1100 lines
51 KiB
Rust
1100 lines
51 KiB
Rust
//! Filling a closed loop of curves with one surface: a non-rational bicubic B-spline height
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//! field over the loop's best-fit plane, fitted to the loop by least squares with a smoothness
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//! term. A loop that cannot be seen as the boundary of a height field over that plane (it
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//! crosses itself, folds back or runs along the plane's normal when seen along that normal) is
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//! refused, never filled with wrong geometry.
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//!
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//! References: Piegl & Tiller, *The NURBS Book* (2nd ed.) — §9.4.1 and §9.4.3 (least squares
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//! approximation: minimise the sum of squared residuals, eq. 9.62, through the normal
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//! equations, eq. 9.65), §3.4 (tensor product B-spline surfaces), §3.3 (derivative of a
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//! B-spline curve, eq. 3.4: control values at the knot averages reproduce a linear function),
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//! A2.1 (span search).
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use crate::Vec3;
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use crate::bspline::{Dir, HPoint, NurbsSurface, Side, SplineError, SplineResult, find_span, invalid, local_basis_array};
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/// Most loop points a fill takes.
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pub const MAX_LOOP_POINTS: usize = 8192;
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/// Most spans of the height field in each direction.
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pub const MAX_FILL_SPANS: usize = 32;
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/// How full a patch's middle may be made (`crown`): 0 the least-bending fill, positive fuller,
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/// negative flatter, in fractions of the loop's size. Outside this range the value is refused.
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pub const MIN_CROWN: f64 = -0.9;
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pub const MAX_CROWN: f64 = 3.0;
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/// Fewest spans of the height field in each direction (the first fit tried).
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const MIN_FILL_SPANS: usize = 4;
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/// The smoothness energy's terms: `(order in u, order in v, weight)` of `∫∫ (∂ h)²`.
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const ENERGY: [(usize, usize, f64); 3] = [(2, 0, 1.0), (1, 1, 2.0), (0, 2, 1.0)];
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/// Four-point Gauss–Legendre nodes and weights on `[−1, 1]` (exact to degree 7).
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const GAUSS4: [(f64, f64); 4] = [
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(-0.861_136_311_594_052_6, 0.347_854_845_137_453_8),
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(-0.339_981_043_584_856_3, 0.652_145_154_862_546_2),
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(0.339_981_043_584_856_3, 0.652_145_154_862_546_2),
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(0.861_136_311_594_052_6, 0.347_854_845_137_453_8),
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];
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/// The smoothness term's weight per fitted point (see [`fill_loop`]).
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pub(crate) const SMOOTHNESS: f64 = 1e-7;
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/// A loop step shorter than this fraction of its length in space, seen along the normal, runs
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/// along the normal (steeper than about 87° from the plane).
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const MIN_STEP_SLOPE: f64 = 0.05;
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/// Consecutive steps closer to reversed than this cosine fold the loop back on itself.
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const FOLD_COS: f64 = -0.999;
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/// A loop's fill: the surface (`S(u, v) = o + u·x + v·y + h(u, v)·n` over the plane with normal
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/// `n`, so `∂S/∂u × ∂S/∂v` points along `n`), the plane's frame, whether the loop runs counter-
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/// clockwise about it, the largest distance from a sampled loop point to the surface along `n`,
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/// and how many points of the loop were evaluated.
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#[derive(Clone, Debug, PartialEq)]
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pub struct LoopFill {
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pub surface: NurbsSurface,
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/// The plane's origin and its axes `x`, `y` (the surface's parameter directions).
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pub origin: Vec3,
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pub axes: [Vec3; 2],
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pub normal: Vec3,
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pub counter_clockwise: bool,
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pub deviation: f64,
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pub evaluations: usize,
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/// The crown the fill was asked for, how many control points took its bump, and how far the
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/// bump moved the surface at the sampled loop points (always exactly 0: the bump vanishes
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/// along the loop, so the boundary is untouched).
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pub crown: f64,
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pub crowned: usize,
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pub crown_drift: f64,
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}
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/// Fill the closed loop of `sides` curves (side `k` given by `point(k, t)`, `t` from 0 at its
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/// start to 1 at its end, which is where side `k+1` starts) with one surface passing within
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/// `tol` of the loop.
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///
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/// - Sampling: each side at `t = i/s` (fitted) and halfway between (checked), `s` from 16
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/// doubling while the fit needs denser points, at most [`MAX_LOOP_POINTS`] points in all. The
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/// sample parameters are the same run either way, so the fill does not depend on where the
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/// loop starts or which way it runs.
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/// - The plane: through the fitted points' centroid, normal by Newell's method, turned so that
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/// its largest component is positive; its first axis points to the fitted point farthest from
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/// the centroid in the plane (the highest of points as far). So the plane and its axes, and with them the grid and the fill,
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/// move with the loop (a reflected axis gives the same grid).
