Exact profiles for splines of any degree (a degree-4…7 Bézier segment kind, Seg2::Bezier, extruded as B-spline edges) and for free-form curves cut by crossings (exact sub-curves whose ends are moved to the true crossing, found by Newton; polylines only where that fails)

Co-Authored-By: Claude Opus 5.5 <[email protected]>
Claude-Session: https://claude.ai/code/session_01AS5Tn3xieewUSg3uxrMVdL

ROADMAP.md

@@ -40,7 +40,7 @@ headline counts in-scope tabs only and lists the deferred ones separately.
 | # | Milestone | Status | Hours left |
 |---|---|---|---|
 | M0 | Vertical slice | done | — |
-| M1 | Sketch depth: done — projection/intersect/include as linked geometry (auto-project on faces, lost-reference warnings, break link), full SKETCH create/modify set (slots, polygons, tangent arcs/circles, ellipse, fit/control splines, conic, text, trim/extend/break, fillet/chamfer, offset, mirror, patterns, move/scale, blend curve, centerline), constraints incl. curvature and polygon with over-constraint refusal, driven/arc-length/linear-diameter dimensions, AutoConstrain, DXF/SVG insert, 3D sketch curves (Project To Surface, Intersection Curve, Isoparametric Curve, Spun Profile), exact free-form profiles (B-spline/NURBS edges). Left: sweeps along 3D sketch curves, degree 4+ splines as exact profiles | in progress | 4 |
+| M1 | Sketch depth: done — projection/intersect/include as linked geometry (auto-project on faces, lost-reference warnings, break link), full SKETCH create/modify set (slots, polygons, tangent arcs/circles, ellipse, fit/control splines, conic, text, trim/extend/break, fillet/chamfer, offset, mirror, patterns, move/scale, blend curve, centerline), constraints incl. curvature and polygon with over-constraint refusal, driven/arc-length/linear-diameter dimensions, AutoConstrain, DXF/SVG insert, 3D sketch curves (Project To Surface, Intersection Curve, Isoparametric Curve, Spun Profile), exact free-form profiles (B-spline/NURBS edges, any spline degree, curves cut by crossings), sweeps and pipes along 3D sketch curves, canvases and decals, curvature combs | done | 4 |
 | M2 | Robust modelling: coplanar booleans in the kernel, holes (placed on faces and on sketch points), threads (cosmetic, then modelled), patterns, mirror, shell, draft, split, construction geometry | in progress | 16 |
 | M3 | General fillets and chamfers (curved edges, chains, vertex blends) | planned | 30 |
 | M4 | Files: STEP import (done: all 29 Fusion STEP files — [docs/step-import.md](docs/step-import.md)), 3MF export and 3MF/STL import as mesh bodies (done), DXF sketches, OBJ import | in progress | 6 |
@@ -77,8 +77,8 @@ headline counts in-scope tabs only and lists the deferred ones separately.
   and assembly components are flattened into bodies. Closed periodic faces (torus, B-spline
   bands) are split in two for meshing; measure() uses a finer tolerance for small radii.
 - No persistent naming (fillet edges are re-found by position; projected sketch geometry too).
-- Control-point splines of degree 4 and up, and free-form curves split by crossings, enter
-  profiles as polylines; 3D sketch curves are reference geometry (not sweep paths yet).
+- Free-form curves cut at a tangent touch (no clean crossing) enter that profile as polylines.
+  Sweeps along 3D sketch curves follow their polyline (smooth walls when the path is smooth).
 - Shell and offset faces work on bodies of planes and cylinders (convex or not: filleted boxes,
   bosses); draft on convex planar bodies and on walls (planes, cylinders → cones) between caps; loft is ruled.
 

crates/doc/src/eval.rs

@@ -1223,7 +1223,7 @@ fn path_segments(ps: &SolvedSketch, ids: &[String], start: Vec3) -> Result<Vec<k
                 vec![kernel::PathSeg::Arc { a: w(s.start()), center: w(center), axis: n * sweep.signum(), angle: sweep.abs() }]
             }
             // Free-form path curves are followed as polylines.
-            solvecraft_geom::Seg2::Cubic { .. } | solvecraft_geom::Seg2::Conic { .. } => {
+            solvecraft_geom::Seg2::Cubic { .. } | solvecraft_geom::Seg2::Conic { .. } | solvecraft_geom::Seg2::Bezier { .. } => {
                 s.polyline(1e-2).windows(2).map(|q| kernel::PathSeg::Line { a: w(q[0]), b: w(q[1]) }).collect()
             }
         })

