brepkit-math 4.0.48

//! Point projection onto NURBS curves and surfaces.
//!
//! Finds the closest point on a curve or surface to a given point in space.
//! Used for Boolean classification, snapping, distance queries, and
//! tessellation refinement.
//!
//! Algorithms follow NURBS Book A6.1–A6.6: subdivision for initial guess
//! followed by Newton–Raphson refinement.

use crate::MathError;
use crate::nurbs::curve::NurbsCurve;
use crate::nurbs::surface::NurbsSurface;
use crate::vec::Point3;

/// Maximum Newton iterations before declaring convergence failure.
const MAX_ITERATIONS: usize = 50;

/// Number of grid subdivisions per direction for surface coarse search.
const SURFACE_GRID_SIZE: usize = 8;

// ---------------------------------------------------------------------------
// Public result types
// ---------------------------------------------------------------------------

/// Result of projecting a point onto a curve.
#[derive(Debug, Clone, Copy)]
pub struct CurveProjection {
    /// Parameter value at the closest point.
    pub parameter: f64,
    /// The closest point on the curve.
    pub point: Point3,
    /// Distance from the input point to the closest point.
    pub distance: f64,
}

/// Result of projecting a point onto a surface.
#[derive(Debug, Clone, Copy)]
pub struct SurfaceProjection {
    /// Parameter value u at the closest point.
    pub u: f64,
    /// Parameter value v at the closest point.
    pub v: f64,
    /// The closest point on the surface.
    pub point: Point3,
    /// Distance from the input point to the closest point.
    pub distance: f64,
}

// ---------------------------------------------------------------------------
// Curve projection
// ---------------------------------------------------------------------------

/// Find the closest point on a NURBS curve to the given point.
///
/// Samples each Bezier segment (knot span) for initial guesses, then
/// Newton–Raphson refinement (NURBS Book A6.1 + A6.3–A6.4).
///
/// # Errors
///
/// None for a curve that constructed: the `Result` stays in the public
/// signature, which callers across the workspace match on.
#[allow(clippy::unnecessary_wraps)]
pub fn project_point_to_curve(
    curve: &NurbsCurve,
    point: Point3,
    tolerance: f64,
) -> Result<CurveProjection, MathError> {
    let u_min = curve.knots()[curve.degree()];

    let candidates = curve_coarse_search(curve, point);

    // Run Newton from each candidate and keep the globally closest result.
    let mut best_u = u_min;
    let mut best_pt = curve.evaluate(u_min);
    let mut best_dist = (best_pt - point).length();

    for (u_guess, lo, hi) in candidates {
        let (u_refined, pt_refined) = curve_newton_refine(curve, point, u_guess, lo, hi, tolerance);
        let dist = (pt_refined - point).length();
        if dist < best_dist {
            best_dist = dist;
            best_u = u_refined;
            best_pt = pt_refined;
        }
    }

    Ok(CurveProjection {
        parameter: best_u,
        point: best_pt,
        distance: best_dist,
    })
}

/// Coarse search: sample each knot span (each of the curve's Bezier
/// segments) to find multiple candidate parameter values for Newton
/// refinement. The spans are sampled on the curve itself: decomposing it
/// into segments first costs knot insertions, on every projection, for the
/// same points.
///
/// Returns the candidates with the span each came from, as
/// `(u, span_start, span_end)`. Newton runs inside its seed's span, where
/// the curve is one polynomial piece: at a corner (a knot of multiplicity
/// `p`) the other piece's derivatives would carry it away from the corner.
/// So the seeds are the closest sample of each of the closest spans (a
/// short span's crowded samples take one seed, not all of them, and a knot
/// ends one span and starts the next, so it can seed both) and the closest
/// local minima of the sampled distance along the curve (several basins in
/// one span). An interior span ends one step below its knot, since the
/// curve is read there by the next piece.
#[allow(clippy::cast_precision_loss)]
fn curve_coarse_search(curve: &NurbsCurve, point: Point3) -> Vec<(f64, f64, f64)> {
    type Span = (f64, f64, Vec<(f64, f64)>);
    const SEEDS: usize = 5;
    let knots = curve.knots();
    let p = curve.degree();
    let (lo, hi) = (knots[p], knots[knots.len() - p - 1]);

