docs: add polylabel part label positioning design spec

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

docs/superpowers/specs/2026-03-16-polylabel-part-labels-design.md

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+# Polylabel Part Label Positioning
+
+**Date:** 2026-03-16
+**Status:** Approved
+
+## Problem
+
+Part ID labels in `PlateView` are drawn at `PathPoints[0]` — the first point of the graphics path, which sits on the part contour edge. This causes labels to overlap adjacent parts and be unreadable, especially in dense nests.
+
+## Solution
+
+Implement the polylabel algorithm (pole of inaccessibility) to find the point inside each part's polygon with maximum distance from all edges. Draw the part ID label centered on that point.
+
+## Design
+
+### Part 1: Polylabel Algorithm
+
+Add `PolyLabel` static class in `OpenNest.Geometry` namespace (file: `OpenNest.Core/Geometry/PolyLabel.cs`).
+
+**Public API:**
+
+```csharp
+public static class PolyLabel
+{
+    public static Vector Find(Polygon polygon, double precision = 1.0);
+}
+```
+
+**Algorithm:**
+
+1. Compute bounding box of the polygon.
+2. Divide into a grid of cells (cell size = shorter bbox dimension).
+3. For each cell, compute signed distance from cell center to nearest polygon edge (negative if outside polygon).
+4. Track the best interior point found so far.
+5. Use a priority queue (sorted list) ordered by maximum possible distance for each cell.
+6. Subdivide promising cells that could beat the current best; discard the rest.
+7. Stop when the best cell's potential improvement over the current best is less than the precision tolerance.
+
+**Dependencies within codebase:**
+
+- `Polygon.Contains(Vector)` or ray-casting point-in-polygon test (already exists via `Intersect`).
+- Point-to-segment distance calculation (already exists via `Line`/`Intersect`).
+
+**No external dependencies.**
+
+### Part 2: Label Rendering in LayoutPart
+
+Modify `LayoutPart` in `OpenNest/LayoutPart.cs`.
+
+**Changes:**
+
+1. Add a cached `PointF? _labelPoint` field, invalidated when `IsDirty` is set.
+2. In `Draw(Graphics g, string id)`:
+   - If `_labelPoint` is null, compute it:
+     - Convert the part's `Program` to geometry via `ConvertProgram.ToGeometry`.
+     - Build shapes via `ShapeBuilder.GetShapes`.
+     - Convert the outer shape to a `Polygon`.
+     - Run `PolyLabel.Find()` on the polygon.
+     - Offset by `BasePart.Location`.
+     - Transform through the view matrix.
+     - Cache the resulting `PointF`.
+   - Draw the ID string centered on `_labelPoint` using `StringFormat` with `Alignment = Center` and `LineAlignment = Center`.
+3. Invalidate `_labelPoint` when `IsDirty` is set (already triggers recompute on next draw).
+
+**Coordinate pipeline:** polylabel runs in program-local coordinates (pre-transform), result is offset by `Location`, then transformed by the view matrix — same pipeline the existing `Path` uses.
+
+## Scope
+
+- **In scope:** polylabel algorithm, label positioning change in `LayoutPart.Draw`.
+- **Out of scope:** changing part origins, modifying the nesting engine, any changes to `Part`, `Drawing`, or `Program` classes.
+
+## Testing
+
+- Unit tests for `PolyLabel.Find()` with known polygons: square, L-shape, C-shape, triangle.
+- Verify the returned point is inside the polygon and has the expected distance from edges.