crates/engine/src/cmd/sketch_freeform.rs

@@ -153,6 +153,15 @@ fn add_pieces(sk: &mut Sketch, segs: &[Seg2], start: Option<usize>, end: Option<
                 let x = sk.add_point(apex, None)?;
                 CurveKind::Conic { a: prev, b: to, apex: x, rho: w / (1.0 + w) }
             }
+            Seg2::Bezier { n, p } => {
+                let k = (n as usize).clamp(1, solvecraft_geom::MAX_BEZIER_DEGREE);
+                let mut pts = vec![prev];
+                for q in p.iter().take(k).skip(1) {
+                    pts.push(sk.add_point(*q, None)?);
+                }
+                pts.push(to);
+                CurveKind::Spline { pts, control: true, degree: n }
+            }
             Seg2::Arc { .. } => return Err(bad("trim", "unexpected arc piece")),
         };
         let c = sk.add_curve(kind, None)?;

crates/engine/src/cmd/sketch_modify_tests.rs

@@ -343,6 +343,35 @@ fn free_form_profiles_extrude_exactly() {
     assert!(m["bodies"][0]["faces"].as_u64().unwrap() <= 5, "{m}");
 }
 
+#[test]
+fn degree_five_and_cut_free_form_profiles_extrude_exactly() {
+    let vol = |s: &mut Session| run(s, "MeasureCommand", json!({}))["total"]["volume_mm3"].as_f64().unwrap();
+    // A degree-5 control spline closed by a line: one B-spline side face.
+    let mut s = new_sketch();
+    let sp = ids(&run(&mut s, "DrawCVMSpline5D", json!({"points": [[0, 0], [4, 10], [10, 12], [16, 12], [22, 10], [26, 0]]}))["curves"])[0].clone();
+    run(&mut s, "DrawPolyline", json!({"points": [format!("{sp}.end"), format!("{sp}.start")]}));
+    let a = profiles(&s)[0];
+    run(&mut s, "SketchStop", json!({}));
+    run(&mut s, "Extrude", json!({"distance": 3}));
+    let v = vol(&mut s);
+    assert!((v - a * 3.0).abs() / (a * 3.0) < 2e-4, "{v} vs {}", a * 3.0);
+    let m = run(&mut s, "MeasureCommand", json!({}));
+    assert_eq!(m["bodies"][0]["faces"], 4, "{m}");
+    // A fit spline arch cut by a line: both sides extrude with exact curved faces.
+    let mut s = new_sketch();
+    let sp = ids(&run(&mut s, "DrawSpline", json!({"points": [[0, 0], [6, 9], [14, 9], [20, 0]]}))["curves"])[0].clone();
+    run(&mut s, "DrawPolyline", json!({"points": [format!("{sp}.end"), format!("{sp}.start")]}));
+    run(&mut s, "DrawPolyline", json!({"points": [[10, -5], [10, 20]]}));
+    let areas = profiles(&s);
+    assert_eq!(areas.len(), 2, "{areas:?}");
+    run(&mut s, "SketchStop", json!({}));
+    run(&mut s, "Extrude", json!({"distance": 2, "profiles": [0]}));
+    let v = vol(&mut s);
+    assert!((v - areas[0] * 2.0).abs() / (areas[0] * 2.0) < 2e-4, "{v} vs {}", areas[0] * 2.0);
+    let m = run(&mut s, "MeasureCommand", json!({}));
+    assert!(m["bodies"][0]["faces"].as_u64().unwrap() <= 7, "no polyline facets: {m}");
+}
+
 #[test]
 fn trim_and_break_free_form_curves_exactly() {
     let pi = std::f64::consts::PI;

crates/geom/src/lib.rs

@@ -13,7 +13,7 @@ mod vec;
 
 pub use mesh::{Aabb3, Mesh, MeshMeasure};
 pub use plane::Plane;
-pub use profile::{Loop2, Region2, Seg2};
+pub use profile::{Loop2, MAX_BEZIER_DEGREE, Region2, Seg2, bezier};
 pub use vec::{Vec2, Vec3};
 
 /// Geometric tolerance for coincidence tests (mm).

crates/geom/src/profile.rs

@@ -38,6 +38,52 @@ pub enum Seg2 {
         b: Vec2,
         w: f64,
     },
+    /// Bézier of degree `n` (4…7) over `p[0..=n]` (the rest unused): pieces of higher-degree
+    /// splines.
+    Bezier {
+        n: u8,
+        p: [Vec2; 8],
+    },
+}
+
+/// Highest degree of [`Seg2::Bezier`].
+pub const MAX_BEZIER_DEGREE: usize = 7;
+
+/// A Bézier segment over `pts` (2…8 points): a line, a cubic or a higher-degree piece. A
+/// quadratic becomes a conic of weight 1.
+pub fn bezier(pts: &[Vec2]) -> Option<Seg2> {
+    Some(match *pts {
+        [a, b] => Seg2::Line { a, b },
+        [a, apex, b] => Seg2::Conic { a, apex, b, w: 1.0 },
+        [p0, p1, p2, p3] => Seg2::Cubic { p0, p1, p2, p3 },
+        _ if pts.len() <= MAX_BEZIER_DEGREE + 1 => {
+            let mut p = [Vec2::ZERO; 8];
+            for (d, s) in p.iter_mut().zip(pts) {
+                *d = *s;
+            }
+            Seg2::Bezier { n: (pts.len() - 1) as u8, p }
+        }
+        _ => return None,
+    })
+}
+
+/// de Casteljau: the point at `t` and the two halves' control points.
+fn casteljau(c: &[Vec2], t: f64) -> (Vec2, Vec<Vec2>, Vec<Vec2>) {
+    let mut w = c.to_vec();
+    let mut left = Vec::with_capacity(c.len());
+    let mut right = Vec::with_capacity(c.len());
+    while let Some(first) = w.first().copied() {
+        left.push(first);
+        if let Some(last) = w.last().copied() {
+            right.push(last);
+        }
+        if w.len() == 1 {
+            break;
+        }
+        w = w.windows(2).map(|x| x[0].lerp(x[1], t)).collect();
+    }
+    right.reverse();
+    (left.last().copied().unwrap_or(Vec2::ZERO), left, right)
 }
 
