implicit, torus: evaluates_as's bound derived term by term; the recognised torus's sweep frame from its form (#3185, #3199) (#13743)

Moved from tangency-labs/solidcore#3224.

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

solidcore/crates/solidcore-convert/tests/flat_parity.rs

@@ -7,8 +7,8 @@
 //! kind, by what it became; a number within [`ROUNDING`] of its size (relative, no floor) is
 //! counted as rounding, with its count and the largest.
 //!
-//! On the whole corpus (1,332 bodies, at #3187): 1,256 come back bit for bit, 36 within
-//! rounding (the largest 4.9e-16 relative), and 40 differ, each in exactly the classes
+//! On the whole corpus (1,333 bodies, at #3199 with kernel case 11991 (#3231) in): 1,259 come
+//! back bit for bit, 36 within rounding (the largest 4.9e-16 relative), and 38 differ, each in exactly the classes
 //! [`pinned`] lists for it.
 #![allow(
     clippy::unwrap_used,
@@ -390,8 +390,8 @@ fn pinned() -> BTreeMap<String, BTreeSet<String>> {
             ],
         ),
         // a label its form is not proved on goes as the form alone (#2540): cylinders whose
-        // frames are about 1e-14 rad off their forms (two of them at the rounding of a derived
-        // bound, #3185's thread)
+        // frames are off their forms past the rounding `evaluates_as` derives (margins 1.23 to
+        // 2.03, tests/issue_3185_label_margins.rs)
         (
             label("cylinder -> rational_spline", &["radius"]),
             &[
@@ -401,18 +401,14 @@ fn pinned() -> BTreeMap<String, BTreeSet<String>> {
                 "fillets/2903-original-rim-round.json:b0",
             ],
         ),
-        // tori whose frame and fitted sweep are about 1e-14 rad off their forms, rightly
-        // refused (#3185)
+        // a torus label whose frame is off its form's own first sweep row past the rounding
+        // `evaluates_as` derives (margin 1.69, tests/issue_3185_label_margins.rs)
         (
             label(
                 "torus -> rational_spline",
                 &["major_radius", "minor_radius"],
             ),
-            &[
-                "sweeps/3897-sweep-ring-along-a-short-arc-joined-to-a-collar.json:b6",
-                "sweeps/4280-circular-pattern-of-sweep-cut-groove-on-hub.json:b46",
-                "sweeps/4280-circular-pattern-of-sweep-cut-groove-on-hub.json:b85",
-            ],
+            &["sweeps/4280-circular-pattern-of-sweep-cut-groove-on-hub.json:b46"],
         ),
         // an extruded circle the flat hop left a spline, proved a cylinder
         (