    // Each span's (start, end) and its (parameter, distance_sq) samples, in
    // order along the curve.
    let n_samples = (p + 1).max(5) * 2;
    let mut spans: Vec<Span> = Vec::new();
    for span in knots.windows(2) {
        let (u_start, u_end) = (span[0], span[1]);
        if u_end <= u_start || u_start < lo || u_end > hi {
            continue;
        }
        let top = if u_end < hi { u_end.next_down() } else { u_end };
        let samples = (0..=n_samples)
            .map(|i| {
                let t = i as f64 / n_samples as f64;
                let u = t.mul_add(u_end - u_start, u_start).min(top);
                (u, (curve.evaluate(u) - point).length_squared())
            })
            .collect();
        spans.push((u_start, top, samples));
    }

    let closest = |samples: &[(f64, f64)]| {
        samples
            .iter()
            .copied()
            .min_by(|a, b| a.1.total_cmp(&b.1))
            .unwrap_or((0.0, f64::INFINITY))
    };
    let mut nearest: Vec<(f64, f64, f64, f64)> = spans
        .iter()
        .map(|(u_start, top, samples)| {
            let (u, d) = closest(samples);
            (d, u, *u_start, *top)
        })
        .collect();
    nearest.sort_by(|a, b| a.0.total_cmp(&b.0));
    nearest.truncate(SEEDS);

    // A sample's neighbours along the curve; a span's end sample is the
    // next span's start, so it looks past it. A run of equal minima counts
    // once.
    let mut minima: Vec<(f64, f64, f64, f64)> = Vec::new();
    for (s, (u_start, top, samples)) in spans.iter().enumerate() {
        let last = samples.len() - 1;
        let mut in_run = false;
        for (i, &(u, d)) in samples.iter().enumerate() {
            let prev = if i > 0 {
                Some(samples[i - 1].1)
            } else {
                s.checked_sub(1).map(|r| {
                    let before = &spans[r].2;
                    before[before.len() - 2].1
                })
            };
            let next = if i < last {
                Some(samples[i + 1].1)
            } else {
                spans.get(s + 1).map(|after| after.2[1].1)
            };
            let minimum = prev.is_none_or(|q| d <= q) && next.is_none_or(|q| d <= q);
            if minimum && !in_run {
                minima.push((d, u, *u_start, *top));
            }
            in_run = minimum;
        }
    }
    minima.sort_by(|a, b| a.0.total_cmp(&b.0));
    minima.truncate(SEEDS);

    let mut seeds: Vec<(f64, f64, f64)> = Vec::new();
    for (_, u, u_start, top) in nearest.into_iter().chain(minima) {
        if !seeds
            .iter()
            .any(|s| s.0.to_bits() == u.to_bits() && s.1.to_bits() == u_start.to_bits())
        {
            seeds.push((u, u_start, top));
        }
    }
    seeds
}

/// Newton–Raphson refinement for curve point projection.
///
/// Finds parameter u in `[u_min, u_max]` that minimizes ||C(u) - P||
/// starting from `u_init`, and returns the closest point it evaluated.
/// Each step starts from the closest iterate so far, and a step that lands
/// farther is halved back toward it (a backtracking line search): a plain
/// Newton step can overshoot the minimum, clamp to the interval's end and
/// cycle there.
#[allow(clippy::suspicious_operation_groupings)]
fn curve_newton_refine(
    curve: &NurbsCurve,
    point: Point3,
    u_init: f64,
    u_min: f64,
    u_max: f64,
    tolerance: f64,
) -> (f64, Point3) {
    let tol_sq = tolerance * tolerance;
    let mut u = u_init;
    let mut best: Option<(f64, Point3, f64)> = None;

    for _ in 0..MAX_ITERATIONS {
        let ders = curve.derivatives(u, 2);
        let c_pt = Point3::new(ders[0].x(), ders[0].y(), ders[0].z());
        let c_prime = ders[1]; // C'(u)
        let c_double_prime = ders[2]; // C''(u)
        let diff = c_pt - point; // C(u) - P

        let dist_sq = diff.length_squared();

        if let Some((best_u, _, best_dist_sq)) = best
            && dist_sq >= best_dist_sq
        {
            if (u - best_u).abs() < tolerance * (1.0 + best_u.abs()) {
                break;
            }
            u = 0.5 * (u + best_u);
            continue;
        }
        best = Some((u, c_pt, dist_sq));

        // Convergence check 1: point coincidence.
        if dist_sq < tol_sq {
            break;
        }

        // f(u) = C'(u) · (C(u) - P)
        let f_val = c_prime.dot(diff);

