Fix imp_spline_2d for non-convex polygons via signed-distance construction

[Copilot]

imp_spline_2d evaluated non-convex polygons by multiplying per-edge half-plane fields ∏ H(L_i), where L_i is the signed distance to the infinite line through each edge. At any interior point where an edge's line extension passes through it, L_i ≤ 0 and the product collapses to 0. For non-convex polygons this means the field degenerates to the polygon kernel rather than filling the full shape.

Root fix — imp_spline_2d non-convex path

is_convex(P) selects the code path:

• Convex → original ∏ H(L_i) (unchanged, backward-compatible)
• Non-convex → signed-distance construction:
  1. d_min = min unsigned distance to nearest boundary segment (not line)
  2. sign = +1 inside (ray-casting), −1 outside
  3. B = H(signed_dist, δ, n)

# Before: L-shape interior near reflex vertex incorrectly gives 0
L = np.array([[0,0],[2,0],[2,1],[1,1],[1,2],[0,2]], dtype=float)
imp_spline_2d(0.5, 1.0, L, delta=0.05)  # was: 0.0  now: >0.9
imp_spline_2d(1.5, 1.5, L, delta=0.05)  # exterior notch: still 0.0 ✓

New polygon utilities

• is_convex(P) — cross-product sign test; selects code path in imp_spline_2d
• polygon_validate(P) — removes repeated closing vertex, deduplicates consecutive vertices, rejects degenerate edges and self-intersecting polygons, normalizes to CCW
• triangulate_polygon(P) — ear-clipping triangulation for simple polygons (convex or concave); area of triangles sums exactly to polygon area

All three are exported from __init__.py.

Demo (Section 7)

Replaced the single convex-decomposition panel with a 4-panel figure: (a) direct corrected field via imp_spline_2d, (b) convex decomposition via convex_decomp_field, (c) absolute error map |direct − decomp| with shared decomposition edge annotated, (d) 3D surface. Error is non-zero at shared decomposition edges (piece product gives 0 at its own boundary; direct signed-distance gives the correct interior value) and near polygon corners (different decay profiles). Delta-evolution panel uses the direct path.

Known construction difference

The two approaches are not numerically identical across the full grid. They agree at deep interior points (both → 1) and exterior points (both → 0), but diverge:

• At shared decomposition edges: piece product = 0; direct ≈ 1 (correct)
• Near polygon corners: product of two small H values vs. H of the diagonal distance

This is documented in tests and the demo error panel rather than hidden.

[Copilot]

Pull request overview

This PR fixes imp_spline_2d evaluation for non-convex simple polygons by switching from the convex-only per-edge half-plane product to a signed-distance construction when is_convex(P) is false. It also introduces new polygon utilities (is_convex, polygon_validate, triangulate_polygon), expands tests, and updates the demo to compare direct vs decomposition behavior.

Changes:

• Add convexity detection and a signed-distance path in imp_spline_2d for non-convex polygons.
• Introduce and export polygon utilities: is_convex, polygon_validate, triangulate_polygon.
• Add comprehensive tests for the new utilities + non-convex behavior; update the demo visualization accordingly.

Reviewed changes

Copilot reviewed 4 out of 4 changed files in this pull request and generated 7 comments.

File	Description
python/implicit_spline/core.py	Adds geometry helpers/utilities and modifies imp_spline_2d to support non-convex polygons via signed distance.
python/implicit_spline/__init__.py	Exports the new polygon utility functions.
tests/test_implicit_spline.py	Adds unit tests for is_convex, polygon_validate, triangulate_polygon, and non-convex imp_spline_2d behavior.
examples/demo.py	Updates Section 7 demo to show direct non-convex evaluation vs convex decomposition, including error visualization.

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