solidcore/crates/solidcore-convert/tests/issue_3185_label_margins.rs

@@ -0,0 +1,258 @@
+//! (#3185) An analytic label is kept on its NURBS form when the kernel's analytic surface,
+//! carrying the label's numbers, evaluates as the form within the rounding
+//! `AnalyticSupport::build` derives term by term (`surface::implicit`'s `evaluates_as`):
+//! [`AnalyticSupport::parametrisation_margin`] is the largest ratio of distance to bound over
+//! the form's Greville points, at most 1 where the label stands.
+//!
+//! - The corpus labels #3137's parity run refused under the old constant `gamma(16 (p + q +
+//!   2)) size`, each pinned to its margin as measured under the derived bound: the three tori
+//!   of b6 and b85 (0.02 to 0.50) stand; b46's torus (1.69) and the four cylinders (b94 1.23,
+//!   b43 1.40, the 2903 rims 1.52 and 2.03) do not: their frames are off their forms past
+//!   the rounding at their size. Since #3199 the kernel's recognised sweep frame is the form's
+//!   own (its end meridians), so a label is measured against the form, not against a fit
+//!   that shared the label's error.
+//! - A torus the kernel makes at the groove's sizes stands on its own form, and the same label
+//!   turned about its axis by four times what the bound allows is refused.
+#![allow(
+    clippy::unwrap_used,
+    clippy::expect_used,
+    clippy::panic,
+    clippy::indexing_slicing,
+    clippy::print_stderr,
+    clippy::disallowed_methods
+)] // test (the turn's cos and sin)
+
+use std::fmt::Write as _;
+use std::io::Read as _;
+use std::path::{Path, PathBuf};
+
+use serde_json::Value;
+use solidcore::blend::fillet_section::station::as_b_spline;
+use solidcore::geom::axis2_placement::Axis2Placement;
+use solidcore::geom::extent2d::Extent2d;
+use solidcore::geom::vector3d::{Point3, Vec3};
+use solidcore::kernel::{Kernel, KernelOptions, UpstreamFixes};
+use solidcore::surface::from_parameters::from_parameters;
+use solidcore::surface::implicit::AnalyticSupport;
+use solidcore::surface::surface::{Surface, SurfaceKind};
+use solidcore::surface::torus::{Torus, TorusOptions};
+use solidcore_record::brep::Brep as Record;
+
+fn corpus() -> PathBuf {
+    Path::new(env!("CARGO_MANIFEST_DIR")).join("../solidcore/tests/kernel_cases")
+}
+
+/// The corpus's kernel, its fixes on (as the parity run and SolidCraft run it).
+fn kernel() -> Kernel {
+    let v: Value =
+        serde_json::from_str(&std::fs::read_to_string(corpus().join("fixes.json")).unwrap())
+            .unwrap();
+    let mut fixes = UpstreamFixes::default();
+    for flag in v.as_array().unwrap() {
+        *fixes.flag_mut(flag.as_str().unwrap()).unwrap() = true;
+    }
+    let mut k = Kernel::new(&KernelOptions::default());
+    k.fixes = fixes;
+    k
+}
+
+fn body(case: &str, id: &str) -> Record {
+    let gz = corpus().join(format!("{case}.bodies.json.gz"));
+    let mut text = String::new();
+    flate2::read::GzDecoder::new(std::fs::File::open(&gz).unwrap())
+        .read_to_string(&mut text)
+        .unwrap();
+    let records: Value = serde_json::from_str(&text).unwrap();
+    serde_json::from_value(records[id].clone()).unwrap()
+}
+
+fn placement(f: &solidcore::eval::Frame) -> Axis2Placement {
+    let v = |a: [f64; 3]| Vec3::new(a[0], a[1], a[2]);
+    Axis2Placement::from_stored(
+        Point3::new(f.origin[0], f.origin[1], f.origin[2]),
+        v(f.x_axis),
+        v(f.y_axis),
+    )
+}
+
+/// A corpus face's label as the kernel's support, and its NURBS form.
+fn label_and_form(
+    k: &Kernel,
+    case: &str,
+    id: &str,
+    face: usize,
+) -> (
+    AnalyticSupport,
+    solidcore::nurbs::b_spline_surface::BSplineSurface,
+) {
+    let r = body(case, id);
+    let f = &r.faces[face];
+    let nurbs = solidcore_convert::surface(f.nurbs.as_ref().unwrap()).unwrap();
+    let (s, _) = from_parameters(k, &nurbs, f.domain).unwrap();
+    let SurfaceKind::BSpline(form) = s.kind() else {
+        panic!("{case} {id} {face}: the form is not a B-spline")
+    };
+    let support = match solidcore_convert::surface(&f.surface).unwrap() {
+        solidcore::eval::Surface::Torus {
+            frame,
+            major_radius,
+            minor_radius,
+            ..
+        } => AnalyticSupport::Torus {
+            placement: placement(&frame),
+            major_radius,
+            minor_radius,
+        },
+        solidcore::eval::Surface::Cylinder { frame, radius } => AnalyticSupport::Cylinder {
+            placement: placement(&frame),
+            radius,
+        },
+        other => panic!("{case} {id} {face}: {other:?}"),
+    };
+    (support, form.copy())
+}
+
+/// (case, body, face, the margin measured at #3185, whether the label stands)
+const CASES: &[(&str, &str, usize, f64, bool)] = &[
+    (
+        "sweeps/3897-sweep-ring-along-a-short-arc-joined-to-a-collar",
+        "b6",
+        1,
+        0.0201,
+        true,
+    ),
+    (
+        "sweeps/4280-circular-pattern-of-sweep-cut-groove-on-hub",
+        "b46",
+        1,
+        1.6943,
+        false,
+    ),
+    (
+        "sweeps/4280-circular-pattern-of-sweep-cut-groove-on-hub",
+        "b85",
+        15,
+        0.4995,
+        true,
+    ),
+    (
+        "sweeps/4280-circular-pattern-of-sweep-cut-groove-on-hub",
+        "b85",
+        21,
+        0.4245,
+        true,
+    ),
+    (
+        "booleans/3456-deep-cut-through-drilled-taper-leaves-base",
+        "b94",
+        3,
+        1.2339,
+        false,
+    ),
+    (
+        "crashes/2243-step-export-flanged-enclosure",
+        "b43",
+        1,
+        1.4008,
+        false,
+    ),
+    ("fillets/2903-original-rim-chamfer", "b0", 1, 1.5243, false),
+    ("fillets/2903-original-rim-chamfer", "b0", 3, 2.0275, false),
+    ("fillets/2903-original-rim-round", "b0", 1, 1.5243, false),
+    ("fillets/2903-original-rim-round", "b0", 3, 2.0275, false),
+];
+
+/// Each corpus label's margin, pinned to 1 % of what #3185 measured, on its side of 1, and
+/// `build` agreeing with it.
+#[test]
+fn the_corpus_labels_stand_or_fall_by_the_derived_bound() {
+    let k = kernel();
+    let mut report = String::new();
+    let mut wrong = Vec::new();
+    for &(case, id, face, pinned, stands) in CASES {
+        let (support, form) = label_and_form(&k, case, id, face);
+        let m = support.parametrisation_margin(&k, &form).unwrap().unwrap();
+        let _ = writeln!(report, "{case} {id} face {face}: margin {m:.4}");
+        let built = support.build(&k, form).is_ok();
+        if (m <= 1.0) != stands || built != stands || (m - pinned).abs() > 0.01 * pinned {
+            wrong.push(format!(
+                "{case} {id} face {face}: margin {m} (pinned {pinned}, stands {stands}), build {built}"
+            ));
+        }
+    }
+    eprintln!("{report}");
+    assert!(wrong.is_empty(), "{wrong:#?}");
+}
+
+/// A torus at the groove's sizes (#3185's b46: centre (19.375, -33.558, 37.5), R 33.657, r 2,
+/// a 43.6 degree sweep of the whole tube), made by the kernel: its own label stands on its
+/// form; turned about its axis by four times the turn the bound allows (the margin is linear
+/// in a small turn: a turn of `t` reads `m(t)`, so `4 t / m(t)` reads 4), it is refused.
+#[test]
+fn a_torus_label_turned_past_its_bound_is_refused() {
+    let k = kernel();
+    let o = Point3::new(19.374_999_999_999_975, -33.558_484_396_647_01, 37.5);
+    let x = Vec3::new(
+        -0.427_099_277_807_218_7,
+        0.739_757_649_038_078_7,
+        -0.519_946_946_895_745_4,
+    );
+    let y = Vec3::new(
+        0.259_973_473_447_872_36,
+        -0.450_287_264_631_874_2,
+        -0.854_198_555_614_438_5,
+    );
+    let domain = Extent2d::from_bounds(0.0, -180.0, 43.602_818_972_703_645, 180.0);
+    let made = Torus::new(
+        &k,
+        o,
+        x,
+        y,
+        domain,
+        33.657_280_044_590_65,
+        2.0,
+        TorusOptions::default(),
+    )
+    .unwrap();
+    let s = Surface::from(made);
+    let form = as_b_spline(&k, &s).unwrap().unwrap().into_owned();
+    let own = AnalyticSupport::of(&s).unwrap();
+    let m0 = own.parametrisation_margin(&k, &form).unwrap().unwrap();
+    assert!(m0 <= 1.0, "own label: {m0}");
+    assert!(own.build(&k, form.copy()).is_ok());
+    let turned = |t: f64| {
+        let AnalyticSupport::Torus {
+            placement,
+            major_radius,
+            minor_radius,
+        } = own
+        else {
+            panic!()
+        };
+        let (c, sn) = (t.cos(), t.sin());
+        let (px, py) = (placement.x_axis(), placement.y_axis());
+        AnalyticSupport::Torus {
+            placement: Axis2Placement::from_stored(
+                placement.origin(),
+                px * c + py * sn,
+                py * c - px * sn,
+            ),
+            major_radius,
+            minor_radius,
+        }
+    };
+    let probe = 1e-11;
+    let m1 = turned(probe)
+        .parametrisation_margin(&k, &form)
+        .unwrap()
+        .unwrap();
+    assert!(m1 > 1.0, "a turn of {probe} rad reads {m1}");
+    let four = 4.0 * probe / m1;
+    let m4 = turned(four)
+        .parametrisation_margin(&k, &form)
+        .unwrap()
+        .unwrap();
+    eprintln!("own {m0:.4}; turn {probe:e}: {m1:.3}; turn {four:e}: {m4:.3}");
+    assert!(m4 > 1.0, "a turn of {four:e} rad reads {m4}");
+    assert!(turned(four).build(&k, form).is_err());
+}

