Mesh: the triangle primitive for three-cornered sweep regions

A cross-section region with three corners (a fillet's quarter disk, a
pocket corner cut off by a partition) was left to the free all-quad
mesher. It now gets midpoint subdivision's triangle primitive (Tam and
Armstrong 1991; Li, Armstrong, McKeag 1997): lines from a centre to a
node on each side split it into three mapped blocks, the interior then
smoothed and every quadrilateral checked (a fold falls back to free).
Interval assignment gains dominance constraints (a side shorter than
the other two together) so the split always exists.

Co-Authored-By: Claude Opus 5.5 <[email protected]>
Claude-Session: https://claude.ai/code/session_01UhhZMBTviDQxZvwnyP89fo

crates/mesh/src/hex/interval.rs

@@ -9,7 +9,9 @@
 //!   face;
 //! - **parity**: `Σ_L x` even on every loop of a face meshed by the free all-quad mesher (a
 //!   multiple of 4 around a disk meshed as an O-grid);
-//! - **minimum sums** `Σ_L x ≥ m`: a loop needs enough points to be meshed.
+//! - **minimum sums** `Σ_L x ≥ m`: a loop needs enough points to be meshed;
+//! - **dominance** `Σ_A x ≥ Σ_B x + d`: a side of a triangle primitive shorter than the other two
+//!   together.
 //!
 //! Mitchell's *incremental interval assignment* (Mitchell, *Incremental interval assignment by
 //! integer linear algebra*, International Meshing Roundtable 2021; Computer-Aided Design 158:103485, 2023) starts from a
@@ -37,6 +39,8 @@ pub(crate) struct Intervals {
     pub multiple: Vec<(Vec<usize>, usize)>,
     /// `Σ vars ≥ min`.
     pub at_least: Vec<(Vec<usize>, usize)>,
+    /// `Σ lhs ≥ Σ rhs + d`.
+    pub exceed: Vec<(Vec<usize>, Vec<usize>, usize)>,
     /// Variables held at their goal (an exact edge seed), with the seed in words; empty or
     /// `None`: free.
     pub fixed: Vec<Option<String>>,
@@ -109,6 +113,7 @@ impl Intervals {
         if !self.equal.iter().all(|(a, b)| check(a) && check(b))
             || !self.multiple.iter().all(|(l, m)| check(l) && *m >= 2)
             || !self.at_least.iter().all(|(l, _)| check(l))
+            || !self.exceed.iter().all(|(a, b, _)| check(a) && check(b))
         {
             return Err(Error::Failed("interval assignment: a constraint names an unknown curve".into()));
         }
@@ -131,7 +136,7 @@ impl Intervals {
             error(i, c + 1) - error(i, c)
         };
         let total: usize = x.iter().sum();
-        let constraints = self.equal.len() + self.multiple.len() + self.at_least.len();
+        let constraints = self.equal.len() + self.multiple.len() + self.at_least.len() + self.exceed.len();
         let max_steps = total.saturating_mul(8).saturating_add(64 * (n + constraints)).saturating_add(1000);
         for _ in 0..max_steps {
             let sum = |v: &[usize], x: &[usize]| v.iter().map(|&i| x.get(i).copied().unwrap_or(0)).sum::<usize>();
@@ -170,6 +175,28 @@ impl Intervals {
                     }
                 }
             }
+            if raise.is_none() {
+                for (a, b, d) in &self.exceed {
+                    let (sa, sb) = (sum(a, &x), sum(b, &x));
+                    if sa < sb.saturating_add(*d) {
+                        let net = |i: usize| a.iter().filter(|&&k| k == i).count() as i64 - b.iter().filter(|&&k| k == i).count() as i64;
+                        let cands: Vec<usize> = a.iter().copied().filter(|&i| net(i) > 0).collect();
+                        if cands.is_empty() {
+                            return Err(Error::Failed("interval assignment: a dominance constraint cannot be met".into()));
+                        }
+                        let free: Vec<usize> = cands.into_iter().filter(|&i| !self.is_fixed(i)).collect();
+                        if free.is_empty() {
+                            let all: Vec<usize> = a.iter().chain(b).copied().collect();
+                            return Err(self.conflict(
+                                &all,
+                                &format!("a side of a three-sided face needs fewer divisions than the other two ({sb} against {sa})"),
+                            ));
+                        }
+                        raise = Some(free);
+                        break;
+                    }
+                }
+            }
             if raise.is_none() {
                 for (l, m) in &self.multiple {
                     if !sum(l, &x).is_multiple_of(*m) {
@@ -290,6 +317,21 @@ mod tests {
         assert!(r.solve().unwrap_err().to_string().contains("multiple of 2"));
     }
 