        // Convergence check 2: zero cosine (perpendicularity).
        // cos²(angle) = (C'·diff)² / (|C'|² · |diff|²) < tol²
        let c_prime_len_sq = c_prime.length_squared();
        if c_prime_len_sq > 1e-30 && dist_sq > tol_sq {
            let cos_sq = (f_val * f_val) / (c_prime_len_sq * dist_sq);
            if cos_sq < tol_sq {
                break;
            }
        }

        // f'(u) = C''(u) · (C(u) - P) + |C'(u)|². Where it is not positive
        // the Newton step climbs toward a farthest point, and the
        // Gauss-Newton step (|C'(u)|² alone) descends instead.
        let mut f_prime = c_double_prime.dot(diff) + c_prime_len_sq;
        if f_prime <= 0.0 {
            f_prime = c_prime_len_sq;
        }

        // Guard against zero denominator.
        if f_prime.abs() < 1e-30 {
            break;
        }

        let delta_u = f_val / f_prime;
        let u_new = (u - delta_u).clamp(u_min, u_max);

        // Guard NaN.
        if u_new.is_nan() {
            break;
        }

        // Convergence check 3: parameter step negligible.
        if (u_new - u).abs() < tolerance * (1.0 + u.abs()) {
            let pt = curve.evaluate(u_new);
            let d_sq = (pt - point).length_squared();
            if d_sq < dist_sq {
                best = Some((u_new, pt, d_sq));
            }
            break;
        }

        u = u_new;
    }

    best.map_or_else(|| (u_init, curve.evaluate(u_init)), |b| (b.0, b.1))
}

// ---------------------------------------------------------------------------
// Surface projection
// ---------------------------------------------------------------------------

/// Find the closest point on a NURBS surface to the given point.
///
/// Uses grid evaluation for initial guess, then 2D Newton–Raphson
/// refinement (NURBS Book A6.2 + A6.5–A6.6).
///
/// # Errors
///
/// Returns [`MathError::ConvergenceFailure`] if Newton iteration does not
/// converge within the maximum number of iterations.
pub fn project_point_to_surface(
    surface: &NurbsSurface,
    point: Point3,
    tolerance: f64,
) -> Result<SurfaceProjection, MathError> {
    let (u_guess, v_guess) = surface_coarse_search(surface, point);

    let knots_u = surface.knots_u();
    let knots_v = surface.knots_v();
    let pu = surface.degree_u();
    let pv = surface.degree_v();
    let u_min = knots_u[pu];
    let u_max = knots_u[knots_u.len() - pu - 1];
    let v_min = knots_v[pv];
    let v_max = knots_v[knots_v.len() - pv - 1];

    // Wrapping lets Newton cross a closed direction's seam from a seed on
    // the far copy of it; where the seam has a kink it can bounce across
    // without converging, so a failed wrapped solve retries clamped.
    let wraps = surface.is_periodic_u() || surface.is_periodic_v();
    let (u_final, v_final, pt_final) = surface_newton_refine(
        surface, point, u_guess, v_guess, u_min, u_max, v_min, v_max, tolerance, wraps,
    )
    .or_else(|err| {
        if wraps {
            surface_newton_refine(
                surface, point, u_guess, v_guess, u_min, u_max, v_min, v_max, tolerance, false,
            )
        } else {
            Err(err)
        }
    })?;
    let dist = (pt_final - point).length();

    Ok(SurfaceProjection {
        u: u_final,
        v: v_final,
        point: pt_final,
        distance: dist,
    })
}

/// Coarse search: evaluate surface on a uniform grid and find the closest
/// grid point.
#[allow(clippy::cast_precision_loss)]
fn surface_coarse_search(surface: &NurbsSurface, point: Point3) -> (f64, f64) {
    let knots_u = surface.knots_u();
    let knots_v = surface.knots_v();
    let pu = surface.degree_u();
    let pv = surface.degree_v();
    let u_min = knots_u[pu];
    let u_max = knots_u[knots_u.len() - pu - 1];
    let v_min = knots_v[pv];
    let v_max = knots_v[knots_v.len() - pv - 1];

    let mut best_u = u_min;
    let mut best_v = v_min;
    let mut best_dist_sq = f64::INFINITY;

    let n = SURFACE_GRID_SIZE;
    for i in 0..=n {
        let u = (i as f64 / n as f64).mul_add(u_max - u_min, u_min);
        for j in 0..=n {
            let v = (j as f64 / n as f64).mul_add(v_max - v_min, v_min);
            let pt = surface.evaluate(u, v);
            let d_sq = (pt - point).length_squared();
            if d_sq < best_dist_sq {
                best_dist_sq = d_sq;
                best_u = u;
                best_v = v;
            }
        }
    }