solidcore/crates/solidcore/src/surface/implicit.rs

@@ -1123,12 +1123,37 @@ impl AnalyticSupport {
         // a surface of revolution: the one the kernel recognises in the form, which must be
         // this one, take the form as its NURB and evaluate as the form at the form's own
         // parameters
-        let not_form = || {
-            SolidError::invalid_input(
-                OP_BUILD,
-                "a support's analytic surface is not its NURBS form's parametrisation",
-            )
-        };
+        let surface = self.on_form(k, &form)?;
+        if !evaluates_as(k, &surface, &form)? {
+            return Err(not_form());
+        }
+        Ok(surface)
+    }
+
+    /// The largest ratio, over the form's Greville points, of this support's analytic
+    /// surface's distance from `form` to the rounding [`evaluates_as`] derives there (#3185):
+    /// at most 1 where [`Self::build`]'s parametrisation test passes. The surface is the one
+    /// `build` makes (the kernel's recognition of `form`, carrying this support's numbers);
+    /// `lies_on` and the placement's axes are not checked here. `None` for a plane, or a
+    /// surface the bound does not cover.
+    ///
+    /// # Errors
+    ///
+    /// `InvalidInput` where the kernel recognises no surface of this kind in `form`; those of
+    /// the recognisers, the conversions and the evaluations.
+    pub fn parametrisation_margin(&self, k: &Kernel, form: &BSplineSurface) -> Result<Option<f64>> {
+        if matches!(self, Self::Plane { .. }) {
+            return Ok(None);
+        }
+        let surface = self.on_form(k, form)?;
+        evaluation_margin(k, &surface, form)
+    }
+
+    /// The surface of revolution of this support's kind the kernel recognises in `form`,
+    /// carrying this support's own numbers and `form` as its NURB ([`Self::build`] after its
+    /// checks of the point set, before the parametrisation's).
+    fn on_form(&self, k: &Kernel, form: &BSplineSurface) -> Result<Surface> {
+        let form = form.copy();
         let mut surface = recognise(k, self.kind(), &form)?.ok_or_else(not_form)?;
         // the recognised surface must be of this support's kind; the numbers recognition
         // recomputed are not compared with the support's, because it recomputes them from
@@ -1178,13 +1203,18 @@ impl AnalyticSupport {
             }
             _ => return Err(not_form()),
         }
-        if !evaluates_as(k, &surface, &form)? {
-            return Err(not_form());
-        }
         Ok(surface)
     }
 }
 