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/// - Refused ("not supported yet"): a loop that, seen along the normal, has a step much shorter
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/// than in space (it runs along the normal), turns straight back, or crosses or touches
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/// itself; such a loop does not bound a height field over the plane.
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/// - The height field: a bicubic B-spline with uniform knots over the projected loop's box
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/// (widened by 1%); its control points sit at the knot averages in the plane, so its surface
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/// parameters are plane coordinates, and their heights are the unknowns. The heights minimise
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/// `Σ (h(uₖ, vₖ) − hₖ)² + λ·A · ∫∫ h_uu² + 2·h_uv² + h_vv²` over the fitted points — least
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/// squares (eq. 9.62, normal equations eq. 9.65) plus the thin-plate bending energy over the
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/// box, `A` the area of one grid cell (so each cell's energy counts like a squared second
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/// difference of heights, and the balance is the same at any size). The energy decides the
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/// inside, where there are no points; it is zero only for planes, so a plane through the
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/// loop is reproduced exactly, and any other rim gets the least-bending surface on it, which
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/// is free to turn at the rim (it is not the surface that made the rim). It does not depend
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/// on the grid's directions. `λ = 10⁻⁷ ·` (fitted points) `/` (control points): just enough
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/// to fix the inside, small enough not to pull the surface off the loop.
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/// - Grids of 4, 8, 16, 32 spans across the box's longer side are tried, with as many across
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/// the shorter side as keep the cells about square; a grid whose cells are under twice the
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/// fitted points' largest step in a direction (cells between points would be left to the
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/// energy alone) waits for denser sampling. The first grid within `tol` of every sampled
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/// point (measured along the normal, which bounds the distance) is the fill; otherwise the
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/// sampling doubles, and when that runs out, "not supported yet" with the closest deviation
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/// reached.
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/// - `crown` (0 for the plain fill, between [`MIN_CROWN`] and [`MAX_CROWN`]) fills the middle:
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/// the surface gains `crown · size · 16·s·(1−s)·t·(1−t)` (`s`, `t` the box's coordinates scaled
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/// to 0…1, `size` the box's diagonal: 1 at the middle, 0 on the box's boundary), applied to the
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/// control points whose support keeps clear of the loop, so the bump vanishes along the loop
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/// and the boundary does not move at all. The fitted net may be too coarse to hold such a
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/// point; it is then refined by knot insertion, which leaves the surface itself as it is. A
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/// loop whose fill leaves no room inside it even so is refused.
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pub fn fill_loop(sides: usize, point: &mut dyn FnMut(usize, f64) -> Option<Vec3>, tol: f64, crown: f64) -> SplineResult<LoopFill> {
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if sides == 0 || sides.saturating_mul(16) > MAX_LOOP_POINTS / 2 {
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return Err(SplineError::TooLarge(format!("a loop of {sides} sides (at most {})", MAX_LOOP_POINTS / 32)));
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}
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if !(tol.is_finite() && tol > 0.0) {
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return invalid("the fill tolerance must be positive");
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}
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// A range test also throws out NaN.
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if !(MIN_CROWN..=MAX_CROWN).contains(&crown) {
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return invalid("the crown must be between -0.9 and 3");
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}
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let mut evaluations = 0usize;
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let mut eval = |k: usize, t: f64| -> SplineResult<Vec3> {
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evaluations += 1;
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point(k, t).filter(|p| p.is_finite()).ok_or_else(|| SplineError::Invalid(format!("side {k} has no finite point at {t}")))
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};
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let mut per = 16usize;
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// Fitted points: side by side, per side at i/per (i < per).
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let mut fit: Vec<Vec3> = Vec::with_capacity(sides * per);
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for k in 0..sides {
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for i in 0..per {
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fit.push(eval(k, i as f64 / per as f64)?);
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}
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}
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let mut deviation = f64::INFINITY;
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loop {
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let mut check = Vec::with_capacity(fit.len());
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for k in 0..sides {
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for i in 0..per {
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check.push(eval(k, (i as f64 + 0.5) / per as f64)?);
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}
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}
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// The loop in order: fitted and checked points alternating.
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let all: Vec<Vec3> = fit.iter().zip(&check).flat_map(|(a, b)| [*a, *b]).collect();
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match fit_once(&fit, &all, tol, crown)? {
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Fit::Done(mut fill) => {
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fill.evaluations = evaluations;
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return Ok(*fill);
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}
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Fit::Denser(dev) => deviation = deviation.min(dev),
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Fit::Exhausted(dev) => {
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deviation = deviation.min(dev);
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break;
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}
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}
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if all.len() * 2 > MAX_LOOP_POINTS {
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break;
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}
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fit = all;
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per *= 2;
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}
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Err(SplineError::Unsupported(format!("filling a loop this wavy with one surface (it stays {deviation:e} off the loop, tolerance {tol:e})")))
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}
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/// What one round of [`fill_loop`] gave.