    (best_u, best_v)
}

/// 2D Newton–Raphson refinement for surface point projection.
///
/// Solves the 2×2 system at each step to find the (u, v) that minimizes
/// ||S(u,v) - P||.
#[allow(clippy::too_many_arguments, clippy::similar_names)]
#[allow(clippy::suspicious_operation_groupings)]
fn surface_newton_refine(
    surface: &NurbsSurface,
    point: Point3,
    u_init: f64,
    v_init: f64,
    u_min: f64,
    u_max: f64,
    v_min: f64,
    v_max: f64,
    tolerance: f64,
    wrap_closed: bool,
) -> Result<(f64, f64, Point3), MathError> {
    let mut u = u_init;
    let mut v = v_init;
    // Along a closed direction the step wraps across the seam instead of
    // stopping at the domain end: a seed on the far copy of the seam (the
    // coarse grid samples both ends) must still reach a point just short of
    // it.
    let advance = |x: f64, delta: f64, lo: f64, hi: f64, closed: bool| -> (f64, f64) {
        if closed && hi > lo {
            (lo + (x + delta - lo).rem_euclid(hi - lo), delta)
        } else {
            let next = (x + delta).clamp(lo, hi);
            (next, next - x)
        }
    };
    let (closed_u, closed_v) = (
        wrap_closed && surface.is_periodic_u(),
        wrap_closed && surface.is_periodic_v(),
    );

    for _ in 0..MAX_ITERATIONS {
        let ders = surface.derivatives(u, v, 1);
        let s_pt = Point3::new(ders[0][0].x(), ders[0][0].y(), ders[0][0].z());
        let deriv_u = ders[1][0]; // ∂S/∂u
        let deriv_v = ders[0][1]; // ∂S/∂v
        let r = s_pt - point; // S(u,v) - P

        // Convergence check 1: point coincidence.
        let dist = r.length();
        if dist < tolerance {
            return Ok((u, v, s_pt));
        }

        // Convergence check 2: zero cosine in both directions.
        let du_len = deriv_u.length();
        let dv_len = deriv_v.length();
        let dot_du_r = deriv_u.dot(r);
        let dot_dv_r = deriv_v.dot(r);
        if du_len > 0.0 && dv_len > 0.0 {
            let cos_u = dot_du_r.abs() / (du_len * dist);
            let cos_v = dot_dv_r.abs() / (dv_len * dist);
            if cos_u < tolerance && cos_v < tolerance {
                return Ok((u, v, s_pt));
            }
        }

        // Build the 2×2 Jacobian and right-hand side.
        // J = [S_u · S_u,  S_u · S_v]
        //     [S_v · S_u,  S_v · S_v]
        let j00 = deriv_u.dot(deriv_u);
        let j01 = deriv_u.dot(deriv_v);
        let j11 = deriv_v.dot(deriv_v);
        // rhs = [-S_u · r, -S_v · r]
        let rhs0 = -dot_du_r;
        let rhs1 = -dot_dv_r;

        // Solve 2×2 system via Cramer's rule: det = j00*j11 - j01²
        // Use a relative threshold so the singularity test stays meaningful
        // near surface poles / cone apex where both derivatives shrink to zero.
        let det = j00.mul_add(j11, -(j01 * j01));
        let (delta_u, delta_v) = if det.abs() < (j00 + j11).max(1e-30) * 1e-12 {
            // Near-singular: apply Tikhonov (Levenberg–Marquardt) regularisation
            // by adding λI to the normal equations.  This yields a step biased
            // toward zero rather than blowing up, preserving convergence near
            // poles and cone apices.
            let lambda = (j00 + j11).max(1e-10) * 1e-4;
            let j00r = j00 + lambda;
            let j11r = j11 + lambda;
            let det_r = j00r.mul_add(j11r, -(j01 * j01));
            if det_r.abs() < 1e-30 {
                // Still singular even after regularisation — fall back to a 1-D
                // search along whichever parameter axis has more gradient.
                if j00 > j11 {
                    (rhs0 / j00.max(1e-30), 0.0)
                } else if j11 > 1e-30 {
                    (0.0, rhs1 / j11.max(1e-30))
                } else {
                    return Ok((u, v, s_pt));
                }
            } else {
                (
                    rhs0.mul_add(j11r, -(rhs1 * j01)) / det_r,
                    j00r.mul_add(rhs1, -(j01 * rhs0)) / det_r,
                )
            }
        } else {
            (
                rhs0.mul_add(j11, -(rhs1 * j01)) / det,
                j00.mul_add(rhs1, -(j01 * rhs0)) / det,
            )
        };

        let (u_new, step_u) = advance(u, delta_u, u_min, u_max, closed_u);
        let (v_new, step_v) = advance(v, delta_v, v_min, v_max, closed_v);