+    #[test]
+    fn dominance_raises_the_other_sides() {
+        // A triangle's sides: 6 against 1 + 2 needs the short sides raised to 8 together.
+        let mut p = Intervals::default();
+        let a = p.var(6.0, 1);
+        let b = p.var(1.0, 1);
+        let c = p.var(2.0, 1);
+        p.exceed.push((vec![b, c], vec![a], 2));
+        p.exceed.push((vec![a, c], vec![b], 2));
+        p.exceed.push((vec![a, b], vec![c], 2));
+        let x = p.solve().unwrap();
+        assert_eq!(x[a], 6);
+        assert!(x[b] + x[c] >= 8, "{x:?}");
+    }
+
     #[test]
     fn infeasible_systems_are_errors_not_loops() {
         let mut p = Intervals::default();

crates/mesh/src/hex/region.rs

@@ -9,6 +9,16 @@
 //!   counts and the ring between them is a periodic structured grid, each radial line joining a
 //!   node of the outer loop to its partner on the inner loop (the rotation that pairs them most
 //!   closely) — the O-grid of a tube's cross-section.
+//! - **Triangle**: one loop with three corners (a fillet's quarter disk, the corner of a
+//!   pocket cut off by a partition): the triangle primitive of midpoint subdivision (Tam and
+//!   Armstrong, *2D finite element mesh generation by medial axis subdivision*, Advances in
+//!   Engineering Software 13(5–6):313–324, 1991; Li, Armstrong, McKeag, *Quad mesh generation for
+//!   k-sided faces and hex mesh generation for trivalent polyhedra*, Finite Elements in Analysis
+//!   and Design 26(4):279–301, 1997): lines from a centre to one node on each side split the
+//!   region into three mapped quadrilateral blocks. Side `k` with `n_k` divisions is split into
+//!   `m_{k-1}` and `m_{k+1}` divisions, the counts of the lines that end on the sides before
+//!   and after it, so `m_k = (n_{k+1} + n_{k+2} − n_k) / 2`: the loop's count is even and each
+//!   side shorter than the other two together.
 //! - **Disk**: one smooth loop (no corners): an O-grid ("butterfly"): a mapped square in the
 //!   middle, its corners halfway from the centroid to the loop's quarter points, and rings of
 //!   quadrilaterals from the square out to the loop. The loop's count is a multiple of four.
@@ -31,10 +41,15 @@ type Q = [f64; 2];
 /// The most nodes the annulus pairing compares (its cost is quadratic in the loop length).
 const MAX_ANNULUS_NODES: usize = 4096;
 
+/// Smoothing sweeps over the interior of a triangle primitive.
+const TRIANGLE_SWEEPS: usize = 20;
+
 #[derive(Debug, Clone, Copy, PartialEq, Eq)]
 pub(crate) enum Scheme {
     /// Corner positions (curve uses) in the single loop.
     Mapped([usize; 4]),
+    /// Corner positions (curve uses) in the single loop.
+    Triangle([usize; 3]),
     Annulus,
     Disk,
     Free,
@@ -78,7 +93,11 @@ pub(crate) fn classify(loops: &[Vec<Vec<Q>>]) -> Scheme {
             if corners.is_empty() {
                 return Scheme::Disk;
             }
-            corners.try_into().map(Scheme::Mapped).unwrap_or(Scheme::Free)
+            match corners[..] {
+                [a, b, c] => Scheme::Triangle([a, b, c]),
+                [a, b, c, d] => Scheme::Mapped([a, b, c, d]),
+                _ => Scheme::Free,
+            }
         }
         [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));
@@ -88,14 +107,34 @@ pub(crate) fn classify(loops: &[Vec<Vec<Q>>]) -> Scheme {
     }
 }
 