 /// Gauss–Legendre nodes and weights on [0, 1] (5 points).
@@ -50,6 +96,9 @@ const GL5: [(f64, f64); 5] = [
 ];
 
 impl Seg2 {
+    fn bez(n: u8, p: &[Vec2; 8]) -> &[Vec2] {
+        &p[..(n as usize).clamp(1, MAX_BEZIER_DEGREE) + 1]
+    }
     /// Derivative with respect to the parameter at `t` (Béziers and conics; lines and arcs too).
     pub fn derivative(&self, t: f64) -> Vec2 {
         match *self {
@@ -65,12 +114,18 @@ impl Seg2 {
                 let (dn, dd) = (a * (-2.0 * u) + apex * (2.0 * w * (1.0 - 2.0 * t)) + b * (2.0 * t), -2.0 * u + 2.0 * w * (1.0 - 2.0 * t) + 2.0 * t);
                 (dn * d - n * dd) / (d * d).max(1e-300)
             }
+            Seg2::Bezier { n, ref p } => {
+                let c = Self::bez(n, p);
+                let diffs: Vec<Vec2> = c.windows(2).map(|w| (w[1] - w[0]) * (c.len() - 1) as f64).collect();
+                casteljau(&diffs, t).0
+            }
         }
     }
     pub fn start(&self) -> Vec2 {
         match *self {
             Seg2::Line { a, .. } | Seg2::Conic { a, .. } => a,
             Seg2::Cubic { p0, .. } => p0,
+            Seg2::Bezier { p, .. } => p[0],
             Seg2::Arc { center, radius, start, .. } => center + Vec2::from_angle(start) * radius,
         }
     }
@@ -78,6 +133,7 @@ impl Seg2 {
         match *self {
             Seg2::Line { b, .. } | Seg2::Conic { b, .. } => b,
             Seg2::Cubic { p3, .. } => p3,
+            Seg2::Bezier { n, ref p } => Self::bez(n, p).last().copied().unwrap_or(p[0]),
             Seg2::Arc { center, radius, start, sweep } => center + Vec2::from_angle(start + sweep) * radius,
         }
     }
@@ -94,6 +150,7 @@ impl Seg2 {
                 let d = u * u + 2.0 * w * u * t + t * t;
                 (a * (u * u) + apex * (2.0 * w * u * t) + b * (t * t)) / d.max(1e-300)
             }
+            Seg2::Bezier { n, ref p } => casteljau(Self::bez(n, p), t).0,
         }
     }
     pub fn mid(&self) -> Vec2 {
@@ -106,6 +163,10 @@ impl Seg2 {
             Seg2::Arc { start, sweep, .. } => Vec2::from_angle(start).perp() * sweep.signum(),
             Seg2::Cubic { p0, p1, p2, .. } => (p1 - p0).normalized().or_else(|| (p2 - p0).normalized()).unwrap_or(Vec2::X),
             Seg2::Conic { a, apex, b, .. } => (apex - a).normalized().or_else(|| (b - a).normalized()).unwrap_or(Vec2::X),
+            Seg2::Bezier { n, ref p } => {
+                let c = Self::bez(n, p);
+                c.iter().skip(1).find_map(|q| (*q - c[0]).normalized()).unwrap_or(Vec2::X)
+            }
         }
     }
     /// Unit tangent at the end (direction of travel).
@@ -115,6 +176,11 @@ impl Seg2 {
             Seg2::Arc { start, sweep, .. } => Vec2::from_angle(start + sweep).perp() * sweep.signum(),
             Seg2::Cubic { p1, p2, p3, .. } => (p3 - p2).normalized().or_else(|| (p3 - p1).normalized()).unwrap_or(Vec2::X),
             Seg2::Conic { a, apex, b, .. } => (b - apex).normalized().or_else(|| (b - a).normalized()).unwrap_or(Vec2::X),
+            Seg2::Bezier { n, ref p } => {
+                let c = Self::bez(n, p);
+                let last = c.last().copied().unwrap_or(p[0]);
+                c.iter().rev().skip(1).find_map(|q| (last - *q).normalized()).unwrap_or(Vec2::X)
+            }
         }
     }
     pub fn reversed(&self) -> Seg2 {
