; it is exact distance for a planar floor and a conservative bound otherwise. Combine the tube and cap as an intersection, then carve using Core’s positive-air convention. Do not smooth-blend the floor edge or apply wall relief at the floor: either can shift the walk surface. The cave-network paper uses this slope-bounded height-field form and notes that arbitrary cross-section sweeps along high-curvature curves do not generally yield a continuous SDF. Paris et al., Synthesizing Geologically Coherent Cave Networks

A circular sweep needs no frame. For an asymmetric cross-section, precompute a rotation-minimising frame along the arc-length table; the double-reflection method is a practical low-cost choice and avoids the Frenet frame’s flips at inflections. Wang et al., Computation of Rotation Minimizing Frames

Cost and culling

Build an arc-length lookup table, spline-span bounds and optional frames once when resolving the layout. At sample time, traverse a BVH ordered by the conservative lower bound distance(point, spanBounds) - maxRadius. The current passage BVH already uses the analogous AABB distance - maximum radius bound; spline bounds must include each cubic span’s control hull, not just its endpoints.

Use the chord projection to seed a bracketed Newton or similar local solve on each promising span, include span endpoints, and retain the lowest result globally. A single seed can converge to the wrong local minimum near a bend or helix. For exact refinement, isolate all roots of the quintic stationary equation on the few surviving spans; variable radius adds a derivative term and makes the solve more costly. The NURBS distance method is a relevant example of clipping candidate regions before numerical refinement.

The exact minimum of 1-Lipschitz sphere fields is itself 1-Lipschitz. An approximate solver’s sampled value is not automatically a safe culling bound: it may overestimate the minimum. Keep a certified lower bound from the BVH/subdivision search for skip decisions, or stop with a known error gap and subtract that error. Report node visits, candidate spans, refinements and bound gaps per sample; a 256-span curve has about eight BVH levels before pruning, but tight overlapping coils may leave many candidates alive. Unreal’s spline API exposes closest-key queries, distance-along-spline queries and a reparameterisation table, matching the useful precomputed lookup pattern for game workloads. Epic’s USplineComponent API

Floor ownership and code fit

For overlapping coils, do not use the closest 3D centerline to choose the floor. R3 recommends considering only floor candidates within half a voxel (0.125 m) vertically and inside the bore radius, then choosing the nearest floor point with a stable ID tie-break. Its synthetic helix showed roughly 3.37 m peak-to-peak error for closest-centerline ownership versus 0.029 m for floor-near ownership; it did not prove the rule fixes the real L38 ripple, so the real FI Style 2 route still needs measurement. R3 floor ownership · R12 graph-field recommendation

Put a shared evaluator in a Core internal sweep helper, with a resolved curve record carrying stable ID, cubic controls, arc-length attributes, floor/radius profiles and conservative bounds.

• Passages: adapt ApplyResolvedStratePassageCarves to call it; retain bridge, mouth and support policy around the shared geometry query. Update BuildPassageNearestNode and its lower bounds for spline hulls.
• L45 ramp: replace the repeated ramp carve/support requests in MakeLandmarkChamberDemoRequests with one floor-capped curve record. Route its evaluation through the shared composition/stamp path in ShapeWorldInternal.h.
• R12 graph edges: store the same curve/floor data on each resolved edge and evaluate it from BindResolvedStratumFields; include blend, relief and floor-support reach in tile and shape-free bounds.

Failure modes and suggested checks

The main risks are missed local minima, spline overshoot, curvature causing a swept cross-section to fold or overlap, variable radius changing the floor, frame flips, and hard floor-owner changes across nearby branches. Validate radius slope and curvature; use monotone floor interpolation where route height must not overshoot. Sphere-swept canal surfaces also become degenerate when radius changes as fast as the spine advances. Canal surface conditions

For a zero-relief tight S-bend with nonuniform knot spacing, sample the floor crossing every 0.125 m of arc length against its analytic (h(s)): the analytic ripple should be 0, with measured error only from the solver/voxel tolerance and no peak at the knot spacing. Then run an overlapping helix with floor-near ownership, and the real L35/L36/L45 controls. Track floor peak-to-peak error, the 0.5–8 m ripple spectrum, culling false negatives, solver gaps and candidate counts. L35 and L36 documented the passage wall/floor beats, while L45 records the 57-carve/107-support ramp ribs. L35 · L36 · L45