+/// The curve variables of each side of a loop between its corners (positions in increasing
+/// order); the last side wraps round to the first corner.
+fn loop_sides<const N: usize>(lp: &[usize], c: [usize; N]) -> [Vec<usize>; N] {
+    std::array::from_fn(|k| {
+        let (a, b) = (c[k], c.get(k + 1).copied().unwrap_or(c[0] + lp.len()));
+        (a..b).filter_map(|i| lp.get(i % lp.len()).copied()).collect()
+    })
+}
+
 /// Add the scheme's constraints on the loops' curve variables.
 pub(crate) fn constrain(ia: &mut Intervals, loops: &[Vec<usize>], scheme: Scheme) {
     match scheme {
         Scheme::Mapped(c) => {
             let Some(lp) = loops.first() else { return };
-            let side = |a: usize, b: usize| (a..b).filter_map(|k| lp.get(k).copied()).collect::<Vec<_>>();
+            let [s0, s1, s2, s3] = loop_sides(lp, c);
-            ia.equal.push((side(c[0], c[1]), side(c[2], c[3])));
+            ia.equal.push((s0, s2));
-            ia.equal.push((side(c[1], c[2]), side(c[3], lp.len()).into_iter().chain(side(0, c[0])).collect()));
+            ia.equal.push((s1, s3));
+        }
+        Scheme::Triangle(c) => {
+            // Even, and each side at least two divisions shorter than the other two together
+            // (each line from the centre at least one division).
+            let Some(lp) = loops.first() else { return };
+            let sides = loop_sides(lp, c);
+            for k in 0..3 {
+                let others = sides[(k + 1) % 3].iter().chain(&sides[(k + 2) % 3]).copied().collect();
+                ia.exceed.push((others, sides[k].clone(), 2));
+            }
+            ia.multiple.push((lp.clone(), 2));
         }
         Scheme::Annulus => {
             if let (Some(o), Some(i)) = (loops.first(), loops.get(1)) {
@@ -134,6 +173,11 @@ pub(crate) fn mesh(loops: &[Vec<Q>], starts: &[Vec<usize>], scheme: Scheme, size
                 _ => None,
             }
         }
+        Scheme::Triangle(c) => {
+            let s0 = starts.first();
+            let pos = c.map(|k| s0.and_then(|s| s.get(k)).copied().unwrap_or(0));
+            loops.first().map(|b| triangle(b, pos, opts)).transpose()?.flatten()
+        }
         Scheme::Annulus => annulus(loops, size, opts)?,
         Scheme::Disk => disk(loops, size, opts)?,
         Scheme::Free => None,
@@ -144,6 +188,120 @@ pub(crate) fn mesh(loops: &[Vec<Q>], starts: &[Vec<usize>], scheme: Scheme, size
     }
 }
 