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enum Fit {
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Done(Box<LoopFill>),
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/// Within tolerance only with cells finer than the points allow: sample denser.
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Denser(f64),
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/// The finest grid tried is not within tolerance.
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Exhausted(f64),
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}
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/// One round of [`fill_loop`]: fit the points `fit`, check every point of `all` (the loop in
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/// order, `fit` among them).
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fn fit_once(fit: &[Vec3], all: &[Vec3], tol: f64, crown: f64) -> SplineResult<Fit> {
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let m = fit.len();
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let o = fit.iter().fold(Vec3::ZERO, |a, p| a + *p) / m as f64;
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let at = |k: usize| fit.get(k % m).copied().unwrap_or(Vec3::ZERO);
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let area = (0..m).fold(Vec3::ZERO, |a, k| a + (at(k) - o).cross(at(k + 1) - o));
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let Some(normal) = area.normalized() else { return invalid("the loop encloses no area") };
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let largest = [normal.x, normal.y, normal.z].into_iter().fold(0.0f64, |a, c| if c.abs() > a.abs() { c } else { a });
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let counter_clockwise = largest > 0.0;
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let n = if counter_clockwise { normal } else { normal * -1.0 };
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// The first axis towards the fitted point farthest from the centroid in the plane: the axes,
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// the grid and so the fill move with the loop.
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let flat = |p: &Vec3| {
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let d = *p - o;
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d - n * d.dot(n)
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};
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// Points as far as each other (within rounding) are told apart by their height, and points
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// alike in both by their coordinates: the same point is chosen however the loop is given,
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// and as the loop moves, but for ties its symmetry makes.
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let key = |p: &Vec3| (flat(p).len(), (*p - o).dot(n));
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let reach = fit.iter().map(|p| key(p).0).fold(0.0, f64::max);
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let near = 1e-9 * reach;
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let far = fit.iter().fold(None::<&Vec3>, |best, p| match best {
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None => Some(p),
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Some(b) => {
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let ((db, hb), (dp, hp)) = (key(b), key(p));
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let better = if (dp - db).abs() > near {
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dp > db
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} else if (hp - hb).abs() > near {
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hp > hb
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} else {
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(p.x, p.y, p.z) > (b.x, b.y, b.z)
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};
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Some(if better { p } else { b })
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}
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});
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let far = far.map(flat).unwrap_or(Vec3::ZERO);
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let x = far.normalized().ok_or_else(|| SplineError::Invalid("the loop has no extent in its plane".into()))?;
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let y = n.cross(x);
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let project = |pts: &[Vec3]| -> (Vec<[f64; 2]>, Vec<f64>) {
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(pts.iter().map(|p| [(*p - o).dot(x), (*p - o).dot(y)]).collect(), pts.iter().map(|p| (*p - o).dot(n)).collect())
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};
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let (uv, h) = project(all);
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check_projectable(all, &uv)?;
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let (fuv, fh) = project(fit);
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let (mut lo, mut hi) = ([f64::INFINITY; 2], [f64::NEG_INFINITY; 2]);
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for q in &uv {
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for d in 0..2 {
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lo[d] = lo[d].min(q[d]);
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hi[d] = hi[d].max(q[d]);
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}
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}
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for d in 0..2 {
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let pad = 0.01 * (hi[d] - lo[d]);
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lo[d] -= pad;
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hi[d] += pad;
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}
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if !(lo[0] < hi[0] && lo[1] < hi[1] && (hi[0] - lo[0]).is_finite() && (hi[1] - lo[1]).is_finite()) {
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return invalid("the loop's extent is not finite");
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}
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// Spans per direction: `level` across the longer side of the box, as many across the
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// shorter as keep the cells about square (at least one). A grid finer than the fitted
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// points can hold in a direction (cells under twice their largest step along it) is tried
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// only after denser sampling; the coarsest is always tried.
