Kernel tests: the tabbed plate's rim rounds take their exact volume

fillet_tabbed_plate_outer_loop_and_partial_selection_after_hole only
checked that every subset of the R25 tabbed plate's seven top-outline edges
builds and stays inside the plate. It now first holds the hole's rim round
and the three selections that end only at the plate's sharp concave
corners (the R25 arc's two halves, the tab's tangent run of sides, R2
corners and top, and both) to their exact removed volume; the sweep over
all 127 subsets stays as it was.

Proof. The outline's area A(da, dt), the arc moved in by da and the tab's
run by dt, each element parallel to itself, by Green's theorem: an arc
about c of radius rho from a to b gives
1/2 (rho^2 (b - a) + rho (cx (sin b - sin a) - cy (cos b - cos a))), a
segment p -> q gives 1/2 p x q. At a concave corner (+-4, sqrt(609)) where
only one side moves, the elements meet where they cross (the round runs
into the other wall: the section is the union of the two sides' half
regions); where both move the corner is swept, an arc of radius d about the
corner, traversed clockwise from the arc's normal to the side's (the
owner's ruling of 2026-10-30 for rounds meeting at a concave corner beside
an unpicked sharp edge: the profile swept round the corner edge). Checks:
A(0, 0) = 312.5 (pi + 2 theta) + 4 y + 8 (cy - y) + 8 + 2 pi (disc sector,
triangle, tab, R2 corners), and A(0, 0) - A(d, d) = L d - pi d^2 by the
Steiner formula for the swept offset (total smooth turning 2 pi + 2 theta,
the two swept corners -theta each), agreeing to 1e-15 at d = 0.1, 0.3,
0.5; A(0, 0) - A(0, d) agrees with a 0.004 mm grid to 2e-6. The round's
section at height r sin(phi) into it is the outline moved in by
d = r (1 - cos phi), so it removes the integral of
(A(0, 0) - A(d)) r cos(phi) dphi over [0, pi/2] (midpoint rule, 8192
steps, converged to 1e-7 against 4096). The hole's rim round is Pappus:
(1 - pi/4) r^2 about the axis at 1.5 + r (10 - 3 pi) / (3 (4 - pi)).
Graded at 1e-5 of the removal, the bound the D prism's rounds are held to
(the blend surfaces are SMLib's fits).

Today the three selections build with these volumes (aarch64, SMLib); the
sweep still fails where a subset ends a round at a tangent joint (the x = 4
side without its R2 corner): SMLib cannot end a blend inside a tangent
chain, "the blend cannot end there".

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

crates/kernel/src/tests.rs

@@ -1949,6 +1949,29 @@ fn fillet_sharp_arc_and_arc_loop() {
     assert!(rel(removed, want) < 1e-3, "{removed} vs {want}");
 }
 