        // Convergence check 3: parameter step negligible.
        let step = (deriv_u * step_u + deriv_v * step_v).length();
        if step < tolerance {
            let pt = surface.evaluate(u_new, v_new);
            return Ok((u_new, v_new, pt));
        }

        u = u_new;
        v = v_new;
    }

    Err(MathError::ConvergenceFailure {
        iterations: MAX_ITERATIONS,
    })
}

// ---------------------------------------------------------------------------
// Tests
// ---------------------------------------------------------------------------

#[cfg(test)]
#[allow(clippy::expect_used)]
mod tests {
    use super::*;
    use crate::vec::Vec3;

    const TOL: f64 = 1e-8;

    /// A simple line from (0,0,0) to (10,0,0) as a degree-1 NURBS.
    fn line_curve() -> NurbsCurve {
        NurbsCurve::new(
            1,
            vec![0.0, 0.0, 1.0, 1.0],
            vec![Point3::new(0.0, 0.0, 0.0), Point3::new(10.0, 0.0, 0.0)],
            vec![1.0, 1.0],
        )
        .expect("valid line")
    }

    /// Quarter circle arc as a rational NURBS (degree 2).
    fn quarter_circle() -> NurbsCurve {
        let w = std::f64::consts::FRAC_1_SQRT_2;
        NurbsCurve::new(
            2,
            vec![0.0, 0.0, 0.0, 1.0, 1.0, 1.0],
            vec![
                Point3::new(1.0, 0.0, 0.0),
                Point3::new(1.0, 1.0, 0.0),
                Point3::new(0.0, 1.0, 0.0),
            ],
            vec![1.0, w, 1.0],
        )
        .expect("valid quarter circle")
    }

    /// Cubic Bezier curve.
    fn cubic_bezier() -> NurbsCurve {
        NurbsCurve::new(
            3,
            vec![0.0, 0.0, 0.0, 0.0, 1.0, 1.0, 1.0, 1.0],
            vec![
                Point3::new(0.0, 0.0, 0.0),
                Point3::new(1.0, 2.0, 0.0),
                Point3::new(3.0, 2.0, 0.0),
                Point3::new(4.0, 0.0, 0.0),
            ],
            vec![1.0, 1.0, 1.0, 1.0],
        )
        .expect("valid cubic")
    }

    /// Bilinear flat patch (z=0 plane, from (0,0) to (1,1)).
    fn flat_patch() -> NurbsSurface {
        NurbsSurface::new(
            1,
            1,
            vec![0.0, 0.0, 1.0, 1.0],
            vec![0.0, 0.0, 1.0, 1.0],
            vec![
                vec![Point3::new(0.0, 0.0, 0.0), Point3::new(1.0, 0.0, 0.0)],
                vec![Point3::new(0.0, 1.0, 0.0), Point3::new(1.0, 1.0, 0.0)],
            ],
            vec![vec![1.0, 1.0], vec![1.0, 1.0]],
        )
        .expect("valid flat patch")
    }

    // -- Curve tests -------------------------------------------------------

    #[test]
    fn project_to_line() {
        let c = line_curve();
        // Point (5, 3, 0) — closest point should be (5, 0, 0) at u=0.5.
        let res =
            project_point_to_curve(&c, Point3::new(5.0, 3.0, 0.0), TOL).expect("should converge");
        assert!((res.parameter - 0.5).abs() < TOL, "u={}", res.parameter);
        assert!((res.point.x() - 5.0).abs() < TOL);
        assert!((res.point.y()).abs() < TOL);
        assert!((res.distance - 3.0).abs() < TOL, "dist={}", res.distance);
    }

    #[test]
    #[allow(clippy::suboptimal_flops)]
    fn project_to_circle() {
        let c = quarter_circle();
        // Point (2, 2, 0) — closest point should be on the unit circle at 45°.
        let res =
            project_point_to_curve(&c, Point3::new(2.0, 2.0, 0.0), TOL).expect("should converge");
        let expected = std::f64::consts::FRAC_1_SQRT_2;
        assert!(
            (res.point.x() - expected).abs() < 1e-6,
            "x={} expected={}",
            res.point.x(),
            expected
        );
        assert!(
            (res.point.y() - expected).abs() < 1e-6,
            "y={} expected={}",
            res.point.y(),
            expected
        );
        // Distance from (2,2) to unit circle at 45° = sqrt(8) - 1.
        let expected_dist = 2.0_f64.hypot(2.0) - 1.0;
        assert!(
            (res.distance - expected_dist).abs() < 1e-6,
            "dist={} expected={}",
            res.distance,
            expected_dist
        );
    }