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let extent = [hi[0] - lo[0], hi[1] - lo[1]];
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let mut step = [0.0f64; 2];
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for k in 0..m {
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let (p, q) = (fuv.get(k).copied().unwrap_or([0.0; 2]), fuv.get((k + 1) % m).copied().unwrap_or([0.0; 2]));
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for d in 0..2 {
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step[d] = step[d].max((q[d] - p[d]).abs());
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}
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}
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let longer = extent[0].max(extent[1]);
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let mut deviation = f64::INFINITY;
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let mut level = MIN_FILL_SPANS;
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while level <= MAX_FILL_SPANS {
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let spans = [0, 1].map(|d| ((extent[d] / longer * level as f64).ceil() as usize).clamp(1, level));
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let holds = (0..2).all(|d| extent[d] / spans[d] as f64 >= 2.0 * step[d]);
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if !holds && level > MIN_FILL_SPANS {
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return Ok(Fit::Denser(deviation));
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}
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let grid = Grid::new(spans, lo, hi);
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let heights = grid.fit(&fuv, &fh)?;
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let dev = uv.iter().zip(&h).map(|(q, hk)| (grid.height(&heights, *q) - hk).abs()).fold(0.0, f64::max);
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if !dev.is_finite() {
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return invalid("the fit is not finite");
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}
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deviation = deviation.min(dev);
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if dev <= tol && holds {
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let surface = grid.surface(&heights, o, x, y, n)?;
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let (surface, crowned, crown_drift) = crown_fill(surface, &uv, crown, n)?;
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let fill = LoopFill {
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surface,
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origin: o,
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axes: [x, y],
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normal: n,
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counter_clockwise,
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deviation: dev,
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evaluations: 0,
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crown,
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crowned,
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crown_drift,
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};
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return Ok(Fit::Done(Box::new(fill)));
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}
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if !holds {
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return Ok(Fit::Denser(deviation));
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}
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level *= 2;
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}
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Ok(Fit::Exhausted(deviation))
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}
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/// The fitted surface with the crown's bump (see [`fill_loop`]): `crown == 0` gives it back as it
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/// is. A control point whose support rectangle keeps clear of the loop takes the bump's blossom
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/// coefficients, so the surface between the perturbed points is the bubble exactly and the
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/// surface over the loop is not moved at all (those points' basis functions are zero all along
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/// it). A fitted net is often too coarse to hold one; it is then refined by knot insertion, which
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/// leaves the surface itself as it is, until it does. Returns the crowned surface, how many
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/// control points took the bump, and the farthest the bump moved the surface at a sampled loop
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/// point (0).
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fn crown_fill(surface: NurbsSurface, uv: &[[f64; 2]], crown: f64, normal: Vec3) -> SplineResult<(NurbsSurface, usize, f64)> {
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if crown == 0.0 {
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return Ok((surface, 0, 0.0));
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}
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if !normal.is_finite() {
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return invalid("the loop's extent is not finite");
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}
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crown_fill_along(surface, uv, crown, &|_| Some(normal), MAX_FILL_SPANS)
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}
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/// [`crown_fill`] moving each control point along `along(ξ, η)` (at its Greville point, the mean
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/// of its knots each way) rather than one normal: what a fill that is not a height field takes,
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/// with its own normal there. Its net may be refined up to `max_spans` each way.
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pub(crate) fn crown_fill_along(
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surface: NurbsSurface,
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uv: &[[f64; 2]],
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crown: f64,
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along: &dyn Fn([f64; 2]) -> Option<Vec3>,
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max_spans: usize,
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) -> SplineResult<(NurbsSurface, usize, f64)> {
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if crown == 0.0 {
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return Ok((surface, 0, 0.0));
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}
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let gap = loop_gap(uv);
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let mut base = surface;
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loop {
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if let Some((crowned, points)) = crown_net(&base, uv, gap, crown, along)? {
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let mut drift = 0.0f64;
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for q in uv {
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if let (Ok(a), Ok(b)) = (base.eval(q[0], q[1]), crowned.eval(q[0], q[1])) {
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drift = drift.max(a.dist(b));
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}
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}
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if drift > 0.0 {
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return Err(SplineError::Unsupported("crowning a patch whose loop the fill cannot keep clear".into()));
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}
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return Ok((crowned, points, drift));
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}
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// No room inside the loop yet: halve the cells (the surface itself does not change).
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let spans = [0, 1].map(|d| base.count()[d].saturating_sub(3));
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if spans[0] >= max_spans && spans[1] >= max_spans {
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return Err(SplineError::Unsupported("crowning a patch whose loop leaves no room inside it".into()));
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}
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for (d, &s) in spans.iter().enumerate() {
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if s < max_spans {
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base = subdivide(&base, d)?;
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}
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}
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}
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}
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/// The crown's bump in the net `base`: a control point whose support rectangle keeps clear of the
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/// loop takes `crown · size · B(ξ, η)`, `B` the bubble's blossom at the point's knots (so the
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/// surface between the perturbed points is exactly the bubble, not a fit of it) and `size` the
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/// box's diagonal. `None` when the net has no such point.
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fn crown_net(
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base: &NurbsSurface,
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uv: &[[f64; 2]],
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gap: [f64; 2],
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crown: f64,
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along: &dyn Fn([f64; 2]) -> Option<Vec3>,
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) -> SplineResult<Option<(NurbsSurface, usize)>> {
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let [cu, cv] = base.count();
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let degree = base.degree();
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let knots = base.knots();
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let domain = base.domain();
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let size = (domain[0][1] - domain[0][0]).hypot(domain[1][1] - domain[1][0]);
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if !(size > 0.0 && size.is_finite()) {
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return invalid("the loop's extent is not finite");
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}
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// The bubble `4·s·(1−s)` per direction, as a quadratic in the plane coordinate.
