mesh: an angle exactly on the 135-degree threshold is not a corner, whichever way it rounds

A region's corners are its interior angles below 135 degrees, and an
O-grid's 45-degree cuts meet straight sides at exactly that angle. The
computed angle then rounds either way with the last bits of its points,
which differ between platforms' sin and cos and between tessellations:
one ulp in three points of the slot test's arcs, Windows' sin against
glibc's, turned a mapped region into a triangle and the whole blocking
into a different one with a corner scaled Jacobian of 0.375 for 0.707.
Angles within 1e-9 rad of the corner and reflex thresholds now count as
on them, and on a threshold is smooth (neither a corner nor reflex),
which keeps the slot's intended O-grid of 9 regions and 848 quads on
Linux, macOS and Windows. The region classification and the O-grid's
own corner tests share the rule, the slot test pins the blocking, and
its arc helper uses the mesher's portable trig.

Part of #556.

crates/mesh/src/hex/decompose.rs

@@ -916,8 +916,8 @@ fn assignable(loops: &[Poly], cuts: &[Vec<Q>], s: &Settings) -> bool {
 fn o_grid(loops: &[Poly], s: &Settings, budget: &mut usize) -> Vec<Vec<Q>> {
     let (Some(outer), Some(hole)) = (loops.first(), loops.get(1)) else { return Vec::new() };
     let c = centroid(&hole.pts);
-    let corner = |p: &Poly, i: usize| p.is_end(i) && p.angle(i) <= 0.75 * std::f64::consts::PI;
-    let concave = |p: &Poly, i: usize| p.is_end(i) && p.angle(i) >= 1.25 * std::f64::consts::PI;
+    let corner = |p: &Poly, i: usize| p.is_end(i) && region::is_corner(p.angle(i));
+    let concave = |p: &Poly, i: usize| p.is_end(i) && region::is_reflex(p.angle(i));
     // A cut from loop point (l, i) along d that must end on loop `target`.
     let ray = |l: usize, i: usize, d: Q, target: usize| -> Option<Vec<Q>> {
         let o = loops.get(l)?.at(i);
@@ -1134,8 +1134,9 @@ mod tests {
         let n = ((a1 - a0).abs() / 5.0).ceil().max(1.0) as usize;
         (0..=n)
             .map(|k| {
+                // The mesher's portable trig, so every platform gets the same points (#556).
                 let t = (a0 + (a1 - a0) * k as f64 / n as f64).to_radians();
-                [c[0] + r * t.cos(), c[1] + r * t.sin()]
+                [c[0] + r * libm::cos(t), c[1] + r * libm::sin(t)]
             })
             .collect()
     }
@@ -1308,6 +1309,10 @@ mod tests {
         ]);
         let m = mesh_profile(&profile, &[], 2.5);
         check(&m, &profile, 0.5);
+        // The intended O-grid on every platform (#556): Windows' last-bit-different arcs used to
+        // give 10 regions and 2063 quads with a 0.375 corner.
+        assert_eq!(m.section.schemes.len(), 9, "{:?}", m.section.schemes);
+        assert_eq!(m.section.quads.iter().map(Vec::len).sum::<usize>(), 848);
     }
 
     #[test]

crates/mesh/src/hex/region.rs

@@ -47,6 +47,24 @@ const MAX_ANNULUS_NODES: usize = 4096;
 /// Smoothing sweeps over the interior of a triangle primitive.
 const TRIANGLE_SWEEPS: usize = 20;
 
