kernel-ladder: tier 07 corner cases grade their geometry

010, 011 and 012 passed on one valid body alone. 010 (r7 then an r3 chain
round its arc) and 011 (two r5 rounds in a mitre) now check closed-form
volumes, counts and bbox; 012's variable round takes only its edge and
runs on into the vertical round's column, so its volume is graded as the
interval between that end carried on as a constant r5 tube and as its
growing radius. All hold on SMLib (worst 4e-8 relative; 012 at the
growing-radius end).

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

tests/kernel-cases/ladder/07-fillets/010-corner-unequal-r7-r3/expected.json

@@ -1,7 +1,26 @@
 {
  "sources": [],
  "what": "an r7 vertical round at a corner, then r3 along a top edge, which runs on round the r7 arc to the other top edge",
- "source": "invariants",
- "method": "two radii meeting at a corner over a tangent chain; one valid body",
- "body_count": 1
+ "source": "analytic",
+ "method": "the r7 vertical round removes (1-pi/4) 7^2 20; the r3 chain runs along x = 60 (y 0..33) and y = 40 (x 0..53), straight, removing (1-pi/4) 3^2 each mm, and round the r7 arc's top edge as a torus quarter: Pappus, the section (1-pi/4) 3^2 with its centroid 3 (10-3pi)/(12-3pi) in from the wall, at radius 7 - that, over pi/2 (as 003). Faces: 6 planes + the r7 cylinder + the r3 chain's cylinder, torus, cylinder; edges: 5 round the bottom, 5 round the top, 5 vertical (two r7 tangent lines, the box's three other verticals), the chain's three on the walls and two end arcs, its two seams; vertices from Euler (V - E + F = 2): 14",
+ "body_count": 1,
+ "volume_mm3": 47604.384347169,
+ "faces": 10,
+ "edges": 22,
+ "vertices": 14,
+ "bbox": {
+  "min": [
+   0,
+   0,
+   0
+  ],
+  "max": [
+   60,
+   40,
+   20
+  ]
+ },
+ "tolerance": {
+  "rel": 1e-05
+ }
 }

tests/kernel-cases/ladder/07-fillets/011-chain-ending-at-corner/expected.json

@@ -1,7 +1,26 @@
 {
  "sources": [],
  "what": "r5 rounds on two top edges meeting at a corner whose vertical edge stays sharp",
- "source": "invariants",
- "method": "two rounds ending at a sharp vertical edge; one valid body",
- "body_count": 1
+ "source": "analytic",
+ "method": "the two rounds remove (1-pi/4) 5^2 (60 + 40), less their common part in the corner cube, counted twice: in it (u, v, w from the round axes' corner) both remove where u, v > sqrt(25 - w^2), int_0^5 (5 - sqrt(25 - w^2))^2 dw = 5^3 (5/3 - pi/2). They meet in a mitre (an elliptic arc down to the sharp vertical edge). Faces 6 + 2; edges: 4 bottom, 4 vertical, 4 round the top face, 2 round-wall tangent lines, 2 end arcs, the mitre; vertices 11 (Euler)",
+ "body_count": 1,
+ "volume_mm3": 47475.479200978,
+ "faces": 8,
+ "edges": 17,
+ "vertices": 11,
+ "bbox": {
+  "min": [
+   0,
+   0,
+   0
+  ],
+  "max": [
+   60,
+   40,
+   20
+  ]
+ },
+ "tolerance": {
+  "rel": 1e-05
+ }
 }

tests/kernel-cases/ladder/07-fillets/012-variable-into-rounded-corner/expected.json