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let quad = |d: usize| {
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let (lo, hi) = (domain[d][0], domain[d][1]);
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let w = hi - lo;
|
||
(-4.0 * lo * hi / (w * w), 4.0 * (lo + hi) / (w * w), -4.0 / (w * w))
|
||
};
|
||
let (qu, qv) = (quad(0), quad(1));
|
||
let (Some(knots_u), Some(knots_v)) = (knots.first(), knots.get(1)) else { return invalid("the surface has no knot vectors") };
|
||
let mut ctrl = base.homogeneous().to_vec();
|
||
let mut points = 0usize;
|
||
for j in 0..cv {
|
||
let Some((v_span, v_knots)) = support(knots_v, degree[1], j) else { continue };
|
||
for i in 0..cu {
|
||
let Some((u_span, u_knots)) = support(knots_u, degree[0], i) else { continue };
|
||
if !clear_of_loop(u_span, v_span, uv, gap) {
|
||
continue;
|
||
}
|
||
let bump = crown * size * blossom(qu, u_knots) * blossom(qv, v_knots);
|
||
let mean = |t: [f64; 3]| (t[0] + t[1] + t[2]) / 3.0;
|
||
let Some(normal) = along([mean(u_knots), mean(v_knots)]).filter(|n| n.is_finite()) else { continue };
|
||
let Some(slot) = ctrl.get_mut(j * cu + i) else { continue };
|
||
let h = *slot;
|
||
let Some(p) = h.project() else { continue };
|
||
*slot = HPoint::weighted(p + normal * bump, h.w);
|
||
points += 1;
|
||
}
|
||
}
|
||
if points == 0 {
|
||
return Ok(None);
|
||
}
|
||
Ok(Some((NurbsSurface::from_homogeneous(degree, knots.clone(), ctrl)?, points)))
|
||
}
|
||
|
||
/// The same surface with a knot inserted in the middle of every span of direction `d` (0 = u):
|
||
/// the surface itself does not change, only its net is finer, which is what gives the crown room
|
||
/// inside the loop.
|
||
fn subdivide(surface: &NurbsSurface, d: usize) -> SplineResult<NurbsSurface> {
|
||
let (degree, count) = (surface.degree()[d], surface.count()[d]);
|
||
let knots = surface.knots()[d].clone();
|
||
let mut mids = Vec::new();
|
||
for i in degree..count {
|
||
let (a, b) = (knots.get(i).copied().unwrap_or(0.0), knots.get(i + 1).copied().unwrap_or(0.0));
|
||
if b > a {
|
||
mids.push(0.5 * (a + b));
|
||
}
|
||
}
|
||
let mut out = surface.clone();
|
||
for m in mids {
|
||
out = out.insert_knot(if d == 0 { Dir::S } else { Dir::T }, m)?;
|
||
}
|
||
Ok(out)
|
||
}
|
||
|
||
/// A control point's knot interval — where its basis is non-zero — and its knot triple, whose
|
||
/// blossom is the value the point must take for the spline to reproduce a quadratic exactly.
|
||
fn support(knots: &[f64], degree: usize, i: usize) -> Option<([f64; 2], [f64; 3])> {
|
||
let interval = [*knots.get(i)?, *knots.get(i + degree + 1)?];
|
||
let triple = [*knots.get(i + 1)?, *knots.get(i + 2)?, *knots.get(i + 3)?];
|
||
(interval[1] >= interval[0]).then_some((interval, triple))
|
||
}
|
||
|
||
/// A quadratic `a + b·x + c·x²`'s blossom at a knot triple.
|
||
fn blossom(quad: (f64, f64, f64), t: [f64; 3]) -> f64 {
|
||
let (a, b, c) = quad;
|
||
let mean = (t[0] + t[1] + t[2]) / 3.0;
|
||
let pair = (t[0] * t[1] + t[1] * t[2] + t[0] * t[2]) / 3.0;
|
||
a + b * mean + c * pair
|
||
}
|
||
|
||
/// Whether the parameter rectangle `u` × `v` keeps clear of the sampled loop: its middle is
|
||
/// inside the loop and no loop segment comes within `gap` of it. A control point whose support
|
||
/// keeps clear takes the crown's bump without moving the surface over the loop.
|
||
fn clear_of_loop(u: [f64; 2], v: [f64; 2], uv: &[[f64; 2]], gap: [f64; 2]) -> bool {
|
||
if !inside_loop([(u[0] + u[1]) / 2.0, (v[0] + v[1]) / 2.0], uv) {
|
||
return false;
|
||
}
|
||
let lo = [u[0] - gap[0], v[0] - gap[1]];
|
||
let hi = [u[1] + gap[0], v[1] + gap[1]];
|
||
let m = uv.len();
|
||
(0..m).all(|k| {
|
||
let (Some(a), Some(b)) = (uv.get(k).copied(), uv.get((k + 1) % m).copied()) else { return false };
|
||
!segment_meets_rect(a, b, lo, hi)
|
||
})
|
||
}
|
||
|
||
/// The largest step between neighbouring samples of the loop, per direction: a rectangle this far
|
||
/// clear of the sampled loop cannot be entered by the edges between the samples either.