+/// How close (radians) an interior angle must come to a classification threshold to count as on
+/// it. Exact geometry lands on the thresholds (an O-grid's 45° cuts meet straight sides at
+/// exactly 135°), and the computed angle is then rounded either way by the last bits of its
+/// points: those differ between platforms' `sin`/`cos` and between tessellations, and the
+/// scheme must not (#556). An angle within this of a threshold is taken as exactly on it, and
+/// on a threshold is smooth, neither a corner nor reflex: that keeps the slot's intended O-grid.
+const ANGLE_TOLERANCE: f64 = 1e-9;
+
+/// A corner: an interior angle clearly below 135° (exactly 135° is not one).
+pub(crate) fn is_corner(angle: f64) -> bool {
+    angle < 0.75 * PI - ANGLE_TOLERANCE
+}
+
+/// A reflex turn no mapped or triangle region can take: clearly above 225°.
+pub(crate) fn is_reflex(angle: f64) -> bool {
+    angle > 1.25 * PI + ANGLE_TOLERANCE
+}
+
 #[derive(Debug, Clone, Copy, PartialEq, Eq)]
 pub(crate) enum Scheme {
     /// Corner positions (curve uses) in the single loop.
@@ -89,10 +107,10 @@ pub(crate) fn classify(loops: &[Vec<Vec<Q>>]) -> Scheme {
     match loops {
         [lp] => {
             let Some(a) = angles(lp, ccw(lp)) else { return Scheme::Free };
-            if a.iter().any(|&x| x >= 1.25 * PI) {
+            if a.iter().any(|&x| is_reflex(x)) {
                 return Scheme::Free;
             }
-            let corners: Vec<usize> = (0..a.len()).filter(|&k| a.get(k).is_some_and(|&x| x <= 0.75 * PI)).collect();
+            let corners: Vec<usize> = (0..a.len()).filter(|&k| a.get(k).is_some_and(|&x| is_corner(x))).collect();
             if corners.is_empty() {
                 return Scheme::Disk;
             }
@@ -103,7 +121,7 @@ pub(crate) fn classify(loops: &[Vec<Vec<Q>>]) -> Scheme {
             }
         }
         [outer, inner] => {
-            let smooth = |lp: &Vec<Vec<Q>>| angles(lp, ccw(lp)).is_some_and(|a| a.iter().all(|&x| x > 0.75 * PI && x < 1.25 * PI));
+            let smooth = |lp: &Vec<Vec<Q>>| angles(lp, ccw(lp)).is_some_and(|a| a.iter().all(|&x| !is_corner(x) && !is_reflex(x)));
             if smooth(outer) && smooth(inner) { Scheme::Annulus } else { Scheme::Free }
         }
         _ => Scheme::Free,
@@ -568,6 +586,30 @@ mod tests {
     use super::*;
     use crate::size::UniformSize;
 
+    /// A corner of exactly 135°, the threshold, is not a corner however the last bits of its
+    /// points round (Windows' `sin` against glibc's, or two tessellations): a trapezoid with one
+    /// stays a triangle rather than flipping to mapped when that vertex moves by an ulp (#556).
+    #[test]
+    fn a_corner_on_the_threshold_is_one_whichever_way_it_rounds() {
+        // (0, 0) 90°, (3, 0) 45°, (2, 1) 135°, (0, 1) 90°: three corners and a 135° bend.
+        let trapezoid = |p: Q| {
+            let pts = [[0.0, 0.0], [3.0, 0.0], p, [0.0, 1.0]];
+            let curves: Vec<Vec<Q>> = (0..4).map(|k| vec![pts[k], pts[(k + 1) % 4]]).collect();
+            classify(&[curves])
+        };
+        let steps: [fn(f64) -> f64; 3] = [|v| v, f64::next_up, f64::next_down];
+        for nx in steps {
+            for ny in steps {
+                let mut p = [nx(2.0), ny(1.0)];
+                // A few ulps further each way as well.
+                for _ in 0..3 {
+                    assert_eq!(trapezoid(p), Scheme::Triangle([0, 1, 3]), "{p:?}");
+                    p = [nx(p[0]), ny(p[1])];
+                }
+            }
+        }
+    }
+
     fn circle(r: f64, n: usize, ccw: bool) -> Vec<Q> {
         (0..n)
             .map(|k| {