    #[test]
    fn project_endpoint() {
        let c = cubic_bezier();
        // Project a point very close to the start endpoint.
        let res =
            project_point_to_curve(&c, Point3::new(0.0, 0.01, 0.0), TOL).expect("should converge");
        assert!(res.distance < 0.02, "dist={}", res.distance);
        assert!(res.parameter < 0.1, "u={}", res.parameter);
    }

    #[test]
    fn project_far_point() {
        let c = cubic_bezier();
        // A point far away should still converge.
        let res =
            project_point_to_curve(&c, Point3::new(2.0, 100.0, 0.0), TOL).expect("should converge");
        // The closest point should be roughly at the top of the curve (y ≈ 1.5).
        assert!(res.point.y() > 0.0);
        assert!(res.distance < 100.0);
    }

    #[test]
    fn project_on_curve() {
        let c = cubic_bezier();
        // Evaluate a point on the curve, then project it back.
        let u_orig = 0.3;
        let pt_on = c.evaluate(u_orig);
        let res = project_point_to_curve(&c, pt_on, TOL).expect("should converge");
        assert!(res.distance < TOL, "dist={}", res.distance);
        assert!(
            (res.parameter - u_orig).abs() < 1e-4,
            "u={} expected={}",
            res.parameter,
            u_orig
        );
    }

    /// A cubic B-spline over four spans: points on it at and beside its
    /// interior knots, and points off it, project to the closest of 100,001
    /// samples or closer.
    #[test]
    fn project_across_knot_spans() {
        let c = NurbsCurve::new(
            3,
            vec![0.0, 0.0, 0.0, 0.0, 0.25, 0.5, 0.75, 1.0, 1.0, 1.0, 1.0],
            vec![
                Point3::new(0.0, 0.0, 0.0),
                Point3::new(1.0, 2.0, 0.0),
                Point3::new(2.0, -1.0, 0.5),
                Point3::new(3.0, 2.5, 0.0),
                Point3::new(4.0, 0.0, -0.5),
                Point3::new(5.0, 1.5, 0.0),
                Point3::new(6.0, 0.0, 0.0),
            ],
            vec![1.0; 7],
        )
        .expect("valid cubic");
        let dense = dense_samples(&c);
        for u in [0.25, 0.25 + 1e-3, 0.5 - 1e-3, 0.5, 0.75, 0.75 + 1e-3] {
            let on = c.evaluate(u);
            let res = project_point_to_curve(&c, on, TOL).expect("should converge");
            assert!(res.distance < TOL, "u={u}: dist {}", res.distance);
            for off in [Vec3::new(0.0, 0.3, 0.2), Vec3::new(0.1, -0.4, 0.0)] {
                let p = on + off;
                let res = project_point_to_curve(&c, p, TOL).expect("should converge");
                let brute = closest_sample(&dense, p);
                assert!(
                    res.distance <= brute + 1e-9,
                    "u={u}: {} > {brute}",
                    res.distance
                );
            }
        }
    }

    /// 100,001 points evenly spaced in the parameter of a curve on `[0, 1]`.
    fn dense_samples(c: &NurbsCurve) -> Vec<Point3> {
        (0..=100_000)
            .map(|k| c.evaluate(f64::from(k) / 100_000.0))
            .collect()
    }

    fn closest_sample(dense: &[Point3], p: Point3) -> f64 {
        dense
            .iter()
            .map(|q| (*q - p).length())
            .fold(f64::INFINITY, f64::min)
    }