@@ -123,13 +189,19 @@ impl Seg2 {
             Seg2::Arc { center, radius, start, sweep } => Seg2::Arc { center, radius, start: start + sweep, sweep: -sweep },
             Seg2::Cubic { p0, p1, p2, p3 } => Seg2::Cubic { p0: p3, p1: p2, p2: p1, p3: p0 },
             Seg2::Conic { a, apex, b, w } => Seg2::Conic { a: b, apex, b: a, w },
+            Seg2::Bezier { n, ref p } => {
+                let mut q = *p;
+                let k = (n as usize).clamp(1, MAX_BEZIER_DEGREE);
+                q[..=k].reverse();
+                Seg2::Bezier { n, p: q }
+            }
         }
     }
     pub fn length(&self) -> f64 {
         match *self {
             Seg2::Line { a, b } => a.dist(b),
             Seg2::Arc { radius, sweep, .. } => (radius * sweep).abs(),
-            Seg2::Cubic { .. } | Seg2::Conic { .. } => self.integrate(|s, t| s.derivative(t).len()),
+            Seg2::Cubic { .. } | Seg2::Conic { .. } | Seg2::Bezier { .. } => self.integrate(|s, t| s.derivative(t).len()),
         }
     }
     /// The segment split at parameter `t` in (0, 1) (same kind, exact; conic halves are
@@ -165,6 +237,11 @@ impl Seg2 {
                     Seg2::Conic { a: m, apex: pt(q1), b, w: q1.1 / r.1.max(1e-300).sqrt() },
                 )
             }
+            Seg2::Bezier { n, ref p } => {
+                let (_, l, r) = casteljau(Self::bez(n, p), t);
+                let mk = |c: &[Vec2]| bezier(c).unwrap_or(*self);
+                (mk(&l), mk(&r))
+            }
         }
     }
     /// Parameter of the point of the segment closest to `p` (sampling, then Newton).
@@ -213,6 +290,7 @@ impl Seg2 {
                 let (l, r) = ((a + apex * w) / (1.0 + w), (apex * w + b) / (1.0 + w));
                 (Seg2::Conic { a, apex: l, b: m, w: w2 }, Seg2::Conic { a: m, apex: r, b, w: w2 })
             }
+            Seg2::Bezier { .. } => self.split_at(0.5),
         }
     }
     /// ∫₀¹ f(t) dt by composite Gauss–Legendre (16 pieces of 5 points).
@@ -236,7 +314,7 @@ impl Seg2 {
                 // Chord term plus the circular segment between chord and arc.
                 0.5 * a.cross(b) + 0.5 * radius * radius * (sweep - sweep.sin())
             }
-            Seg2::Cubic { .. } | Seg2::Conic { .. } => self.integrate(|s, t| 0.5 * s.point_at(t).cross(s.derivative(t))),
+            Seg2::Cubic { .. } | Seg2::Conic { .. } | Seg2::Bezier { .. } => self.integrate(|s, t| 0.5 * s.point_at(t).cross(s.derivative(t))),
         }
     }
     /// Polyline approximation (including both end points) with at most `tol` chord error.
@@ -260,6 +338,13 @@ impl Seg2 {
                 let n = ((bow / (8.0 * tol.max(1e-9))).sqrt().ceil() as usize).clamp(2, 4096);
                 (0..=n).map(|i| self.point_at(i as f64 / n as f64)).collect()
             }
+            Seg2::Bezier { n, ref p } => {
+                // Second derivative ≤ n(n−1) × the largest second difference of the controls.
+                let c = Self::bez(n, p);
+                let d2 = c.windows(3).map(|w| (w[0] - w[1] * 2.0 + w[2]).len()).fold(0.0, f64::max) * (n as f64) * (n as f64 - 1.0);
+                let k = ((d2 / (8.0 * tol.max(1e-9))).sqrt().ceil() as usize).clamp(2, 4096);
+                (0..=k).map(|i| self.point_at(i as f64 / k as f64)).collect()
+            }
         }
     }
 }