+/// [`AnalyticSupport::build`]'s refusal of a surface that is not its form's parametrisation.
+fn not_form() -> SolidError {
+    SolidError::invalid_input(
+        OP_BUILD,
+        "a support's analytic surface is not its NURBS form's parametrisation",
+    )
+}
+
 /// A cylinder's or cone's recognised parametrisation carried by the support's own
 /// `placement`: its heights are measured along the axis from the placement's origin, so the
 /// recognised heights move by how far along the axis the recognised origin lies from it.
@@ -1259,41 +1289,314 @@ fn recognise(k: &Kernel, kind: SupportKind, form: &BSplineSurface) -> Result<Opt
 }
 
 /// Whether `surface` evaluates as `form` at the form's Greville parameters (each control
-/// point's), within the rounding of the two evaluations: `gamma_K` times the form's size,
-/// `K = 16 (p + q + 2)` for the rational evaluation of degrees `(p, q)` and the analytic
-/// one's conversion of the parameters and its trigonometry, a few ulps per operation.
+/// point's): at each, the analytic surface's point and the form's agree within the bound
+/// [`Rounding::at`] derives for that point, term by term below. The analytic side is the
+/// **STEP** evaluation of the converted parameters ([`Surface::convert_uv_from_nurbs_to_step`]
+/// and [`Surface::evaluate_step_point`]), which a cone, sphere, torus and surface of
+/// revolution compute from their placement, radii, semi-angle and generator.
+/// [`Surface::evaluate_point`] would not do: a sphere, a torus and a surface of revolution
+/// take it from their NURBS base, and that base is the `form` the caller has just installed,
+/// so it would compare the form with itself and read nothing of what the caller imposed
+/// (#2588).
 ///
-/// The analytic side is the **STEP** evaluation of the converted parameters
-/// ([`Surface::convert_uv_from_nurbs_to_step`] and [`Surface::evaluate_step_point`]),
-/// which a cone, sphere, torus and surface of revolution compute from their placement,
-/// radii, semi-angle and generator. [`Surface::evaluate_point`] would not do: a sphere, a
-/// torus and a surface of revolution take it from their NURBS base, and that base is the
-/// `form` the caller has just installed, so it would compare the form with itself and read
-/// nothing of what the caller imposed (#2588; the cone already converted and evaluated
-/// analytically, which is why only the other kinds went unchecked).
+/// What the bound forgives and what it tests (#3185): it forgives the rounding of both
+/// evaluations and of the two parameter conversions, nothing else. It does not forgive the
+/// label's frame turned about the axis against the converter's, its axis, origin or radii,
+/// a form whose parameter is not the analytic surface's (a reparametrised arc), nor the
+/// form's distance from the analytic surface: those are what the comparison proves.
+///
+/// `u` is the unit roundoff, `gamma_n = n u / (1 - n u)`. Stated assumption: the libm
+/// `sin`, `cos`, `tan` and `atan2` are within one ulp (relative `2 u`).
+///
+/// 1. **The form** (`Surface::eval`, Cox-de Boor then two blends): each basis value is a
+///    nonnegative combination over `p` levels of at most 6 operations (two knot differences,
+///    a sum, a division, a product and a sum), so relative `gamma_6p`; the blends sum
+///    `q + 1` then `p + 1` products. With positive weights the homogeneous point's error is
+///    at most `gamma_(7 (p + q) + 4) W size` and the weight's `gamma_(7 (p + q) + 4) W`, so
+///    the Euclidean point's, after the division, `E_form = (2 gamma_(7 (p + q) + 4) + u)
+///    size` (`gamma_(7 (p + q) + 4) size` when polynomial), `size` the net's largest point.
+/// 2. **The STEP point**, `O + m (cos u X + sin u Y) + h Z`, `m` and `h` the generator's
+///    radial and axial parts (the torus `R + r cos v`, `r sin v`; the sphere `r cos v`,
+///    `r sin v`; the cone `r0 + v tan a`, `v`). An angle `x` in degrees goes to radians in
+///    two operations of the rounded `PI`: `gamma_3 |x|`; with one ulp of the cosine and sine,
+///    each is off by `dc(x) = 2 u + gamma_3 |x|`. Per coordinate (unit axes) the error is at
+///    most `gamma_3 (|O| + M + H) + M (gamma_2 + 2 dc(u)) + dm + dh`, `M`, `H` the sizes of
+///    `m`, `h` and `dm`, `dh` their errors: the torus `gamma_3 (R + r) + 2 r dc(v)`, the
+///    sphere `gamma_1 r + 2 r dc(v)`, the cone `gamma_2 (|r0| + |v t|) + |v| dt` (`dt =
+///    (1 + t^2) gamma_3 |a| + 2 u |t|` for the tangent of the half angle `a`); a general
+///    surface of revolution rotates its generator's point `G` (a B-spline: item 1's bound of
+///    its own net, `E_gen`) about the axis, `gamma_8 (|O| + 3 D) + 2 D dc(u) + E_gen`, `D`
+///    the net's reach from `O`. `E_step` is `sqrt 3` times the coordinate bound.
+/// 3. **The generator's parameter**: a circle (torus, sphere) inverts its NURBS point by
+///    `atan2` about its centre `c`: at its radius `r` the error is that point's (`E_form`)
+///    plus `gamma_4 (size + |c|)` and `gamma_4 r |v|` of the angle's own rounding; a line
+///    (cone) maps it affinely in 6 operations, `gamma_6 2 (|v0| + |v1|)` along the axis and
+///    its slope, times `1 + |t|`; a B-spline generator's parameter is its NURBS's (none).
+/// 4. **The sweep's parameter**: the polar converter evaluates its circle's NURBS at the
+///    parameter (item 1's bound of that NURBS's own net, `E_polar`) and inverts the point by