    /// A rational quadratic with a corner (a double knot) at `u = 0.5`:
    /// points on it beside the corner project to themselves, and points
    /// below the corner to the closest of 100,001 samples or closer. A run
    /// seeded at the corner reads the far piece's derivatives there unless
    /// it stays in its own span.
    #[test]
    fn project_beside_a_rational_corner() {
        let c = NurbsCurve::new(
            2,
            vec![0.0, 0.0, 0.0, 0.5, 0.5, 1.0, 1.0, 1.0],
            vec![
                Point3::new(0.0, 1.0, 0.0),
                Point3::new(0.8, 0.2, 0.0),
                Point3::new(1.0, 0.0, 0.0),
                Point3::new(1.2, 0.2, 0.0),
                Point3::new(2.0, 1.0, 0.0),
            ],
            vec![1.0, 2.0, 1.0, 0.5, 1.0],
        )
        .expect("valid quadratic");
        for u in [0.48, 0.495, 0.4999, 0.5, 0.5001, 0.505, 0.52] {
            let res = project_point_to_curve(&c, c.evaluate(u), TOL).expect("should converge");
            assert!(res.distance < TOL, "u={u}: dist {}", res.distance);
        }
        let corner = project_point_to_curve(&c, Point3::new(1.038, -0.282, 0.0), TOL)
            .expect("should converge");
        assert!(
            (corner.parameter - 0.5).abs() < 1e-9,
            "u={}",
            corner.parameter
        );
        let dense = dense_samples(&c);
        for i in 0..6 {
            for j in 0..6 {
                let p = Point3::new(0.5 + 0.2 * f64::from(i), -1.0 + 0.2 * f64::from(j), 0.0);
                let res = project_point_to_curve(&c, p, TOL).expect("should converge");
                let brute = closest_sample(&dense, p);
                assert!(
                    res.distance <= brute + 1e-9,
                    "{p:?}: {} > {brute}",
                    res.distance
                );
            }
        }
    }

    /// A quadratic with a span 0.0008 long beside a long one: the short
    /// span's samples crowd together, and points on the long span beside it
    /// project to themselves.
    #[test]
    fn project_beside_a_short_span() {
        let pts = [
            (0.369, 2.254, 0.523),
            (1.49, 0.935, 1.239),
            (2.611, 2.353, 0.691),
            (3.864, 0.113, 0.841),
            (4.381, 1.241, 0.004),
            (5.108, 0.051, 1.203),
            (6.235, 2.743, 1.368),
        ];
        let c = NurbsCurve::new(
            2,
            vec![0.0, 0.0, 0.0, 0.404, 0.404, 0.5225, 0.5233, 1.0, 1.0, 1.0],
            pts.iter().map(|&(x, y, z)| Point3::new(x, y, z)).collect(),
            vec![1.0; 7],
        )
        .expect("valid quadratic");
        for u in [0.526, 0.527, 0.528, 0.53, 0.535, 0.54] {
            let res = project_point_to_curve(&c, c.evaluate(u), TOL).expect("should converge");
            assert!(
                res.distance < TOL,
                "u={u}: at {} dist {}",
                res.parameter,
                res.distance
            );
        }
    }

    /// A rational quartic whose long span ends at a double knot: from the
    /// seed at that knot a Newton step overshoots the point on the curve at
    /// `u = 0.847` and the next one clamps back to the knot, a cycle that
    /// halving the steps back toward the closer iterate breaks.
    #[test]
    fn project_where_newton_overshoots() {
        let pts = [
            (0.214, 2.374, 0.023),
            (1.728, 0.844, 0.444),
            (2.048, 2.959, 0.461),
            (3.318, 2.28, 1.146),
            (4.482, 0.889, 1.641),
            (5.768, 1.669, 1.668),
            (6.109, 0.189, 0.556),
            (7.604, 0.684, 1.081),
            (8.086, 1.522, 1.079),
            (9.709, 2.279, 1.852),
            (10.577, 2.794, 1.997),
            (11.147, 0.728, 1.253),
            (12.239, 2.713, 1.874),
        ];
        let c = NurbsCurve::new(
            4,
            vec![
                0.0, 0.0, 0.0, 0.0, 0.0, 0.247, 0.247, 0.247, 0.247, 0.877, 0.877, 0.913, 0.913,
                1.0, 1.0, 1.0, 1.0, 1.0,
            ],
            pts.iter().map(|&(x, y, z)| Point3::new(x, y, z)).collect(),
            vec![
                1.138, 1.092, 0.938, 1.464, 0.77, 1.44, 1.286, 0.536, 1.319, 0.675, 1.002, 0.793,
                1.279,
            ],
        )
        .expect("valid quartic");
        let res = project_point_to_curve(&c, c.evaluate(0.847), TOL).expect("should converge");
        assert!(
            res.distance < TOL,
            "u={}: dist {}",
            res.parameter,
            res.distance
        );
    }

    // -- Surface tests -----------------------------------------------------

    #[test]
    fn project_to_flat_quad() {
        let s = flat_patch();
        // Point (0.5, 0.5, 3.0) — should project to (0.5, 0.5, 0.0).
        let res =
            project_point_to_surface(&s, Point3::new(0.5, 0.5, 3.0), TOL).expect("should converge");
        assert!((res.point.x() - 0.5).abs() < TOL, "x={}", res.point.x());
        assert!((res.point.y() - 0.5).abs() < TOL, "y={}", res.point.y());
        assert!((res.point.z()).abs() < TOL, "z={}", res.point.z());
        assert!((res.distance - 3.0).abs() < TOL, "dist={}", res.distance);
    }