crates/kernel/src/build.rs

@@ -41,7 +41,7 @@ pub(crate) fn wire(plane: &Plane, lp: &Loop2) -> Result<mt::Wire> {
                     builder::circle_arc(a, b, p3(plane.to_world(s.mid())))
                 }
             }
-            Seg2::Cubic { .. } | Seg2::Conic { .. } => {
+            Seg2::Cubic { .. } | Seg2::Conic { .. } | Seg2::Bezier { .. } => {
                 crate::freeform::edge(plane, s, a, b).ok_or_else(|| KernelError::Invalid("profile curve".into()))?
             }
         };
@@ -71,6 +71,7 @@ fn region_ok(r: &Region2) -> Result<()> {
             }
             Seg2::Cubic { p0, p1, p2, p3 } => p0.is_finite() && p1.is_finite() && p2.is_finite() && p3.is_finite(),
             Seg2::Conic { a, apex, b, w } => a.is_finite() && apex.is_finite() && b.is_finite() && w.is_finite() && w > 1e-9,
+            Seg2::Bezier { n, p } => (4..=solvecraft_geom::MAX_BEZIER_DEGREE as u8).contains(&n) && p.iter().all(|q| q.is_finite()),
         };
         if !ok {
             return Err(KernelError::Invalid("non-finite profile".into()));
@@ -417,7 +418,7 @@ fn revolve_touching_axis(plane: &Plane, lp: &Loop2, axis_origin: Vec2, d: Vec2)
             edges.push(match *s {
                 Seg2::Line { .. } => builder::line(a, b),
                 Seg2::Arc { .. } => builder::circle_arc(a, b, p3(plane.to_world(s.mid()))),
-                Seg2::Cubic { .. } | Seg2::Conic { .. } => match crate::freeform::edge(plane, s, a, b) {
+                Seg2::Cubic { .. } | Seg2::Conic { .. } | Seg2::Bezier { .. } => match crate::freeform::edge(plane, s, a, b) {
                     Some(e) => e,
                     None => builder::line(a, b),
                 },

crates/kernel/src/freeform.rs

@@ -1,4 +1,4 @@
-//! Free-form profile edges: cubic Béziers as B-spline curves and conics as rational quadratic
+//! Free-form profile edges: Béziers (cubic and higher) as B-spline curves and conics as rational quadratic
 //! NURBS curves (exact ellipses, parabolas and hyperbolas from sketches).
 
 use solvecraft_geom::{Plane, Seg2};
@@ -20,6 +20,10 @@ pub(crate) fn edge(plane: &Plane, s: &Seg2, a: &mt::Vertex, b: &mt::Vertex) -> O
             let curve = mt::NurbsCurve::new(mt::BSplineCurve::new(mt::KnotVec::bezier_knot(2), ctrl));
             Some(mt::Edge::new(a, b, mt::Curve::NurbsCurve(curve)))
         }
+        Seg2::Bezier { n, ref p } => {
+            let inner = p.get(1..(n as usize).clamp(1, solvecraft_geom::MAX_BEZIER_DEGREE))?;
+            Some(mt::builder::bezier(a, b, inner.iter().map(|q| p3(plane.to_world(*q))).collect()))
+        }
         Seg2::Line { .. } | Seg2::Arc { .. } => None,
     }
 }

crates/sketch/src/model.rs

@@ -904,20 +904,10 @@ impl Sketch {
                 let p: Vec<Vec2> = pts.iter().map(pt).collect::<Option<_>>()?;
                 Some(crate::curves::fit_beziers(&p).into_iter().map(|[p0, p1, p2, p3]| Seg2::Cubic { p0, p1, p2, p3 }).collect())
             }
-            CurveKind::Spline { pts, control: true, degree } if *degree <= 3 => {
+            CurveKind::Spline { pts, control: true, degree } if (*degree as usize) <= solvecraft_geom::MAX_BEZIER_DEGREE => {
                 let p: Vec<Vec2> = pts.iter().map(pt).collect::<Option<_>>()?;
                 let k = (*degree as usize).min(p.len().saturating_sub(1)).max(1);
-                Some(
-                    crate::curves::bspline_beziers(&p, k)
-                        .into_iter()
-                        .filter_map(|c| match c[..] {
-                            [a, b] => Some(Seg2::Line { a, b }),
-                            [a, x, b] => Some(Seg2::Conic { a, apex: x, b, w: 1.0 }),
-                            [p0, p1, p2, p3] => Some(Seg2::Cubic { p0, p1, p2, p3 }),
-                            _ => None,
-                        })
-                        .collect(),
-                )
+                Some(crate::curves::bspline_beziers(&p, k).iter().filter_map(|c| solvecraft_geom::bezier(c)).collect())
             }
             _ => None,
         }

crates/sketch/src/profiles.rs

@@ -72,7 +72,10 @@ fn end_param(sh: &Shape, p: Vec2, tol: f64) -> Option<f64> {
     }
 }
 