@@ -1,7 +1,26 @@
 {
  "sources": [],
  "what": "a vertical r5 round, then a variable round r1 -> r5 along the top back edge into it",
- "source": "invariants",
- "method": "a variable round running into a constant one at a corner; one valid body",
- "body_count": 1
+ "source": "analytic",
+ "method": "a variable round takes only its picked edge (no tangent chain: crates/doc/src/eval.rs), y = 40 from x = 0 (r1) to x = 55 (r5), where it meets the vertical r5 round's tangent line: removed (1-pi/4) 55 (1 + 5 + 25)/3 (005) and (1-pi/4) 5^2 20. Past x = 55 the round's end runs on into the vertical round's quarter column, trimmed by its cylinder: as a constant r5 tube that takes int_0^5 sqrt(25 - v^2)(5 - sqrt(25 - v^2)) dv = 125 pi/4 - 250/3 = 14.841437; with its radius still growing at 4/55 per mm, 15.854054 (numerical, the round's section against the cylinder). V lies between: graded as the interval's middle with its half width as the tolerance. Faces 6 + 2; edges 18, vertices 12 (5 round the bottom, 5 vertical, 5 round the top face, the round's line on the back wall, its end arc on x = 0, its trim on the cylinder)",
+ "body_count": 1,
+ "volume_mm3": 47755.385959062,
+ "faces": 8,
+ "edges": 18,
+ "vertices": 12,
+ "bbox": {
+  "min": [
+   0,
+   0,
+   0
+  ],
+  "max": [
+   60,
+   40,
+   20
+  ]
+ },
+ "tolerance": {
+  "rel": 1.1602121611875055e-05
+ }
 }

