fix(geometry): replace closest-point heuristic with analytical arc-to-line directional distance

ArcToLineClosestDistance used geometric closest-point as a proxy for
directional push distance, which are fundamentally different queries.
The heuristic could overestimate the safe push distance when an arc
faces an inclined line, causing the Compactor to over-push parts into
overlapping positions.

Replace with analytical computation: for each arc/line pair, solve
dt/dθ = 0 to find the two critical angles where the directional
distance is stationary, evaluate both (if within the arc's angular
span), and fire a ray to verify the hit is within the line segment.

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

OpenNest.Core/Geometry/SpatialQuery.cs

@@ -631,19 +631,46 @@ namespace OpenNest.Geometry
         {
             for (var i = 0; i < arcEntities.Count; i++)
             {
-                if (arcEntities[i] is Arc arc)
+                if (arcEntities[i] is not Arc arc)
+                    continue;
+
+                var cx = arc.Center.X;
+                var cy = arc.Center.Y;
+                var r = arc.Radius;
+
+                for (var j = 0; j < lineEntities.Count; j++)
                 {
-                    for (var j = 0; j < lineEntities.Count; j++)
+                    if (lineEntities[j] is not Line line)
+                        continue;
+
+                    var p1x = line.pt1.X;
+                    var p1y = line.pt1.Y;
+                    var ex = line.pt2.X - p1x;
+                    var ey = line.pt2.Y - p1y;
+
+                    var det = ex * dirY - ey * dirX;
+                    if (System.Math.Abs(det) < Tolerance.Epsilon)
+                        continue;
+
+                    // The directional distance from an arc point at angle θ to the
+                    // line is t(θ) = [A + r·(ey·cosθ − ex·sinθ)] / det.
+                    // dt/dθ = 0 at θ = atan2(−ex, ey) and θ + π.
+                    var theta1 = Angle.NormalizeRad(System.Math.Atan2(-ex, ey));
+                    var theta2 = Angle.NormalizeRad(theta1 + System.Math.PI);
+
+                    for (var k = 0; k < 2; k++)
                     {
-                        if (lineEntities[j] is Line line)
+                        var theta = k == 0 ? theta1 : theta2;
-                        {
+
-                            var linePt = line.ClosestPointTo(arc.Center);
+                        if (!Angle.IsBetweenRad(theta, arc.StartAngle, arc.EndAngle, arc.IsReversed))
-                            var arcPt = arc.ClosestPointTo(linePt);
+                            continue;
-                            var d = RayEdgeDistance(arcPt.X, arcPt.Y,
+
-                                line.pt1.X, line.pt1.Y, line.pt2.X, line.pt2.Y,
+                        var qx = cx + r * System.Math.Cos(theta);
-                                dirX, dirY);
+                        var qy = cy + r * System.Math.Sin(theta);
-                            if (d < minDist) { minDist = d; if (d <= 0) return 0; }
+
-                        }
+                        var d = RayEdgeDistance(qx, qy, p1x, p1y, line.pt2.X, line.pt2.Y,
+                            dirX, dirY);
+                        if (d < minDist) { minDist = d; if (d <= 0) return 0; }
                     }
                 }
             }

OpenNest.Tests/Fill/CompactorTests.cs

@@ -8,6 +8,76 @@ namespace OpenNest.Tests.Fill
 {
     public class CompactorTests
     {
+        [Fact]
+        public void DirectionalDistance_ArcVsInclinedLine_DoesNotOverPush()
+        {
+            // Arc (top semicircle) pushed upward toward a 45° inclined line.
+            // The critical angle on the arc gives a shorter distance than any
+            // sampled vertex (endpoints + cardinal extremes).
+            var arc = new Arc(5, 0, 2, 0, System.Math.PI);
+            var line = new Line(new Vector(3, 4), new Vector(7, 6));
+
+            var moving = new List<Entity> { arc };
+            var stationary = new List<Entity> { line };
+            var direction = new Vector(0, 1); // push up
+
+            var dist = SpatialQuery.DirectionalDistance(moving, stationary, direction);
+
+            // Move the arc up by the computed distance, then verify no overlap.
+            // The topmost reachable point on the arc at the critical angle θ ≈ 2.034
+            // (between π/2 and π) should just touch the line.
+            Assert.True(dist < double.MaxValue, "Should find a finite distance");
+            Assert.True(dist > 0, "Should be a positive distance");
+
+            // Verify: after moving, the closest point on the arc should be within
+            // tolerance of the line, not past it.
+            var theta = System.Math.Atan2(
+                line.pt2.X - line.pt1.X, -(line.pt2.Y - line.pt1.Y));
+            theta = OpenNest.Math.Angle.NormalizeRad(theta + System.Math.PI);
+            var qx = arc.Center.X + arc.Radius * System.Math.Cos(theta);
+            var qy = arc.Center.Y + arc.Radius * System.Math.Sin(theta) + dist;
+
+            // The moved point should be on or just touching the line, not past it.
+            // Line equation: (y - 4) / (x - 3) = (6 - 4) / (7 - 3) = 0.5
+            // y = 0.5x + 2.5
+            var lineYAtQx = 0.5 * qx + 2.5;
+            Assert.True(qy <= lineYAtQx + 0.001,
+                $"Arc point ({qx:F4}, {qy:F4}) should not be past line (line Y={lineYAtQx:F4} at X={qx:F4}). " +
+                $"dist={dist:F6}, overshot by {qy - lineYAtQx:F6}");
+        }
+
+        [Fact]
+        public void DirectionalDistance_ArcVsInclinedLine_BetterThanVertexSampling()
+        {
+            // Same geometry — verify the analytical Phase 3 finds a shorter
+            // distance than the Phase 1/2 vertex sampling alone would.
+            var arc = new Arc(5, 0, 2, 0, System.Math.PI);
+            var line = new Line(new Vector(3, 4), new Vector(7, 6));
+
+            // Phase 1/2 vertex-only distance: sample arc endpoints + cardinal extreme.
+            var vertices = new[]
+            {
+                new Vector(7, 0),  // arc endpoint θ=0
+                new Vector(3, 0),  // arc endpoint θ=π
+                new Vector(5, 2),  // cardinal extreme θ=π/2
+            };
+
+            var vertexMin = double.MaxValue;
+            foreach (var v in vertices)
+            {
+                var d = SpatialQuery.RayEdgeDistance(v.X, v.Y,
+                    line.pt1.X, line.pt1.Y, line.pt2.X, line.pt2.Y, 0, 1);
+                if (d < vertexMin) vertexMin = d;
+            }
+
+            // Full directional distance (includes Phase 3 arc-to-line).
+            var moving = new List<Entity> { arc };
+            var stationary = new List<Entity> { line };
+            var fullDist = SpatialQuery.DirectionalDistance(moving, stationary, new Vector(0, 1));
+
+            Assert.True(fullDist < vertexMin,
+                $"Full distance ({fullDist:F6}) should be less than vertex-only ({vertexMin:F6})");
+        }
         private static Drawing MakeRectDrawing(double w, double h)
         {
             var pgm = new OpenNest.CNC.Program();