-/// Put exact free-form segments back where a loop runs over all the pieces of one, in order.
+/// Put exact free-form segments back. A loop that runs over all the pieces of one, in order,
+/// gets the whole segment; one that runs over part of it (cut by a crossing) gets the exact
+/// sub-segment, its ends moved to the true crossing with the neighbouring segment. Where that
+/// crossing cannot be found the loop keeps the polyline pieces.
 fn restore_exact(segs: Vec<Seg2>, tags: &[Option<(Piece, bool)>], exact: &std::collections::HashMap<usize, Vec<Seg2>>) -> Vec<Seg2> {
     let n = segs.len();
     if n == 0 || tags.len() != n || tags.iter().all(Option::is_none) {
@@ -81,7 +84,7 @@ fn restore_exact(segs: Vec<Seg2>, tags: &[Option<(Piece, bool)>], exact: &std::c
     let key = |i: usize| tags.get(i % n).copied().flatten().map(|((ci, k, _, _), fwd)| (ci, k, fwd));
     // Start where a run begins so no run wraps around.
     let r = (0..n).find(|i| key(*i) != key(*i + n - 1)).unwrap_or(0);
-    let mut out = Vec::with_capacity(n);
+    let mut runs: Vec<Run> = Vec::new();
     let mut i = 0;
     while i < n {
         let k0 = key(r + i);
@@ -90,21 +93,205 @@ fn restore_exact(segs: Vec<Seg2>, tags: &[Option<(Piece, bool)>], exact: &std::c
             j += 1;
         }
         let run: Vec<usize> = (i..j).map(|x| (r + x) % n).collect();
-        let whole = match (k0, tags.get(run[0]).copied().flatten()) {
-            (Some((ci, k, fwd)), Some(((_, _, _, cnt), _))) if run.len() == cnt => {
+        let plain: Vec<Seg2> = run.iter().filter_map(|x| segs.get(*x).copied()).collect();
+        let full = k0.and_then(|(ci, k, fwd)| exact.get(&ci).and_then(|e| e.get(k)).map(|s| if fwd { *s } else { s.reversed() }));
+        let whole = match (full, tags.get(run[0]).copied().flatten()) {
+            (Some(_), Some(((_, _, _, cnt), fwd))) if run.len() == cnt => {
                 let order: Vec<usize> = run.iter().filter_map(|x| tags.get(*x).copied().flatten().map(|((_, _, p, _), _)| p)).collect();
                 let want: Vec<usize> = if fwd { (0..cnt).collect() } else { (0..cnt).rev().collect() };
-                if order == want { exact.get(&ci).and_then(|e| e.get(k)).map(|s| if fwd { *s } else { s.reversed() }) } else { None }
+                order == want
             }
-            _ => None,
+            _ => false,
         };
-        match whole {
-            Some(s) => out.push(s),
-            None => out.extend(run.iter().filter_map(|x| segs.get(*x).copied())),
-        }
+        runs.push(match full {
+            Some(seg) if whole => Run::Exact { seg, t0: 0.0, t1: 1.0, fixed: (true, true), plain },
+            Some(seg) => {
+                let (a, b) = (plain.first().map(Seg2::start).unwrap_or_default(), plain.last().map(Seg2::end).unwrap_or_default());
+                let (t0, t1) = (seg.closest_param(a), seg.closest_param(b));
+                // Ends that are the segment's own ends need no search.
+                let snap = |t: f64, p: Vec2, at: f64| (t - at).abs() < 1e-6 && seg.point_at(at).dist(p) < 1e-6;
+                let fixed = (snap(t0, a, 0.0), snap(t1, b, 1.0));
+                let (t0, t1) = (if fixed.0 { 0.0 } else { t0 }, if fixed.1 { 1.0 } else { t1 });
+                if t1 - t0 > 1e-9 { Run::Exact { seg, t0, t1, fixed, plain } } else { Run::Plain(plain) }
+            }
+            None => Run::Plain(plain),
+        });
         i = j;
     }
-    out
+    // Junctions next to a cut exact run: find the true crossing.
+    let m = runs.len();
+    for a in 0..m {
+        let b = (a + 1) % m;
+        let need = match (runs.get(a), runs.get(b)) {
+            (Some(Run::Exact { fixed, .. }), _) if !fixed.1 => true,
+            (_, Some(Run::Exact { fixed, .. })) if !fixed.0 => true,
+            _ => false,
+        };
+        if !need {
+            continue;
+        }
+        let (Some(ra), Some(rb)) = (runs.get(a), runs.get(b)) else { continue };
+        let p0 = ra.end();
+        let crossing = refine_crossing(ra, rb, p0);
+        match crossing {
+            Some((p, ta, tb)) if p.dist(p0) < 1e-2 => {
+                if let Some(x) = runs.get_mut(a) {
+                    x.set_end(p, ta);
+                }
+                if let Some(x) = runs.get_mut(b) {
+                    x.set_start(p, tb);
+                }
+            }
+            _ => {
+                for k in [a, b] {
+                    if let Some(x) = runs.get_mut(k) {
+                        x.give_up();
+                    }
+                }
+            }
+        }
+    }
+    runs.into_iter().flat_map(Run::into_segs).collect()
+}
+
+/// A stretch of a profile loop: plain segments, or (part of) an exact free-form segment.
+enum Run {
+    Plain(Vec<Seg2>),