tools/kernel-ladder/cases.py

@@ -806,17 +806,47 @@ case(T7, "009-corner-three-equal-r5", "r5 rounds on the three edges meeting at t
      {"body_count": 1, "volume_mm3": VB - SPANDREL * 25 * (60 + 40 + 20 - 15) - 125 * (1 - PI / 6), "faces": 10},
      "each edge zone outside the corner cube removes (1-pi/4) r^2 (L - r); the corner cube r^3 keeps a sphere octant pi r^3/6", base=B1)
 case(T7, "010-corner-unequal-r7-r3", "an r7 vertical round at a corner, then r3 along a top edge, which runs on round the r7 arc to the other top edge",
-     [F([corner[2]], 7), F([[60, 15, 20]], 3)], {"body_count": 1},
-     "two radii meeting at a corner over a tangent chain; one valid body", base=B1, source="invariants")
+     [F([corner[2]], 7), F([[60, 15, 20]], 3)],
+     {"body_count": 1, "volume_mm3": VB - SPANDREL * 49 * 20 - SPANDREL * 9 * (33 + 53) - SPANDREL * 9 * (PI / 2) * (7 - XBAR * 3),
+      "faces": 10, "edges": 22, "vertices": 14, "bbox": BBOX, "tolerance": {"rel": 1e-5}},
+     "the r7 vertical round removes (1-pi/4) 7^2 20; the r3 chain runs along x = 60 (y 0..33) and y = 40 (x 0..53), straight, removing (1-pi/4) 3^2 each mm, and round the "
+     "r7 arc's top edge as a torus quarter: Pappus, the section (1-pi/4) 3^2 with its centroid 3 (10-3pi)/(12-3pi) in from the wall, at radius 7 - that, over pi/2 (as 003). "
+     "Faces: 6 planes + the r7 cylinder + the r3 chain's cylinder, torus, cylinder; edges: 5 round the bottom, 5 round the top, 5 vertical (two r7 tangent lines, the box's "
+     "three other verticals), the chain's three on the walls and two end arcs, its two seams; vertices from Euler (V - E + F = 2): 14", base=B1)
 case(T7, "016-corner-three-sequential-r5-r3-r7", "rounds r5 and r3 on the two top edges at a corner, then r7 on the vertical edge under them",
      [F([corner[0]], 5), F([corner[1]], 3, tangent_chain=False), F([[60, 40, 8]], 7)], {"body_count": 1},
      "three unequal radii at one corner, one after another; the vertical edge stays sharp below the two top rounds (x=60, y=40, z from 0 to 15), so it can be picked at z=8; one valid body", base=B1, source="invariants")
 case(T7, "011-chain-ending-at-corner", "r5 rounds on two top edges meeting at a corner whose vertical edge stays sharp",
-     [F([corner[0], corner[1]], 5)], {"body_count": 1},
-     "two rounds ending at a sharp vertical edge; one valid body", base=B1, source="invariants")
+     [F([corner[0], corner[1]], 5)],
+     {"body_count": 1, "volume_mm3": VB - SPANDREL * 25 * 100 + 125 * (5 / 3 - PI / 2), "faces": 8, "edges": 17, "vertices": 11, "bbox": BBOX, "tolerance": {"rel": 1e-5}},
+     "the two rounds remove (1-pi/4) 5^2 (60 + 40), less their common part in the corner cube, counted twice: in it (u, v, w from the round axes' corner) both remove where "
+     "u, v > sqrt(25 - w^2), int_0^5 (5 - sqrt(25 - w^2))^2 dw = 5^3 (5/3 - pi/2). They meet in a mitre (an elliptic arc down to the sharp vertical edge). Faces 6 + 2; edges: "
+     "4 bottom, 4 vertical, 4 round the top face, 2 round-wall tangent lines, 2 end arcs, the mitre; vertices 11 (Euler)", base=B1)
+# 012: the variable round's end carried on past x = 55 into the vertical round's column, trimmed by it.
+def corner_extension(rf, n=400):
+    tot = 0.0
+    for i in range(n):
+        x = 55 + 5 * (i + 0.5) / n
+        vmax = math.sqrt(max(25 - (x - 55) ** 2, 0))
+        r = rf(x)
+        c = 5 - r
+        def above(v):
+            dv = v - c
+            return max(0.0, 5 - (c + math.sqrt(max(r * r - dv * dv, 0)))) if dv < r else 5 - c
+        tot += quad_split(above, c, vmax, 40, 16) * 5 / n if vmax > c else 0.0
+    return tot
+ext_const = 125 * PI / 4 - 250 / 3
+ext_grow = corner_extension(lambda x: 1 + 4 * x / 55)
+v12a = VB - SPANDREL * 25 * 20 - SPANDREL * 55 * 31 / 3
+v12lo, v12hi = v12a - ext_grow, v12a - ext_const
 case(T7, "012-variable-into-rounded-corner", "a vertical r5 round, then a variable round r1 -> r5 along the top back edge into it",
-     [F([corner[2]], 5), F([[30, 40, 20]], 1, type="variable", radius2=5, start=[0, 40, 20])], {"body_count": 1},
-     "a variable round running into a constant one at a corner; one valid body", base=B1, source="invariants")
+     [F([corner[2]], 5), F([[30, 40, 20]], 1, type="variable", radius2=5, start=[0, 40, 20])],
+     {"body_count": 1, "volume_mm3": (v12lo + v12hi) / 2, "faces": 8, "edges": 18, "vertices": 12, "bbox": BBOX, "tolerance": {"rel": (v12hi - v12lo) / (v12hi + v12lo) + 1e-6}},
+     "a variable round takes only its picked edge (no tangent chain: crates/doc/src/eval.rs), y = 40 from x = 0 (r1) to x = 55 (r5), where it meets the vertical r5 round's "
+     "tangent line: removed (1-pi/4) 55 (1 + 5 + 25)/3 (005) and (1-pi/4) 5^2 20. Past x = 55 the round's end runs on into the vertical round's quarter column, trimmed by its "
+     "cylinder: as a constant r5 tube that takes int_0^5 sqrt(25 - v^2)(5 - sqrt(25 - v^2)) dv = 125 pi/4 - 250/3 = %.6f; with its radius still growing at 4/55 per mm, %.6f "
+     "(numerical, the round's section against the cylinder). V lies between: graded as the interval's middle with its half width as the tolerance. Faces 6 + 2; edges 18, "
+     "vertices 12 (5 round the bottom, 5 vertical, 5 round the top face, the round's line on the back wall, its end arc on x = 0, its trim on the cylinder)" % (ext_const, ext_grow), base=B1)
 missing(T7, "013-corner-setback", "three r5 rounds at a corner with setbacks of 8 mm along each edge", "solid.fillet has no corner setback distances",
         "the setback corner patch would be graded by invariants")
 case(T7, "014-curved-edge-r2", "an r2 round on the top edge of a cylinder r10 h20",