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Stratification Theory

How Whitney stratification applies to CAD Boolean operations, and why gradient discontinuities at topological boundaries are a geometric fact rather than a numerical artifact.

Configuration Space of Two Primitives

Consider two primitives A and B parameterized by design variables (radii, centers, axes). As parameters vary, the topological relationship between A and B changes:

Stratum Label Relationship Example (two spheres)
0 Disjoint No overlap d > r1 + r2
1 Intersecting Partial overlap |r1 - r2| < d < r1 + r2
2 A contained in B A entirely inside B d + r1 <= r2
3 B contained in A B entirely inside A d + r2 <= r1

These four regions partition the configuration space into a Whitney stratification: a decomposition into smooth manifolds (strata) satisfying Whitney's regularity conditions (a) and (b) at their boundaries.

Gradient Structure Per Stratum

Each stratum has a distinct analytical gradient formula for Boolean volume. For union:

Stratum V_union formula d(V_union)/d(p)
0 (disjoint) V_a + V_b d(V_a)/d(p) + d(V_b)/d(p)
1 (intersecting) V_a + V_b - V_intersection Requires integration over intersection boundary
2 (A in B) V_b d(V_b)/d(p)
3 (B in A) V_a d(V_a)/d(p)

In strata 0, 2, and 3, the gradient is exact and analytical because the volume is a simple function of individual primitive volumes. In stratum 1, the intersection boundary curve introduces a non-trivial integration that BRepAX approximates via the straight-through estimator (STE).

Gradient Discontinuity at Boundaries

At stratum boundaries (tangent configurations), the gradient is generically discontinuous. This is not a bug but a consequence of the stratification structure:

External tangency (d = r1 + r2): the gradient of union area/volume is actually continuous here because the intersection formula reduces to the disjoint formula as overlap vanishes. The gradient jump is second-order zero, as derived in the TOI derivation.

Internal tangency (d = |r1 - r2|): the gradient is genuinely discontinuous. The contained-stratum gradient depends only on the outer primitive, while the intersecting-stratum gradient depends on both. This is a first-order jump that cannot be smoothed away.

BRepAX Implementation

BRepAX implements stratum-dispatched gradients via jax.custom_vjp:

  1. Forward pass: exact SDF Boolean (min/max) + Heaviside indicator on a grid. The stratum label is detected via grid-based SDF evaluation and saved as a residual.

  2. Backward pass: jnp.where dispatch selects the gradient formula based on the saved label. This is branchless and compatible with jax.vmap.

The boundary convention (ADR-0004) uses the intersecting-stratum gradient at exact tangency, following the principle that the generic (highest-dimensional) stratum provides the most informative gradient signal.

Convergence Properties

Empirical measurement on sphere-sphere and cylinder-sphere pairs shows:

  • Strata 0, 2, 3: exact analytical gradient (zero error)
  • Stratum 1 interior: STE gradient error < 1% (sphere) and < 10% (cylinder) at resolution 128
  • Stratum 1 near-tangent: error increases to 25-542% as the sigmoid kernel width exceeds the intersection feature size

The convergence rate is between O(h) and O(h^2) in the interior, consistent with grid-based integration of a sigmoid-smoothed discontinuous integrand. Near tangency, the rate degrades because the effective feature size shrinks faster than the grid refines.

Despite near-tangent error, optimization converges to correct optima because the gradient sign is always correct and the bias direction (volume overestimation) is consistent.

Comparison

No existing differentiable CSG system provides exact gradients in any stratum. BRepAX's 3/4 exact + 1/4 characterized is a unique positioning:

System Exact strata Gradient method
BRepAX 3/4 Analytical dispatch + STE
DiffCSG 0/4 Goldfeather rasterization + edge AA
Fuzzy Boolean 0/4 t-norm/t-conorm smoothing
TreeTOp 0/4 Continuous relaxation

References

  • Whitney, H. (1965). Tangents to an analytic variety.
  • Goresky, M. & MacPherson, R. (1988). Stratified Morse Theory.
  • Guest, J.K. et al. (2004). Achieving minimum length scale in topology optimization using nodal design variables and projection functions.
  • Bengio, Y. et al. (2013). Estimating or propagating gradients through stochastic neurons for conditional computation.