Inflation — borders, validity, and the Track result

inflate_warp(center, config, valid=None, count=None) is the final pipeline stage. It composes several sub-stages to produce the complete Track named-tuple from the relaxed centerline.

Composition

The function composes in order:

1. resample_uniform — re-uniformize the relaxed bead chain so that frame and curvature estimates are not biased by residual spacing irregularity.

2. frame_curvature (_frame_k) — central-difference unit tangent, left-normal, and Menger curvature at each bead.

3. Constant half-width offset — offset (_offset_build_k + _offset_assign_k) shifts every centerline point by ±w along the normal. The outer border is chosen as the larger-signed-area candidate, so the convention is consistent even for counter-clockwise layouts.

4. validity (_validity_k) — a single per-env kernel that combines all geometric acceptance criteria (see below).

5. _arclength_k — computes cumulative arc length along the centerline for each valid env.

6. Returns a Track named-tuple.

The offset, validity, and arclength stages are count-aware — they operate over each env’s count[e] real points. The fixed-N parity path is count[e] == N.

Validity gate — _validity_k

_validity_k is a single per-env kernel that combines the following checks:

• Generation flag — all True now that generation no longer gates; validity is purely geometric.

• Closed-loop turning ≈ 2π — the net turning number must be approximately one full winding.

• Width floor — minimum track width must exceed the configured floor.

• No NaN — all points in the env’s real-point range must be finite.

• Thickness ≥ (1 − relax_tol) · half_width — the inflated road must not collapse below the required half-width (within tolerance).

• Border self-intersections — only when validity_border_check is set (default off). This check is redundant with the thickness/separation gate: a crossing or fat-band overlap drives separation_min → 0 → thickness < half_width → invalid anyway.

Index-aligned borders

The inner and outer borders returned in Track are index-aligned with the centerline: outer[e, i], center[e, i], and inner[e, i] are all offset from the same centerline bead i, so downstream consumers can index them identically.

Half-width recovery

The half-width at each bead can be recovered as ‖outer[e, i] − center[e, i]‖ = ‖center[e, i] − inner[e, i]‖ = half_width (constant by construction, modulo floating-point rounding). The thickness validity metric checks the minimum over all beads to ensure no inflation collapsed.

Track fields

Field	Shape	Description
outer	[E, N_max, 2]	Outer border points (NaN-padded past count[e]).
center	[E, N_max, 2]	Re-uniformized centerline points.
inner	[E, N_max, 2]	Inner border points (NaN-padded past count[e]).
tangent	[E, N_max, 2]	Unit tangent at each centerline bead.
normal	[E, N_max, 2]	Left-normal (-tangent.y, tangent.x) at each centerline bead; the boundary it faces is winding-dependent (see winding), so read outer/inner to identify the borders.
arclen	[E, N_max]	Cumulative arc length at each bead (0 at bead 0).
length	[E]	Total track perimeter per env.
valid	[E]	Boolean validity mask (all criteria above must pass).
count	[E]	Number of real beads per env (padding is NaN past this index).
winding	[E]	Signed loop winding: +1.0 CCW, -1.0 CW, 0.0 degenerate (sign of the centerline’s signed area).

generate() always returns the same Track instance with stable wp.array pointers. Callers that need a snapshot should use Track.clone().