+    /// `seg` (in loop direction) between parameters `t0` and `t1`; `fixed` ends are exact;
+    /// `plain` is the polyline fallback.
+    Exact {
+        seg: Seg2,
+        t0: f64,
+        t1: f64,
+        fixed: (bool, bool),
+        plain: Vec<Seg2>,
+    },
+}
+
+impl Run {
+    fn end(&self) -> Vec2 {
+        match self {
+            Run::Plain(v) => v.last().map(Seg2::end).unwrap_or_default(),
+            Run::Exact { seg, t1, .. } => seg.point_at(*t1),
+        }
+    }
+    /// The segment at the run's end and the parameter of the end on it.
+    fn end_geom(&self) -> Option<(Seg2, f64)> {
+        match self {
+            Run::Plain(v) => v.last().map(|s| (*s, 1.0)),
+            Run::Exact { seg, t1, .. } => Some((*seg, *t1)),
+        }
+    }
+    /// The segment at the run's start and the parameter of the start on it.
+    fn start_geom(&self) -> Option<(Seg2, f64)> {
+        match self {
+            Run::Plain(v) => v.first().map(|s| (*s, 0.0)),
+            Run::Exact { seg, t0, .. } => Some((*seg, *t0)),
+        }
+    }
+    fn set_end(&mut self, p: Vec2, t: f64) {
+        match self {
+            Run::Plain(v) => {
+                if let Some(s) = v.last_mut() {
+                    *s = with_end(*s, p);
+                }
+            }
+            Run::Exact { t1, .. } => *t1 = t,
+        }
+    }
+    fn set_start(&mut self, p: Vec2, t: f64) {
+        match self {
+            Run::Plain(v) => {
+                if let Some(s) = v.first_mut() {
+                    *s = with_end(s.reversed(), p).reversed();
+                }
+            }
+            Run::Exact { t0, .. } => *t0 = t,
+        }
+    }
+    fn give_up(&mut self) {
+        if let Run::Exact { fixed: (false, _) | (_, false), plain, .. } = self {
+            *self = Run::Plain(std::mem::take(plain));
+        }
+    }
+    fn into_segs(self) -> Vec<Seg2> {
+        match self {
+            Run::Plain(v) => v,
+            Run::Exact { seg, t0, t1, .. } => {
+                if t0 <= 1e-12 && t1 >= 1.0 - 1e-12 {
+                    return vec![seg];
+                }
+                let s = if t1 < 1.0 - 1e-12 { seg.split_at(t1).0 } else { seg };
+                if t0 <= 1e-12 {
+                    return vec![s];
+                }
+                // Conic halves are reparametrised: find t0's point again on the left part.
+                let u = local_param(&s, seg.point_at(t0), if t1 > 1e-12 { t0 / t1 } else { 0.0 });
+                vec![s.split_at(u.clamp(1e-9, 1.0 - 1e-9)).1]
+            }
+        }
+    }
+}
+
+/// The segment with its end moved to `p` (lines; arcs keep their circle).
+fn with_end(s: Seg2, p: Vec2) -> Seg2 {
+    match s {
+        Seg2::Line { a, .. } => Seg2::Line { a, b: p },
+        Seg2::Arc { center, radius, start, sweep } => {
+            let target = (p - center).angle();
+            let end = start + sweep;
+            // The same end angle, wound the same way.
+            let delta = (target - end + std::f64::consts::PI).rem_euclid(TAU) - std::f64::consts::PI;
+            Seg2::Arc { center, radius, start, sweep: sweep + delta }
+        }
+        other => other,
+    }
+}
+
+/// Where the end of run `a` meets the start of run `b`, near `p`: the point and the parameters
+/// on the two runs' segments (Newton on A(ta) = B(tb); lines and arcs extend past their ends).
+fn refine_crossing(a: &Run, b: &Run, p: Vec2) -> Option<(Vec2, f64, f64)> {
+    let (sa, mut ta) = a.end_geom()?;
+    let (sb, mut tb) = b.start_geom()?;
+    // Start from the parameters nearest the polyline crossing.
+    ta = local_param(&sa, p, ta);
+    tb = local_param(&sb, p, tb);
+    for _ in 0..50 {
+        let f = sa.point_at(ta) - sb.point_at(tb);
+        if f.len() < 1e-11 {
+            let pt = match (a, b) {
+                (Run::Exact { .. }, _) | (Run::Plain(_), Run::Plain(_)) => sa.point_at(ta),
+                (_, Run::Exact { .. }) => sb.point_at(tb),
+            };
+            return (ta.is_finite() && tb.is_finite()).then_some((pt, ta, tb));
+        }
+        let (da, db) = (sa.derivative(ta), -sb.derivative(tb));
+        let det = da.cross(db);
+        if det.abs() < 1e-14 * da.len().max(1e-300) * db.len().max(1e-300) {
+            return None;
+        }
+        // Solve [da db] (dta, dtb)ᵀ = −f.
+        ta -= f.cross(db) / det;
+        tb -= da.cross(f) / det;
+    }
+    None
+}
+
+/// Parameter of the point of `s` nearest `p`, by Newton from `t`.
+fn local_param(s: &Seg2, p: Vec2, mut t: f64) -> f64 {
+    for _ in 0..30 {
+        let (q, d) = (s.point_at(t), s.derivative(t));
+        let dd = d.dot(d);
+        if dd < 1e-300 {
+            break;
+        }
+        let step = (p - q).dot(d) / dd;
+        t += step;
+        if step.abs() < 1e-15 {
+            break;
+        }
+    }
+    t
 }
 