+/// The triangle primitive on one loop with corners at node positions `corners` (in loop order),
+/// or `None` when the side counts allow no split or a block would fold.
+fn triangle(boundary: &[Q], corners: [usize; 3], opts: &QuadOptions) -> Result<Option<QuadMesh>> {
+    let n = boundary.len();
+    if !(6..=MAX_ANNULUS_NODES).contains(&n) || corners.iter().any(|&c| c >= n) {
+        return Ok(None);
+    }
+    // Corners in loop order from the first; side k runs from corner k to corner k + 1.
+    let mut c = corners;
+    c.sort_unstable();
+    let len = |k: usize| -> usize {
+        let (a, b) = (c[k % 3], c[(k + 1) % 3]);
+        (b + n - a) % n
+    };
+    let sides = [len(0), len(1), len(2)];
+    let total: usize = sides.iter().sum();
+    if total != n || !total.is_multiple_of(2) {
+        return Ok(None);
+    }
+    // m[k]: divisions of the line from the centre to side k's split node.
+    let mut m = [0usize; 3];
+    for k in 0..3 {
+        let other = sides[(k + 1) % 3] + sides[(k + 2) % 3];
+        if other <= sides[k] {
+            return Ok(None);
+        }
+        m[k] = (other - sides[k]) / 2;
+    }
+    let count = (0..3).map(|k| m[k] * m[(k + 1) % 3]).sum::<usize>();
+    if count > opts.max_elements {
+        return Err(Error::Limit(format!("the mesh would have {count} quadrilaterals; the limit is {}", opts.max_elements)));
+    }
+    let at = |i: usize| boundary.get(i % n).copied().unwrap_or([0.0; 2]);
+    // Side k is split after m[k + 2] divisions (the line count of the block at corner k).
+    let split: [usize; 3] = std::array::from_fn(|k| (c[k] + m[(k + 2) % 3]) % n);
+    let sp: [Q; 3] = split.map(at);
+    let centre = [(sp[0][0] + sp[1][0] + sp[2][0]) / 3.0, (sp[0][1] + sp[1][1] + sp[2][1]) / 3.0];
+    let mut nodes: Vec<Q> = boundary.to_vec();
+    nodes.push(centre);
+    let centre_id = n as u32;
+    // Line k from the split node of side k to the centre, its interior nodes new.
+    let mut lines: Vec<Vec<u32>> = Vec::with_capacity(3);
+    for k in 0..3 {
+        let mut ids = vec![split[k] as u32];
+        for j in 1..m[k] {
+            let t = j as f64 / m[k] as f64;
+            nodes.push([sp[k][0] + t * (centre[0] - sp[k][0]), sp[k][1] + t * (centre[1] - sp[k][1])]);
+            ids.push(nodes.len() as u32 - 1);
+        }
+        ids.push(centre_id);
+        lines.push(ids);
+    }
+    let ccw = signed_area(boundary) > 0.0;
+    let mut quads = Vec::with_capacity(count);
+    // Block k at corner k: bottom corner k → split k (side k), right line k, top line k − 1
+    // (split k − 1 → centre), left corner k → split k − 1 (side k − 1, backwards).
+    for k in 0..3 {
+        let prev = (k + 2) % 3;
+        let (nu, nv) = (m[prev], m[k]);
+        let bottom: Vec<u32> = (0..=nu).map(|j| ((c[k] + j) % n) as u32).collect();
+        let left: Vec<u32> = (0..=nv).map(|j| ((c[k] + n - j) % n) as u32).collect();
+        let (Some(right), Some(top)) = (lines.get(k), lines.get(prev)) else { return Ok(None) };
+        if !fill_block(&mut nodes, [&bottom, right, top, &left], !ccw, &mut quads)? {
+            return Ok(None);
+        }
+    }
+    // The centre and the lines from it are placed by construction only: smooth them, as the
+    // midpoint-subdivision papers do, keeping every quadrilateral valid.
+    crate::quad::smooth::smooth(&mut nodes, &quads, n, TRIANGLE_SWEEPS)?;
+    Ok(Some(QuadMesh { nodes, quads, boundary_nodes: n, stats: QuadStats::default() }))
+}
+
+/// Fill the block bounded by node sequences `bottom` and `top` (left to right) and `left` and
+/// `right` (bottom to top) with a transfinite grid, pushing its interior nodes and its
+/// quadrilaterals (reversed when `flip`). `false` when a quadrilateral would not be valid.
+fn fill_block(nodes: &mut Vec<Q>, [bottom, right, top, left]: [&[u32]; 4], flip: bool, quads: &mut Vec<[u32; 4]>) -> Result<bool> {
+    let (nu, nv) = (bottom.len().saturating_sub(1), left.len().saturating_sub(1));
+    let at = |nodes: &[Q], i: u32| nodes.get(i as usize).copied().unwrap_or([0.0; 2]);
+    let pts = |ids: &[u32], nodes: &[Q]| ids.iter().map(|&i| at(nodes, i)).collect::<Vec<Q>>();
+    let grid = transfinite_grid(&pts(bottom, nodes), &pts(right, nodes), &pts(top, nodes), &pts(left, nodes))?;
+    let mut id = vec![0u32; (nu + 1) * (nv + 1)];
+    for j in 0..=nv {
+        for i in 0..=nu {
+            let v = if j == 0 {
+                bottom.get(i).copied()
+            } else if j == nv {
+                top.get(i).copied()
+            } else if i == 0 {
+                left.get(j).copied()
+            } else if i == nu {
+                right.get(j).copied()
+            } else {
+                nodes.push(grid.get(j * (nu + 1) + i).copied().unwrap_or([0.0; 2]));