+///    `atan2` about the circle's centre `c` in its frame: at the circle's radius `rc` the
+///    angle is off by `(E_polar + gamma_4 (rc + 2 |c|)) / rc`, plus `gamma_4 |s|` of the
+///    angle's own rounding and the degrees' period (`gamma_1 2 pi`), times the STEP point's
+///    distance from that axis (the lever). The converter's ends are its domain's, which
+///    recognition takes from the form's own end meridians (#3199), so they are the inverted
+///    ends to the same rounding.
+///
+/// A surface of another kind, or a general surface of revolution whose generator is not a
+/// B-spline, has no bound here and is not proved.
 ///
 /// # Errors
 ///
 /// Those of the parameter conversion and the evaluations and of the form's knots.
 fn evaluates_as(k: &Kernel, surface: &Surface, form: &BSplineSurface) -> Result<bool> {
+    Ok(evaluation_margin(k, surface, form)?.is_some_and(|m| m <= 1.0))
+}
+
+/// [`evaluates_as`]'s largest ratio of distance to bound over the Greville points (infinite
+/// for a distance that is not a number); `None` for a surface the bound does not cover.
+fn evaluation_margin(k: &Kernel, surface: &Surface, form: &BSplineSurface) -> Result<Option<f64>> {
     let (net, p, q) = euclidean_net(form)?;
     let size = net.iter().flatten().fold(0.0_f64, |a, x| a.max(x.length()));
-    let tol = gamma(16 * (p + q + 2)) * size;
+    let Some(r) = Rounding::of(surface, form.is_rational(), p, q, size, &net)? else {
+        return Ok(None);
+    };
+    let mut worst = 0.0_f64;
     for u in greville(form, true)? {
         for v in greville(form, false)? {
             let uv = Vec2::new(u, v);
             let step_uv = surface.convert_uv_from_nurbs_to_step(k, uv)?;
-            let off =
-                (surface.evaluate_step_point(k, step_uv)? - form.evaluate_point(uv)?).length();
-            if !matches!(
-                off.partial_cmp(&tol),
-                Some(core::cmp::Ordering::Less | core::cmp::Ordering::Equal)
-            ) {
-                return Ok(false);
-            }
+            let s = surface.evaluate_step_point(k, step_uv)?;
+            let f = form.evaluate_point(uv)?;
+            let ratio = (s - f).length() / r.at(step_uv, s, f);
+            worst = if ratio.is_nan() {
+                f64::INFINITY
+            } else {
+                worst.max(ratio)
+            };
         }
     }
-    Ok(true)
+    Ok(Some(worst))
+}
+
+/// A surface of revolution's generator, for [`Rounding`]'s item 2 and 3.
+#[derive(Debug, Clone, Copy)]
+enum Generator {
+    /// `R + r cos v`, `r sin v` about the circle of centre `c`.
+    Torus { major: f64, minor: f64, c: f64 },
+    /// `r cos v`, `r sin v` about the circle of centre `c`.
+    Sphere { radius: f64, c: f64 },
+    /// `r0 + v t`, `v` (the half angle `a` in radians), its STEP `v` over `[v0, v1]`.
+    Cone {
+        r0: f64,
+        t: f64,
+        a: f64,
+        v0: f64,
+        v1: f64,
+    },
+    /// A B-spline generator, its own evaluation's bound.
+    Spline { e_gen: f64 },
+}
+
+/// The pieces of [`evaluates_as`]'s bound that do not depend on the point.
+#[derive(Debug, Clone, Copy)]
+struct Rounding {
+    e_form: f64,
+    origin: Vec3,
+    /// The net's reach from the origin.
+    reach: f64,
+    generator: Generator,
+    /// The sweep converter's circle: centre, axes.
+    polar: (Vec3, Vec3, Vec3, f64, f64),
+}
+
+/// `|v|`'s largest coordinate.
+fn max_coord(v: Vec3) -> f64 {
+    v.x.abs().max(v.y.abs()).max(v.z.abs())
+}
+
+/// A rational (or polynomial) evaluation's bound of degrees summing to `n` over points of
+/// size `size` ([`evaluates_as`] item 1).
+fn eval_bound(rational: bool, n: usize, size: f64) -> f64 {
+    let g = gamma(7 * n + 4);
+    if rational {
+        (2.0 * g + crate::nurbs::math::UNIT) * size
+    } else {
+        g * size
+    }
+}
+
+impl Rounding {
+    /// The bound's data for `surface` (a cone, sphere, torus or surface of revolution with a
+    /// B-spline generator) on its form of degrees `(p, q)`, net `net`; `None` for any other.
+    fn of(
+        surface: &Surface,
+        rational: bool,
+        p: usize,
+        q: usize,
+        size: f64,
+        net: &[Vec<Vec3>],
+    ) -> Result<Option<Self>> {
+        use crate::curve::curve::CurveKind;
+        let e_form = eval_bound(rational, p + q, size);
+        let circle_centre = |rev: &crate::surface::surf_of_revolution::SurfOfRevolution| match rev
+            .gen_curve
+            .as_ref()
+            .map(crate::curve::curve::Curve::kind)
+        {
+            Some(CurveKind::Ellipse(e)) => Some(e.position.origin().length()),
+            _ => None,
+        };
+        let (rev, generator) = match surface.kind() {
+            SurfaceKind::Torus(t) => {
+                let Some(c) = circle_centre(&t.base) else {
+                    return Ok(None);
+                };
+                (
+                    &t.base,
+                    Generator::Torus {
+                        major: t.major_radius,
+                        minor: t.minor_radius,
+                        c,
+                    },
+                )
+            }
+            SurfaceKind::Sphere(sp) => {
+                let Some(c) = circle_centre(&sp.base) else {
+                    return Ok(None);
+                };
+                (
+                    &sp.base,