    #[test]
    fn project_on_surface() {
        let s = flat_patch();
        // Point directly on the surface.
        let res =
            project_point_to_surface(&s, Point3::new(0.3, 0.7, 0.0), TOL).expect("should converge");
        assert!(res.distance < TOL, "dist={}", res.distance);
    }

    #[test]
    fn project_above_surface() {
        let s = flat_patch();
        // Point at height 1 above the center.
        let res =
            project_point_to_surface(&s, Point3::new(0.5, 0.5, 1.0), TOL).expect("should converge");
        assert!(
            (res.distance - 1.0).abs() < TOL,
            "dist={} expected=1.0",
            res.distance
        );
        assert!((res.u - 0.5).abs() < TOL, "u={}", res.u);
        assert!((res.v - 0.5).abs() < TOL, "v={}", res.v);
    }

    /// Bilinear degenerate "cone apex" patch.
    ///
    /// Control grid:
    ///   v=0 row: apex=(0,0,0)  apex=(0,0,0)   ← S_u = 0 everywhere on this row
    ///   v=1 row: (-1,0,1)      (1,0,1)
    ///
    /// Parametric formula: S(u,v) = (v·(2u-1), 0, v)
    ///
    /// The Jacobian is rank-1 at v=0 (both S_u and S_v are degenerate there),
    /// which triggers the LM-regularisation branch in `project_point_to_surface`.
    fn apex_patch() -> NurbsSurface {
        NurbsSurface::new(
            1,
            1,
            vec![0.0, 0.0, 1.0, 1.0],
            vec![0.0, 0.0, 1.0, 1.0],
            vec![
                vec![Point3::new(0.0, 0.0, 0.0), Point3::new(0.0, 0.0, 0.0)], // v=0: apex
                vec![Point3::new(-1.0, 0.0, 1.0), Point3::new(1.0, 0.0, 1.0)], // v=1: base
            ],
            vec![vec![1.0, 1.0], vec![1.0, 1.0]],
        )
        .expect("valid apex patch")
    }

    /// Project a point whose nearest surface location is the degenerate apex.
    ///
    /// The surface S(u,v)=(v(2u−1), 0, v) lies in the xz-plane.  The query
    /// point (0, 1, 0) is displaced only in y, so its nearest surface point is
    /// the apex (0,0,0) — the only point that minimises the xz-distance.
    /// Without LM regularisation the Newton step blows up at v→0; with it the
    /// solver should converge and return (u≈0.5, v≈0, dist≈1).
    #[test]
    fn project_to_apex_singularity() {
        let s = apex_patch();
        let res = project_point_to_surface(&s, Point3::new(0.0, 1.0, 0.0), 1e-6)
            .expect("should converge at cone apex singularity");
        // Nearest point must be the apex.
        assert!(
            res.point.x().abs() < 1e-6 && res.point.y().abs() < 1e-6 && res.point.z().abs() < 1e-6,
            "nearest point should be apex, got ({:.4},{:.4},{:.4})",
            res.point.x(),
            res.point.y(),
            res.point.z()
        );
        assert!(
            (res.distance - 1.0).abs() < 1e-6,
            "distance to apex should be 1.0, got {:.8}",
            res.distance
        );
    }

    /// Project a point off-axis but close to the apex.  The solver must still
    /// converge despite starting near the singularity.
    #[test]
    fn project_near_apex_off_axis() {
        let s = apex_patch();
        // S(0.7, 0.05) = (0.05*(2*0.7-1), 0, 0.05) = (0.05*0.4, 0, 0.05) = (0.02, 0, 0.05)
        // Query close to that surface point but displaced in y.
        let res = project_point_to_surface(&s, Point3::new(0.02, 0.3, 0.05), 1e-6)
            .expect("should converge near apex");
        assert!(
            (res.distance - 0.3).abs() < 0.02,
            "expected distance ≈ 0.3, got {:.6}",
            res.distance
        );
        // Nearest surface point should be close to S(0.7, 0.05) = (0.02, 0, 0.05).
        assert!(
            (res.point.z() - 0.05).abs() < 0.02,
            "nearest point z should be ≈ 0.05, got z={:.4}",
            res.point.z()
        );
    }
}