+                Some(nodes.len() as u32 - 1)
+            };
+            if let (Some(slot), Some(v)) = (id.get_mut(j * (nu + 1) + i), v) {
+                *slot = v;
+            }
+        }
+    }
+    let g = |i: usize, j: usize| id.get(j * (nu + 1) + i).copied().unwrap_or(0);
+    for j in 0..nv {
+        for i in 0..nu {
+            let q = [g(i, j), g(i + 1, j), g(i + 1, j + 1), g(i, j + 1)];
+            let q = if flip { [q[0], q[3], q[2], q[1]] } else { q };
+            if planar_quad_validity(&q.map(|v| at(nodes, v)))? != Validity::Valid {
+                return Ok(false);
+            }
+            quads.push(q);
+        }
+    }
+    Ok(true)
+}
+
 /// The O-grid between two loops of equal length, or `None` when it would fold.
 fn annulus(loops: &[Vec<Q>], size: &dyn SizeField, opts: &QuadOptions) -> Result<Option<QuadMesh>> {
     let [outer, inner] = loops else { return Ok(None) };
@@ -284,37 +442,8 @@ fn disk(loops: &[Vec<Q>], size: &dyn SizeField, opts: &QuadOptions) -> Result<Op
         v
     };
     let (bottom, right, top, left) = (side(0, false), side(1, false), side(2, true), side(3, true));
-    let pts = |ids: &[u32], nodes: &[Q]| ids.iter().map(|&i| at(nodes, i)).collect::<Vec<Q>>();
+    if !fill_block(&mut nodes, [&bottom, &right, &top, &left], false, &mut quads)? {
-    let grid = transfinite_grid(&pts(&bottom, &nodes), &pts(&right, &nodes), &pts(&top, &nodes), &pts(&left, &nodes))?;
+        return Ok(None);
-    let mut id = vec![0u32; (q + 1) * (q + 1)];
-    for j in 0..=q {
-        for i in 0..=q {
-            let v = if j == 0 {
-                bottom.get(i).copied()
-            } else if j == q {
-                top.get(i).copied()
-            } else if i == 0 {
-                left.get(j).copied()
-            } else if i == q {
-                right.get(j).copied()
-            } else {
-                nodes.push(grid.get(j * (q + 1) + i).copied().unwrap_or([0.0; 2]));
-                Some(nodes.len() as u32 - 1)
-            };
-            if let (Some(slot), Some(v)) = (id.get_mut(j * (q + 1) + i), v) {
-                *slot = v;
-            }
-        }
-    }
-    let g = |i: usize, j: usize| id.get(j * (q + 1) + i).copied().unwrap_or(0);
-    for j in 0..q {
-        for i in 0..q {
-            let quad = [g(i, j), g(i + 1, j), g(i + 1, j + 1), g(i, j + 1)];
-            if planar_quad_validity(&quad.map(|v| at(&nodes, v)))? != Validity::Valid {
-                return Ok(None);
-            }
-            quads.push(quad);
-        }
     }
     Ok(Some(QuadMesh { nodes, quads, boundary_nodes: n, stats: QuadStats::default() }))
 }
@@ -384,4 +513,36 @@ mod tests {
         ];
         assert_eq!(classify(&[l]), Scheme::Free);
     }
+
+    /// A fillet's quarter disk (two straight sides and an arc, three corners) gets the triangle
+    /// primitive: three mapped blocks around a centre, every quadrilateral convex, the boundary
+    /// nodes kept and the area filled.
+    #[test]
+    fn quarter_disks_get_the_triangle_primitive() {
+        // Corners at the origin, (5, 0) and (0, 5); sides of 4, 6 and 4 divisions.
+        let mut pts: Vec<Q> = (0..4).map(|k| [5.0 * k as f64 / 4.0, 0.0]).collect();
+        pts.extend((0..6).map(|k| {
+            let t = 0.5 * PI * k as f64 / 6.0;
+            [5.0 * t.cos(), 5.0 * t.sin()]
+        }));
+        pts.extend((0..4).map(|k| [0.0, 5.0 - 5.0 * k as f64 / 4.0]));
+        let polyline = |a: usize, b: usize| (a..=b).map(|k| pts[k % pts.len()]).collect::<Vec<Q>>();
+        let curves = vec![polyline(0, 4), polyline(4, 10), polyline(10, 14)];
+        assert_eq!(classify(&[curves]), Scheme::Triangle([0, 1, 2]));
+        let (m, s) =
+            mesh(std::slice::from_ref(&pts), &[vec![0, 4, 10]], Scheme::Triangle([0, 1, 2]), &UniformSize(1.0), &QuadOptions::default()).unwrap();
+        assert_eq!(s, Scheme::Triangle([0, 1, 2]));
+        assert_eq!(&m.nodes[..14], &pts[..]);
+        // Lines of (6 + 4 − 4) / 2 = 3, (4 + 4 − 6) / 2 = 1 and 3 divisions: 3·1 + 1·3 + 3·3 quads.
+        assert_eq!(m.quads.len(), 15);
+        for q in &m.quads {
+            assert_eq!(planar_quad_validity(&q.map(|i| m.nodes[i as usize])).unwrap(), Validity::Valid);
+        }
+        let area: f64 = m.quads.iter().map(|q| signed_area(&q.map(|i| m.nodes[i as usize]))).sum();
+        assert!((area - signed_area(&pts)).abs() < 1e-9, "{area}");
+        // A side as long as the other two together has no split: the free mesher takes it.
+        let (_, s) =
+            mesh(std::slice::from_ref(&pts), &[vec![0, 1, 2]], Scheme::Triangle([0, 1, 2]), &UniformSize(1.0), &QuadOptions::default()).unwrap();
+        assert_eq!(s, Scheme::Free);
+    }
 }