+                    Generator::Sphere {
+                        radius: sp.radius,
+                        c,
+                    },
+                )
+            }
+            SurfaceKind::Cone(c) => {
+                let a = c.semi_angle_deg * PI / 180.0;
+                let d = c.rev.anal_uv_domain;
+                (
+                    &c.rev,
+                    Generator::Cone {
+                        r0: c.base_radius,
+                        t: c.tan_semi_angle(),
+                        a,
+                        v0: d.v_min(),
+                        v1: d.v_max(),
+                    },
+                )
+            }
+            SurfaceKind::SurfOfRevolution(rev) => {
+                let Some(CurveKind::BSpline(g)) =
+                    rev.gen_curve.as_ref().map(crate::curve::curve::Curve::kind)
+                else {
+                    return Ok(None);
+                };
+                let pts = g.control_points()?;
+                let gsize = pts.iter().fold(0.0_f64, |m, c| {
+                    let w = if c.w == NOW { 1.0 } else { c.w };
+                    m.max((Vec3::new(c.x, c.y, c.z) * (1.0 / w)).length())
+                });
+                let rational = pts.iter().any(|c| c.w != NOW);
+                (
+                    &**rev,
+                    Generator::Spline {
+                        e_gen: eval_bound(rational, g.degree()?, gsize),
+                    },
+                )
+            }
+            _ => return Ok(None),
+        };
+        let Some(pc) = rev.polar_curve() else {
+            return Ok(None);
+        };
+        let e_polar = {
+            let pts = pc.base().control_points()?;
+            let psize = pts.iter().fold(0.0_f64, |m, c| {
+                let w = if c.w == NOW { 1.0 } else { c.w };
+                m.max((Vec3::new(c.x, c.y, c.z) * (1.0 / w)).length())
+            });
+            eval_bound(pts.iter().any(|c| c.w != NOW), pc.base().degree()?, psize)
+        };
+        let origin = rev.position.origin();
+        let reach = net
+            .iter()
+            .flatten()
+            .fold(0.0_f64, |m, x| m.max((*x - origin).length()));
+        Ok(Some(Self {
+            e_form,
+            origin,
+            reach,
+            generator,
+            polar: (
+                pc.position.origin(),
+                pc.position.x_axis(),
+                pc.position.y_axis(),
+                pc.radius_x,
+                e_polar,
+            ),
+        }))
+    }
+
+    /// The bound at the STEP parameters `step` (degrees, and the generator's), the STEP
+    /// point `s` and the form's point `f` ([`evaluates_as`] items 1 to 4).
+    fn at(&self, step: Vec2, s: Vec3, f: Vec3) -> f64 {
+        let u = crate::nurbs::math::UNIT;
+        let rad = |deg: f64| deg * PI / 180.0;
+        let dc = |deg: f64| 2.0 * u + gamma(3) * rad(deg).abs();
+        let o = max_coord(self.origin);
+        let size_f = f.length();
+        // item 2 (per coordinate) and item 3 (as a length)
+        let (coord, generator) = match self.generator {
+            Generator::Torus { major, minor, c } => {
+                let (m, h) = (major + minor, minor);
+                let dm = gamma(3) * (major + minor) + 2.0 * minor * dc(step.y);
+                let dh = gamma(1) * minor + 2.0 * minor * dc(step.y);
+                (
+                    gamma(3) * (o + m + h) + m * (gamma(2) + 2.0 * dc(step.x)) + dm + dh,
+                    self.e_form + gamma(4) * (size_f + c) + gamma(4) * minor * rad(step.y).abs(),
+                )
+            }
+            Generator::Sphere { radius, c } => {
+                let dm = gamma(1) * radius + 2.0 * radius * dc(step.y);
+                (
+                    gamma(3) * (o + 2.0 * radius)
+                        + radius * (gamma(2) + 2.0 * dc(step.x))
+                        + 2.0 * dm,
+                    self.e_form + gamma(4) * (size_f + c) + gamma(4) * radius * rad(step.y).abs(),
+                )
+            }
+            Generator::Cone { r0, t, a, v0, v1 } => {
+                let v = step.y;
+                let m = r0.abs() + (v * t).abs();
+                let dt = (1.0 + t * t) * gamma(3) * a.abs() + 2.0 * u * t.abs();
+                let dm = gamma(2) * m + v.abs() * dt;
+                (
+                    gamma(3) * (o + m + v.abs()) + m * (gamma(2) + 2.0 * dc(step.x)) + dm,
+                    gamma(6) * 2.0 * (v0.abs() + v1.abs()) * (1.0 + t.abs()),
+                )
+            }
+            Generator::Spline { e_gen } => (
+                gamma(8) * (o + 3.0 * self.reach) + 2.0 * self.reach * dc(step.x) + e_gen,
+                0.0,
+            ),
+        };
+        let e_step = 3.0_f64.sqrt() * coord;
+        // item 4: the polar converter's inversion of its own circle's point
+        let (c, x, y, rc, e_polar) = self.polar;
+        let z = x.cross(y);
+        let along = |v: Vec3| v - z * v.dot(z);
+        let lever = along(s - c).length();
+        let sweep = if rc > 0.0 {
+            lever
+                * ((e_polar + gamma(4) * (rc + 2.0 * c.length())) / rc
+                    + gamma(4) * rad(step.x).abs()
+                    + gamma(1) * 2.0 * PI)
+        } else {
+            // a circle of no radius has no angle: any, at the STEP point's lever
+            lever * PI
+        };
+        self.e_form + e_step + generator + sweep
+    }
 }
 