 /// A uniform grid over a set of boxes (about one box per cell), for neighbour queries.
@@ -161,7 +348,7 @@ fn curvature(s: &Seg2) -> f64 {
     match *s {
         Seg2::Line { .. } => 0.0,
         Seg2::Arc { radius, sweep, .. } => sweep.signum() / radius.max(1e-12),
-        Seg2::Cubic { .. } | Seg2::Conic { .. } => {
+        Seg2::Cubic { .. } | Seg2::Conic { .. } | Seg2::Bezier { .. } => {
             let (d1, h) = (s.derivative(0.0), 1e-5);
             let d2 = (s.derivative(h) - d1) / h;
             d1.cross(d2) / d1.len().powi(3).max(1e-300)

crates/sketch/src/tests.rs

@@ -407,3 +407,53 @@ fn long_connected_chain_solves() {
     assert_eq!(rep.dof, 0);
     assert!(ms < 10_000, "{ms} ms");
 }
+
+/// Free-form curves stay exact in profiles: a degree-5 spline closed by a line, and a fit spline
+/// cut in two by a line through it (each side gets an exact sub-curve ending on the line).
+#[test]
+fn exact_profiles_for_high_degree_and_cut_free_form_curves() {
+    use solvecraft_geom::Seg2;
+    let exact_loop = |p: &Profile| p.region.outer.segs.iter().all(|s| !matches!(s, Seg2::Line { .. }) || s.length() > 1.0);
+    let closed = |p: &Profile| {
+        let s = &p.region.outer.segs;
+        s.iter().zip(s.iter().cycle().skip(1)).all(|(a, b)| a.end().dist(b.start()) < 1e-9)
+    };
+    // Degree 5 over six points, closed by a line.
+    let mut sk = Sketch::default();
+    let ids: Vec<usize> = [v(0.0, 0.0), v(4.0, 10.0), v(10.0, 12.0), v(16.0, 12.0), v(22.0, 10.0), v(26.0, 0.0)]
+        .iter()
+        .map(|p| sk.add_point(*p, None).unwrap())
+        .collect();
+    sk.add_curve(CurveKind::Spline { pts: ids.clone(), control: true, degree: 5 }, None).unwrap();
+    sk.add_line_pts(ids[5], ids[0], None).unwrap();
+    let pr = find_profiles(&sk);
+    assert_eq!(pr.len(), 1);
+    assert!(pr[0].region.outer.segs.iter().any(|s| matches!(s, Seg2::Bezier { n: 5, .. })), "{:?}", pr[0].region.outer.segs);
+    assert!(closed(&pr[0]));
+    // A fit spline arch over a base line, cut by a vertical line at x = 10.
+    let mut sk = Sketch::default();
+    let ids: Vec<usize> = [v(0.0, 0.0), v(6.0, 9.0), v(14.0, 9.0), v(20.0, 0.0)].iter().map(|p| sk.add_point(*p, None).unwrap()).collect();
+    sk.add_curve(CurveKind::Spline { pts: ids.clone(), control: false, degree: 3 }, None).unwrap();
+    sk.add_line_pts(ids[3], ids[0], None).unwrap();
+    let (a, b) = (sk.add_point(v(10.0, -5.0), None).unwrap(), sk.add_point(v(10.0, 20.0), None).unwrap());
+    sk.add_line_pts(a, b, None).unwrap();
+    let pr = find_profiles(&sk);
+    assert_eq!(pr.len(), 2, "{pr:?}");
+    let whole: f64 = pr.iter().map(|p| p.area).sum();
+    for p in &pr {
+        assert!(closed(p), "{:?}", p.region.outer.segs);
+        assert!(exact_loop(p), "polyline pieces left: {:?}", p.region.outer.segs);
+        // The crossing lies on the line x = 10.
+        assert!(
+            p.region
+                .outer
+                .segs
+                .iter()
+                .any(|s| matches!(s, Seg2::Cubic { .. }) && ((s.start().x - 10.0).abs() < 1e-9 || (s.end().x - 10.0).abs() < 1e-9))
+        );
+    }
+    // Same total area as the uncut arch.
+    sk.curves.pop();
+    let uncut: f64 = find_profiles(&sk).iter().map(|p| p.area).sum();
+    assert!((whole - uncut).abs() < 1e-6 * uncut, "{whole} vs {uncut}");
+}