Implement additive boundary-based polygon splines for non-convex shapes and conforming partitions

[Copilot]

The current Section 7 and Section 9 implementations diverged from the Li–Tian construction: non-convex polygons were evaluated via signed distance / convex-piece composition, and partition cells were normalized post hoc instead of using the additive boundary formulation. This change restores the intended arbitrary-polygon spline behavior so direct evaluation matches convex decomposition and internal decomposition edges do not introduce seams.

• Core spline construction

  • Replace the public arbitrary-polygon path in imp_spline_2d with a boundary-based additive evaluator assembled from oriented edge contributions.
  • Add decomposition evaluation that cancels artificial internal edges exactly before evaluation, so
    B(whole polygon) == Σ B(convex cells) up to floating-point error.
  • Keep the convex product form available as a low-level legacy helper (convex_product_spline_2d) instead of treating it as the general polygon algorithm.

• Geometry correctness

  • Tighten polygon normalization and validation:
    • repeated closing vertex handled safely
    • np.allclose(..., rtol=0, atol=...) for geometric tolerance checks
    • separate duplicate-removal and degenerate-edge tolerances
    • detect proper crossings, collinear overlaps, and invalid touching
  • Make triangulate_polygon explicitly require boundary-ordered vertices and fail if ear clipping cannot complete or does not return exactly m - 2 triangles.

• Section 7 demo/content

  • Replace the L-shape/signed-distance presentation with two paper-style non-convex control polygons in CCW boundary order.
  • Render dashed control polygons, vertex markers, and the spline iso-contour in the intended paper style.
  • Evaluate each polygon both directly and through convex decomposition using the same additive construction.

• Section 9 partition construction

  • Replace the strip/trapezoid-style partition with a genuine conforming irregular convex-cell partition.
  • Add reusable partition validation for:
    • CCW / positive area
    • shared-edge incidence and endpoint matching
    • no overlaps / gaps / invalid T-junctions
    • outer-domain area coverage
  • Use additive partition fields (partition_basis_fields) so the partition identity is expressed as the sum of polygon bases, not by normalizing arbitrary positive fields.

• Regression coverage

  • Add focused tests for:
    • Section 7 point inclusion/exclusion on both paper-style polygons
    • direct vs decomposition equality
    • equality across multiple decompositions
    • absence of internal-edge seams
    • repeated closing vertex handling
    • partition topology and additive identity
    • PR Fix imp_spline_2d for non-convex polygons via signed-distance construction #9 review defects around tolerances, triangulation guarantees, and plotting safety

Example of the corrected decomposition path:

from implicit_spline.core import imp_spline_2d, triangulate_polygon, convex_decomp_field

P = ...  # CCW non-convex polygon in boundary order
tris = triangulate_polygon(P)

direct = imp_spline_2d(X, Y, P, delta=0.22, n=2)
decomp = convex_decomp_field(X, Y, tris, delta=0.22, n=2)

# direct and decomposition evaluation now match because internal edges cancel

[Copilot]

Pull request overview

Restores the Li–Tian intended behavior for non-convex polygons and conforming partitions by switching the public polygon evaluator to an additive, boundary-integral-based construction where internal edges cancel under decomposition, eliminating seams and making direct evaluation match decomposition.

Changes:

• Replaces imp_spline_2d’s arbitrary-polygon path with an oriented-edge additive evaluator; keeps the convex product form as convex_product_spline_2d.
• Adds edge cancellation for decomposition evaluation and introduces partition additive basis fields + partition topology validation.
• Updates demos/tests to use paper-style Section 7 polygons and a conforming Section 9 partition, with regression coverage.

Reviewed changes

Copilot reviewed 7 out of 7 changed files in this pull request and generated 2 comments.

Show a summary per file

File	Description
tests/test_implicit_spline.py	Replaces prior signed-distance/normalized-sum tests with additive construction regressions for Section 7/9.
python/implicit_spline/core.py	Implements additive boundary-based imp_spline_2d, edge cancellation for decompositions, additive partition fields, and partition validation utilities.
python/implicit_spline/visualization.py	Updates partition surface visualization to plot additive sums (partition_basis_fields).
python/implicit_spline/paper_examples.py	Adds shared reference polygons/partitions used by demos and tests.
python/implicit_spline/init.py	Exposes new public APIs (cancel_internal_edges, convex_product_spline_2d, partition_basis_fields, validate_partition).
examples/README.md	Updates documentation to describe the corrected additive demo outputs and topology requirements.
examples/demo.py	Rewrites demo around additive construction, paper-style Section 7 examples, and conforming Section 9 partition identity.

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