 const OP_DERIVE: &str = "surface::implicit::AnalyticSupport::derive";

solidcore/crates/solidcore/src/surface/torus.rs

@@ -1061,7 +1061,6 @@ impl Torus {
         let ref_frame = ref_iso.reference_frame.ok_or_else(missing)?;
         let other_frame = other_iso.reference_frame.ok_or_else(missing)?;
         let min_rad = other_iso.radius.ok_or_else(missing)?;
-        let sweep_ivl = ref_iso.angles.ok_or_else(missing)?;
         let common_point = other_iso.common_line_point.ok_or_else(missing)?;
         let common_vec = other_iso.common_line_vec.ok_or_else(missing)?;
 
@@ -1069,6 +1068,17 @@ impl Torus {
         let torus_origin =
             common_point + common_vec.dot(other_frame.origin() - common_point) * common_vec;
 
+        // not upstream (#3199): the x axis and the sweep interval from the form's own first
+        // sweep row's ends about the axis (`sweep_frame`), not from `is_arc`'s fit of it
+        let Some((torus_x_axis, torus_y_axis, sweep_ivl)) = sweep_frame(
+            torus_origin,
+            ref_frame.z_axis(),
+            fitted_sides(&ref_frame, ref_iso.angles.ok_or_else(missing)?),
+            meridians(test, swap_uv)?,
+        ) else {
+            return Ok(None);
+        };
+
         // get torus orientation and size
         // not upstream (WR-501, #1470, opt-in; tangency-labs/smlib@6136a04): the major
         // radius runs from the torus origin, not from the first sweep circle's centre,
@@ -1079,9 +1089,7 @@ impl Torus {
             other_frame.origin() - ref_frame.origin()
         };
         let maj_rad = major_vec.length();
-        let torus_x_axis = ref_frame.x_axis(); // the frame where all circles are parallel
-        let torus_y_axis = ref_frame.y_axis();
-        let torus_z_axis = ref_frame.z_axis();
+        let torus_z_axis = ref_frame.z_axis(); // the frame where all circles are parallel
 
         // torus must have positive radii
         if min_rad <= EFF_ZERO || maj_rad <= EFF_ZERO {
@@ -1294,7 +1302,6 @@ impl Torus {
         let mut ref_frame = ref_iso.reference_frame.ok_or_else(missing)?;
         let other_frame = other_iso.reference_frame.ok_or_else(missing)?;
         let min_rad = other_iso.radius.ok_or_else(missing)?;
-        let sweep_ivl = ref_iso.angles.ok_or_else(missing)?;
         let common_point = other_iso.common_line_point.ok_or_else(missing)?;
         let common_vec = other_iso.common_line_vec.ok_or_else(missing)?;
 
@@ -1302,8 +1309,18 @@ impl Torus {
         let torus_origin =
             common_point + common_vec.dot(other_frame.origin() - common_point) * common_vec;
 