crates/mesh/src/hex/sweep.rs

@@ -331,13 +331,22 @@ fn build(ctx: &Ctx, topo: &Topology, refine: u32) -> Result<Raw> {
         })
         .collect();
     // The rings may start mid-loop: give the mapped scheme its corners as ring positions.
+    // Corners (curve uses) as ring positions, in order; `None` unless every one is found.
+    let positions = |c: &[usize]| {
+        let s0 = starts.first().cloned().unwrap_or_default();
+        let mut pos: Vec<usize> = c.iter().filter_map(|&k| s0.get(k).copied()).collect();
+        pos.sort_unstable();
+        (pos.len() == c.len()).then_some(pos)
+    };
     let (scheme, starts) = match scheme {
-        region::Scheme::Mapped(c) => {
+        region::Scheme::Mapped(c) => match positions(&c) {
-            let s0 = starts.first().cloned().unwrap_or_default();
+            Some(pos) => (region::Scheme::Mapped([0, 1, 2, 3]), vec![pos]),
-            let mut pos: Vec<usize> = c.iter().filter_map(|&k| s0.get(k).copied()).collect();
+            None => (region::Scheme::Free, starts),
-            pos.sort_unstable();
+        },
-            if pos.len() == 4 { (region::Scheme::Mapped([0, 1, 2, 3]), vec![pos]) } else { (region::Scheme::Free, starts) }
+        region::Scheme::Triangle(c) => match positions(&c) {
-        }
+            Some(pos) => (region::Scheme::Triangle([0, 1, 2]), vec![pos]),
+            None => (region::Scheme::Free, starts),
+        },
         other => (other, starts),
     };
     let (mesh, used_scheme) = region::mesh(&loops2, &starts, scheme, &plane_field, &ctx.opts.quad)?;
@@ -400,6 +409,7 @@ fn build(ctx: &Ctx, topo: &Topology, refine: u32) -> Result<Raw> {
         mesh.quads.len(),
         match used_scheme {
             region::Scheme::Mapped(_) => "mapped",
+            region::Scheme::Triangle(_) => "triangle",
             region::Scheme::Annulus => "annulus",
             region::Scheme::Disk => "disk",
             region::Scheme::Free => "free",

