[fontanf]

Problem

The general no_fit_polygon (shapes with holes) is the union of the convex NFPs of the convex parts of the shapes. A position on the boundaries of several convex NFPs, but strictly inside none of them, doesn't make the shapes overlap, even though it is strictly inside their union. A MultiShapeWithHoles can't represent these positions of null area, so they were lost:

Fixed shape, orbiting 1×1 square	Expected	Before
3×3 with a 1×1 hole	feasible point (1, 1)	plain 4×4 square
4×3 with a 2×1 hole	feasible segment (1, 1)–(2, 1)	plain 5×4 rectangle
U shape with a slot of width 1	feasible slit	plain 4×4 square

Changes

• no_fit_polygon(const ShapeWithHoles&, const ShapeWithHoles&) now returns a NoFitPolygonOutput:

  • nfp: the regions, as before;
  • feasible_elements: line segments strictly inside nfp whose points don't make the shapes overlap;
  • feasible_points: isolated points strictly inside nfp which don't make the shapes overlap.

  Callers need to use .nfp to get the previous result.

• A feasible line segment lies on the parts of the boundaries of two convex NFPs which overlap in opposite directions (always line segments, the convex NFPs being convex). A feasible point lies at an intersection of the boundaries of convex NFPs. Only these candidates are checked: not strictly inside any convex NFP, and strictly inside nfp. The last check matters because the boundaries of the convex NFPs may be slightly outside of the computed union.

• Before computing the intersections, the elements of the convex NFPs strictly inside the union of the other convex NFPs are removed, by walking along each element from one convex NFP to the next.

• New IntersectionTree::find_shape(point, strict, condition), which stops at the first shape containing the point, with a condition evaluated before the containment check (used here as a cheap bounding-box filter).

Performance

Simple cases are barely affected. On the sawtooth self-pairing test (Issue558SawtoothSelfPairing, 442 vertices, ~13k convex NFPs), no_fit_polygon goes from about 4.6 s to about 10.5 s, mostly spent computing the ~7M intersections between the remaining elements.

Tests

New NoFitPolygonFeasibleTest suite: square and disk in holes, slots and notches of exactly their size, plus a loose-fit control. Each test checks the expected feasible elements and points, that these positions don't make the shapes overlap, and that the positions strictly inside the NFP around them do. All 714 tests pass.