+        // not upstream (#3199): the x axis and the sweep interval from the form's own first
+        // sweep row's ends about the axis (`sweep_frame`), not from `is_arc`'s fit of it
+        let (x_axis, y_axis, sweep_ivl) = sweep_frame(
+            torus_origin,
+            ref_frame.z_axis(),
+            fitted_sides(&ref_frame, ref_iso.angles.ok_or_else(missing)?),
+            meridians(srf, swap_uv)?,
+        )
+        .ok_or_else(|| SolidError::sm_err(OP_REBUILD))?;
+
         // Install the origin into pRefFrame.  // [B328]
-        let _ = ref_frame.set_canonical(torus_origin, ref_frame.x_axis(), ref_frame.y_axis());
+        let _ = ref_frame.set_canonical(torus_origin, x_axis, y_axis);
 
         // get torus orientation and size
         let major_vec = other_frame.origin() - ref_frame.origin();
@@ -1587,6 +1604,82 @@ impl Revolved for Torus {
 
 /// `SmTorus::CreateIsoParametricCurve` of the surface of revolution `rev` of a torus
 /// of `shape` (also called from the base class's rebuilds, where it is virtual).
+/// The control points, Euclidean, of the form's first and last meridians: the columns of its
+/// net at the sweep's two ends (the sweep is along u, or along v with `swap_uv`). A clamped
+/// end's column is that boundary curve's own control polygon, and a meridian lies in a plane
+/// through the axis, so every one of its control points has the meridian's angle about the
+/// axis.
+fn meridians(srf: &BSplineSurface, swap_uv: bool) -> Result<[Vec<Point3>; 2]> {
+    let nurb = srf
+        .gw_nurb()
+        .ok_or_else(|| SolidError::null_pointer(OP_NURB_TORUS))?;
+    let (n, m) = (nurb.n(), nurb.m());
+    let column = |end: bool| -> Result<Vec<Point3>> {
+        if swap_uv {
+            let j = if end { m } else { 0 };
+            (0..=n).map(|i| Ok(nurb.cpt(i, j)?.to_euclid())).collect()
+        } else {
+            let i = if end { n } else { 0 };
+            (0..=m).map(|j| Ok(nurb.cpt(i, j)?.to_euclid())).collect()
+        }
+    };
+    Ok([column(false)?, column(true)?])
+}
+
+/// The directions, about the axis, of the fitted first sweep row's ends (`is_arc`'s frame and
+/// angles): [`sweep_frame`] reads only their sides from them.
+fn fitted_sides(frame: &Axis2Placement, angles: Extent1d) -> [Vec3; 2] {
+    let at = |deg: f64| {
+        let a = deg * PI / 180.0;
+        a.cos() * frame.x_axis() + a.sin() * frame.y_axis()
+    };
+    [at(angles.min()), at(angles.max())]
+}
+
+/// Not upstream (#3199): a recognised torus's x axis, y axis and sweep interval (degrees) from
+/// its form's first and last meridians (`meridians`: their control points, exact data), about
+/// the axis through `origin` along the unit `z`. A meridian lies in a plane through the axis,
+/// so its control points share its angle on the side the tube's centre is on, and the
+/// opposite angle past the axis (a spindle torus's tube crosses it); `sides` (the fitted
+/// row's end directions, [`fitted_sides`]) says which side that is, and nothing else. Each
+/// meridian's angle is read at its control point farthest from the axis on that side: x is
+/// the first meridian's direction square to the axis (so the interval starts at 0), y `z x
+/// x`, and the interval's end the last meridian's angle from x about z (`atan2` in that
+/// frame), a closed sweep (within `SM_EFF_ZERO_SQRT` of 0) at 360, as `is_arc` maps its end.
+/// `is_arc` fitted the first sweep row's circle through three samples, so on a short arc its
+/// centre, and with it the x axis and the end angle, was off the axis by its conditioning
+/// (1e-14 rad on #3185's grooves); the converter centres the sweep circle on the axis, so its
+/// ends and its inside now agree to rounding. `None` when a meridian has no control point off
+/// the axis on its side.
+fn sweep_frame(
+    origin: Point3,
+    z: Vec3,
+    sides: [Vec3; 2],
+    [first, last]: [Vec<Point3>; 2],
+) -> Option<(Vec3, Vec3, Extent1d)> {
+    let radial = |p: &Point3| {
+        let s = *p - origin;
+        s - s.dot(z) * z
+    };
+    let farthest = |ps: &[Point3], side: Vec3| {
+        ps.iter()
+            .map(radial)
+            .filter(|r| r.dot(side) > 0.0)
+            .max_by(|a, b| a.length().total_cmp(&b.length()))
+    };
+    let [first_side, last_side] = sides;
+    let x = farthest(&first, first_side)?.unitize().ok()?;
+    let y = z.cross(x);
+    let e = farthest(&last, last_side)?;
+    let mut end_deg = e.dot(y).atan2(e.dot(x)) * 180.0 / PI;
+    if end_deg.abs() < EFF_ZERO_SQRT {
+        end_deg = 360.0;
+    } else if end_deg < 0.0 {
+        end_deg += 360.0;
+    }
+    Some((x, y, Extent1d::from_min_max(0.0, end_deg)))
+}
+
 fn iso_curve(
     rev: &SurfOfRevolution,
     shape: Shape,