|
||
fn loop_gap(uv: &[[f64; 2]]) -> [f64; 2] {
|
||
let m = uv.len();
|
||
let mut gap = [0.0f64; 2];
|
||
for k in 0..m {
|
||
let (Some(a), Some(b)) = (uv.get(k).copied(), uv.get((k + 1) % m).copied()) else { continue };
|
||
for d in 0..2 {
|
||
gap[d] = gap[d].max((b[d] - a[d]).abs());
|
||
}
|
||
}
|
||
gap
|
||
}
|
||
|
||
/// Whether `p` is inside the closed loop `uv` (ray casting).
|
||
pub(crate) fn inside_loop(p: [f64; 2], uv: &[[f64; 2]]) -> bool {
|
||
let m = uv.len();
|
||
let mut inside = false;
|
||
for k in 0..m {
|
||
let (Some(a), Some(b)) = (uv.get(k).copied(), uv.get((k + 1) % m).copied()) else { continue };
|
||
if (a[1] > p[1]) != (b[1] > p[1]) && p[0] < a[0] + (p[1] - a[1]) / (b[1] - a[1]) * (b[0] - a[0]) {
|
||
inside = !inside;
|
||
}
|
||
}
|
||
inside
|
||
}
|
||
|
||
/// Whether the segment `p`–`q` meets the rectangle `lo`–`hi` (Liang–Barsky clipping).
|
||
fn segment_meets_rect(p: [f64; 2], q: [f64; 2], lo: [f64; 2], hi: [f64; 2]) -> bool {
|
||
let (mut t0, mut t1) = (0.0f64, 1.0f64);
|
||
for d in 0..2 {
|
||
let delta = q[d] - p[d];
|
||
if delta == 0.0 {
|
||
if p[d] < lo[d] || p[d] > hi[d] {
|
||
return false;
|
||
}
|
||
continue;
|
||
}
|
||
let (mut a, mut b) = ((lo[d] - p[d]) / delta, (hi[d] - p[d]) / delta);
|
||
if a > b {
|
||
std::mem::swap(&mut a, &mut b);
|
||
}
|
||
t0 = t0.max(a);
|
||
t1 = t1.min(b);
|
||
if t0 > t1 {
|
||
return false;
|
||
}
|
||
}
|
||
true
|
||
}
|
||
|
||
/// Refuse a loop that does not bound a height field over its plane (see [`fill_loop`]).
|
||
fn check_projectable(points: &[Vec3], uv: &[[f64; 2]]) -> SplineResult<()> {
|
||
let m = uv.len();
|
||
let q = |k: usize| uv.get(k % m).copied().unwrap_or([0.0; 2]);
|
||
let p = |k: usize| points.get(k % m).copied().unwrap_or(Vec3::ZERO);
|
||
let step = |k: usize| {
|
||
let (a, b) = (q(k), q(k + 1));
|
||
[b[0] - a[0], b[1] - a[1]]
|
||
};
|
||
for k in 0..m {
|
||
let d = step(k);
|
||
let flat = d[0].hypot(d[1]);
|
||
let space = p(k).dist(p(k + 1));
|
||
if space > 0.0 && flat < MIN_STEP_SLOPE * space {
|
||
return Err(SplineError::Unsupported("filling a loop that runs along its plane's normal".into()));
|
||
}
|
||
let e = step(k + 1);
|
||
let fe = e[0].hypot(e[1]);
|
||
if flat > 0.0 && fe > 0.0 && (d[0] * e[0] + d[1] * e[1]) < FOLD_COS * flat * fe {
|
||
return Err(SplineError::Unsupported("filling a loop that folds back on itself".into()));
|
||
}
|
||
}
|
||
// Non-adjacent segments must not meet: a sweep over the segments' u extents.
|
||
let mut segs: Vec<(f64, f64, usize)> = (0..m)
|
||
.map(|k| {
|
||
let (a, b) = (q(k), q(k + 1));
|
||
(a[0].min(b[0]), a[0].max(b[0]), k)
|
||
})
|
||
.collect();
|
||
segs.sort_by(|a, b| a.0.total_cmp(&b.0));
|
||
for (i, &(_, hi, k)) in segs.iter().enumerate() {
|
||
for &(lo2, _, l) in segs.get(i + 1..).into_iter().flatten() {
|
||
if lo2 > hi {
|
||
break;
|
||
}
|
||
let adjacent = (k + 1) % m == l || (l + 1) % m == k;
|
||
if !adjacent && segments_meet(q(k), q(k + 1), q(l), q(l + 1)) {
|
||
return Err(SplineError::Unsupported("filling a loop that crosses itself seen along its plane's normal".into()));
|
||
}
|
||
}
|
||
}
|
||
Ok(())
|
||
}
|
||
|
||
/// Whether the closed segments `ab` and `cd` share a point.