crates/mesh/src/quad/mod.rs

@@ -71,7 +71,7 @@ mod cdt;
 mod domain;
 mod free;
 mod guard;
-mod smooth;
+pub(crate) mod smooth;
 pub mod tfi;
 
 pub use domain::{MAX_BOUNDARY_POINTS, MAX_LOOPS};

docs/meshing/literature.md

@@ -991,7 +991,8 @@ record or the publisher; the others are cited by venue only.
   10(4):399–415, 2000). S. A. Mitchell. *Incremental interval assignment by integer linear algebra.*
   International Meshing Roundtable 2021 (Zenodo 5559025); extended in Computer-Aided Design 158:103485, 2023.
 - **Use.** Division counts of all curves together: side equalities of mapped faces and linking faces,
-  even (or multiple-of-four) loop sums, minimum loop sums. `hex::interval` uses incremental repair from the
+  even (or multiple-of-four) loop sums, minimum loop sums, and dominance (`Σ_A ≥ Σ_B + d`: a side of a
+  triangle primitive shorter than the other two together). `hex::interval` uses incremental repair from the
   rounded goals with increments only, chosen by squared relative error; it is bounded and reports an
   infeasible system instead of looping. Mitchell's integer-linear-algebra solver (Hermite normal form plus
   nullspace improvement) is the upgrade if the increment-only scheme proves too coarse.
@@ -1003,6 +1004,19 @@ record or the publisher; the others are cited by venue only.
   their reflex corners to the boundary, which decomposes rectilinear sections into rectangles (an explicit
   form of submapping's virtual decomposition), then mapped.
 
+**B8.6a Midpoint subdivision: Tam and Armstrong 1991; Li, Armstrong, McKeag 1997.** **Verdict: BUILD ON
+(triangle primitive).**
+- **Citations.** T. K. H. Tam, C. G. Armstrong. *2D finite element mesh generation by medial axis
+  subdivision.* Advances in Engineering Software 13(5–6):313–324, 1991. T. S. Li, C. G. Armstrong,
+  R. M. McKeag. *Quad mesh generation for k-sided faces and hex mesh generation for trivalent polyhedra.*
+  Finite Elements in Analysis and Design 26(4):279–301, 1997.
+- **Use.** A cross-section region with three corners (a fillet's quarter disk, a pocket corner cut off by
+  a partition) is split by lines from a centre to one node on each side into three mapped blocks
+  (`hex::region`, scheme `Triangle`): side `k` of `n_k` divisions splits into the counts of the two lines
+  that end beside it, so each line has `m_k = (n_{k+1} + n_{k+2} − n_k)/2` (an even loop, each side
+  shorter than the other two together). The centre is the mean of the split nodes; the interior is then
+  smoothed (`quad::smooth`) and every quadrilateral checked; a fold falls back to the free mesher.
+
 **B8.7 Gordon and Thiel 1982, trivariate transfinite interpolation.** **Verdict: BUILD ON (mapped blocks).**
 - **Citation.** W. J. Gordon, L. C. Thiel. *Transfinite mappings and their application to grid generation.*
   Applied Mathematics and Computation 10–11:171–233, 1982.