+/// The area of the tabbed plate's outline (`fillet_tabbed_plate_outer_loop_and_partial_selection_after_hole`)
+/// with the R25 arc moved in by `da` and the tab's run (its sides, R2 corners and top) by `dt`,
+/// each element parallel to itself, by Green's theorem: an arc about c of radius ρ from a to b
+/// gives ½ (ρ² (b − a) + ρ (cx (sin b − sin a) − cy (cos b − cos a))), a segment p→q ½ (p × q).
+/// Where only one side of a concave corner at (±4, y) moves the elements meet where they cross
+/// (the round runs into the other wall); where both do, the corner is swept, an arc of radius d
+/// about the corner (the owner's ruling for rounds meeting at a concave corner, 2026-10-30).
+fn tabbed_plate_area(da: f64, dt: f64) -> f64 {
+    let (y, cy) = ((625.0_f64 - 16.0).sqrt(), 280.0 / 9.0);
+    let theta = (4.0_f64 / 25.0).acos();
+    let (big, rho) = (25.0 - da, 2.0 - dt);
+    let tab = rho * rho * PI + 2.0 * rho * (2.0 + cy) + 4.0 * (cy + rho);
+    if da > 0.0 && dt > 0.0 {
+        assert!((da - dt).abs() < 1e-15, "a swept corner of one size");
+        let corner = -da * da * theta + da * (4.0 * theta.sin() + y * (1.0 - theta.cos()));
+        0.5 * (big * big * (PI + 2.0 * theta) + 2.0 * corner + tab) + (4.0 - dt) * (cy - y)
+    } else {
+        let yc = (big * big - (4.0 - dt).powi(2)).sqrt();
+        let at = yc.atan2(4.0 - dt);
+        0.5 * (big * big * (PI + 2.0 * at) + tab) + (4.0 - dt) * (cy - yc)
+    }
+}
+
 #[test]
 fn fillet_tabbed_plate_outer_loop_and_partial_selection_after_hole() {
     use solidcraft_geom::Seg2;
@@ -1971,7 +1994,20 @@ fn fillet_tabbed_plate_outer_loop_and_partial_selection_after_hole() {
     };
     let body =
         extrude(&Plane::XY, &[Region2 { outer: outline, holes: vec![Loop2::circle(Vec2::new(0.0, hole_y), 1.5)] }], 0.0, 2.0).unwrap().pop().unwrap();
-    let inner = fillet(&body, &[Vec3::new(1.5, hole_y, 2.0)], 0.5).unwrap();
+    let r = 0.5;
+    // The section a round takes off, (1 − π/4) r², and its centroid's distance from both faces.
+    let (spandrel, centroid) = ((1.0 - PI / 4.0) * r * r, r * (10.0 - 3.0 * PI) / (3.0 * (4.0 - PI)));
+    // The outline's area (Green's theorem, checked against the disc, the tab's rectangle and its
+    // two R2 corners), the plate its prism 2 high less the hole.
+    let outline_area = tabbed_plate_area(0.0, 0.0);
+    let parts = 312.5 * (PI + 2.0 * theta) + 4.0 * y + 8.0 * (corner_y - y) + 8.0 + 2.0 * PI;
+    assert!(rel(outline_area, parts) < 1e-14, "{outline_area} vs {parts}");
+    assert!(rel(measure(&body).unwrap().volume, 2.0 * (outline_area - PI * 2.25)) < 1e-9);
+    // The hole's rim round (Pappus: the spandrel about the hole's axis at 1.5 + its centroid).
+    let inner = fillet(&body, &[Vec3::new(1.5, hole_y, 2.0)], r).unwrap();
+    let rim = spandrel * 2.0 * PI * (1.5 + centroid);
+    let holed = measure(&inner).unwrap().volume;
+    assert!(rel(measure(&body).unwrap().volume - holed, rim) < 1e-5, "{} vs {rim}", measure(&body).unwrap().volume - holed);
     let edges = [
         Vec3::new(-24.91935125502191, -2.0064727829797544, 2.0),
         Vec3::new(24.919351255429795, -2.006472777913911, 2.0),
@@ -1981,6 +2017,30 @@ fn fillet_tabbed_plate_outer_loop_and_partial_selection_after_hole() {
         Vec3::new(-3.4142135648720653, 32.52532467287703, 2.0),
         Vec3::new(-4.0000000025087745, 27.89451823429679, 2.0),
     ];
+    // The selections that end only at the sharp concave corners at (±4, y): the R25 arc (its two
+    // edges, tangent at the bottom), the tab's run (tangent through its R2 corners), and both.
+    // Each takes off the outline's section less the section moved in by d = r (1 − cos φ) at the
+    // height r sin φ into the round, integrated over that height (midpoint rule in φ).
+    let removed = |arc: bool, tab: bool, steps: u32| {
+        let dphi = (PI / 2.0) / f64::from(steps);
+        (0..steps)
+            .map(|i| {
+                let phi = (f64::from(i) + 0.5) * dphi;
+                let d = r * (1.0 - phi.cos());
+                (outline_area - tabbed_plate_area(if arc { d } else { 0.0 }, if tab { d } else { 0.0 })) * r * phi.cos() * dphi
+            })
+            .sum::<f64>()
+    };
+    for (mask, arc, tab) in [(0b0000011_u32, true, false), (0b1111100, false, true), (0b1111111, true, true)] {
+        let want = removed(arc, tab, 8192);
+        assert!((want - removed(arc, tab, 4096)).abs() < 1e-7 * want, "the reference quadrature must converge");
+        let picks: Vec<_> = edges.iter().enumerate().filter(|(i, _)| mask & (1 << i) != 0).map(|(_, p)| *p).collect();
+        let blended = fillet(&inner, &picks, r).unwrap_or_else(|e| panic!("{picks:?}: {e}"));
+        assert!(blended.validity().is_empty(), "{picks:?}: {:?}", blended.validity());
+        // Exact volumes (SMLib's, to 1e-8); the blend's surfaces are SMLib's fits.
+        let got = holed - measure(&blended).unwrap().volume;
+        assert!(rel(got, want) < 1e-5, "arc {arc}, tab {tab}: removed {got}, expected {want}");
+    }
     // All 127 nonempty subsets: the whole rim, each individual edge, and every
     // contiguous or disjoint partial selection of the issue's seven picks.
     for mask in 1_u32..(1 << edges.len()) {