|
||
pub(crate) fn segments_meet(a: [f64; 2], b: [f64; 2], c: [f64; 2], d: [f64; 2]) -> bool {
|
||
let orient = |p: [f64; 2], q: [f64; 2], r: [f64; 2]| (q[0] - p[0]) * (r[1] - p[1]) - (q[1] - p[1]) * (r[0] - p[0]);
|
||
let within =
|
||
|p: [f64; 2], q: [f64; 2], r: [f64; 2]| r[0] >= p[0].min(q[0]) && r[0] <= p[0].max(q[0]) && r[1] >= p[1].min(q[1]) && r[1] <= p[1].max(q[1]);
|
||
let (d1, d2, d3, d4) = (orient(c, d, a), orient(c, d, b), orient(a, b, c), orient(a, b, d));
|
||
if ((d1 > 0.0 && d2 < 0.0) || (d1 < 0.0 && d2 > 0.0)) && ((d3 > 0.0 && d4 < 0.0) || (d3 < 0.0 && d4 > 0.0)) {
|
||
return true;
|
||
}
|
||
(d1 == 0.0 && within(c, d, a)) || (d2 == 0.0 && within(c, d, b)) || (d3 == 0.0 && within(a, b, c)) || (d4 == 0.0 && within(a, b, d))
|
||
}
|
||
|
||
/// A bicubic height field's grid: uniform clamped knots over `[lo, hi]` in each direction,
|
||
/// `count` control points along each.
|
||
pub(crate) struct Grid {
|
||
pub(crate) knots: [Vec<f64>; 2],
|
||
pub(crate) count: [usize; 2],
|
||
}
|
||
|
||
impl LoopFill {
|
||
/// The surface's point over (or under) `p`, along the normal.
|
||
pub fn point_over(&self, p: Vec3) -> SplineResult<Vec3> {
|
||
let d = p - self.origin;
|
||
let [x, y] = self.axes;
|
||
self.surface.eval(d.dot(x), d.dot(y))
|
||
}
|
||
}
|
||
|
||
impl Grid {
|
||
pub(crate) fn new(spans: [usize; 2], lo: [f64; 2], hi: [f64; 2]) -> Grid {
|
||
let knots = [0, 1].map(|d| {
|
||
let (a, b, s) = (lo[d], hi[d], spans[d]);
|
||
let mut k = vec![a; 4];
|
||
k.extend((1..s).map(|i| a + (b - a) * i as f64 / s as f64));
|
||
k.extend([b; 4]);
|
||
k
|
||
});
|
||
Grid { knots, count: spans.map(|s| s + 3) }
|
||
}
|
||
|
||
/// The length of one span in direction `d`.
|
||
pub(crate) fn cell(&self, d: usize) -> f64 {
|
||
let (Some(k), Some(&c)) = (self.knots.get(d), self.count.get(d)) else { return 0.0 };
|
||
(k.last().copied().unwrap_or(0.0) - k.first().copied().unwrap_or(0.0)) / (c - 3) as f64
|
||
}
|
||
|
||
/// The knot average `(kᵢ₊₁ + kᵢ₊₂ + kᵢ₊₃)/3` of control point `i` in direction `d`.
|
||
fn average(&self, d: usize, i: usize) -> f64 {
|
||
self.knots.get(d).map_or(0.0, |k| (1..=3).map(|t| k.get(i + t).copied().unwrap_or(0.0)).sum::<f64>() / 3.0)
|
||
}
|
||
|
||
/// The four non-zero basis functions' `r`-th derivatives and the first index, at `x` in
|
||
/// direction `d`.
|
||
pub(crate) fn basis_derivative(&self, d: usize, x: f64, r: usize) -> Option<(usize, [f64; 4])> {
|
||
let kn = self.knots.get(d)?;
|
||
let x = x.clamp(*kn.first()?, *kn.last()?);
|
||
let span = find_span(kn, 3, *self.count.get(d)?, x)?;
|
||
let side = if x >= *kn.last()? { Side::Left } else { Side::Right };
|
||
let first = span.checked_sub(3)?;
|
||
let mut out = [0.0; 4];
|
||
for (a, o) in out.iter_mut().enumerate() {
|
||
*o = kn.get(first + a..first + a + 5).map_or(0.0, |l| local_basis_array(l, x, side, r).get(r).copied().unwrap_or(0.0));
|
||
}
|
||
Some((first, out))
|
||
}
|
||
|
||
/// The height at `q` of the net `heights` (i fastest).
|
||
fn height(&self, heights: &[f64], q: [f64; 2]) -> f64 {
|
||
let (Some((i0, bu)), Some((j0, bv))) = (self.basis_derivative(0, q[0], 0), self.basis_derivative(1, q[1], 0)) else { return f64::NAN };
|
||
let mut s = 0.0;
|
||
for (b, wv) in bv.iter().enumerate() {
|
||
for (a, wu) in bu.iter().enumerate() {
|
||
s += wu * wv * heights.get((j0 + b) * self.count[0] + i0 + a).copied().unwrap_or(f64::NAN);
|
||
}
|
||
}
|
||
s
|
||
}
|
||
|
||
/// The control heights fitting the points (see [`fill_loop`]): the normal equations, banded
|
||
/// (unknowns i fastest, so coupled unknowns are at most `3·count[0] + 3` apart), solved by
|
||
/// Cholesky factorisation.
|
||
fn fit(&self, uv: &[[f64; 2]], h: &[f64]) -> SplineResult<Vec<f64>> {
|
||
let [cu, cv] = self.count;
|
||
let n = cu * cv;
|
||
let mut band = Band::new(n, 3 * cu + 3);
|
||
let mut rhs = vec![0.0; n];
|
||
let mut fitted = 0usize;
|
||
for (q, hk) in uv.iter().zip(h) {
|
||
let (Some((i0, bu)), Some((j0, bv))) = (self.basis_derivative(0, q[0], 0), self.basis_derivative(1, q[1], 0)) else {
|
||
return invalid("a loop point is off the grid");
|
||
};
|
||
let mut terms = [(0usize, 0.0f64); 16];
|
||
for (t, slot) in terms.iter_mut().enumerate() {
|
||
let (a, b) = (t % 4, t / 4);
|
||
*slot = ((j0 + b) * cu + i0 + a, bu.get(a).copied().unwrap_or(0.0) * bv.get(b).copied().unwrap_or(0.0));
|
||
}
|
||
for &(r, wr) in &terms {
|
||
if let Some(slot) = rhs.get_mut(r) {
|
||
*slot += wr * hk;
|
||
}
|
||
for &(s, ws) in &terms {
|
||
band.add(r, s, wr * ws);
|
||
}
|
||
}
|
||
fitted += 1;
|
||
}
|
||
// The thin-plate energy ∫∫ h_uu² + 2·h_uv² + h_vv² over the grid's box, times the area of
|
||
// one cell: per cell that is the squared second difference of the heights, so it weighs
|
||
// against the data term (squared heights) alike at any size. Each term is a product of
|
||
// one-direction Gram matrices of the basis functions' derivatives.
|
||
let lambda = SMOOTHNESS * fitted as f64 / n as f64 * self.cell(0) * self.cell(1);
|
||
self.add_energy(&mut band, lambda);
|
||
band.solve(rhs)
|
||
}
|
||
|
||
/// Add `lambda` times the thin-plate energy `∫∫ h_uu² + 2·h_uv² + h_vv²` over the grid's box
|
||
/// to the normal equations `band` (unknowns i fastest): each term is a product of
|
||
/// one-direction Gram matrices of the basis functions' derivatives.
|
||
pub(crate) fn add_energy(&self, band: &mut Band, lambda: f64) {
|
||
let cu = self.count[0];
|
||
self.energy(lambda, &mut |(i, j), (k, l), e| band.add(j * cu + i, l * cu + k, e));
|
||
}
|
||
|
||
/// The entries of `lambda` times the thin-plate energy's matrix (see [`Grid::add_energy`]),
|
||
/// control point `(i, j)` against `(k, l)`, for every pair whose supports overlap.
|
||
pub(crate) fn energy(&self, lambda: f64, add: &mut dyn FnMut((usize, usize), (usize, usize), f64)) {
|
||
let [cu, cv] = self.count;
|
||
let gram = |d: usize, r: usize| -> Vec<f64> {
|
||
let c = self.count.get(d).copied().unwrap_or(0);
|
||
let mut g = vec![0.0; c * c];
|
||
let Some(k) = self.knots.get(d) else { return g };
|
||
for w in k.windows(2) {
|
||
let &[a, b] = w else { continue };
|
||
if !(b > a) {
|
||
continue;
|
||
}
|
||
for (gx, gw) in GAUSS4 {
|
||
let x = 0.5 * (a + b) + 0.5 * (b - a) * gx;
|
||
let Some((first, vals)) = self.basis_derivative(d, x, r) else { continue };
|
||
for (i, vi) in vals.iter().enumerate() {
|
||
for (j, vj) in vals.iter().enumerate() {
|
||
if let Some(slot) = g.get_mut((first + i) * c + first + j) {
|
||
*slot += 0.5 * (b - a) * gw * vi * vj;
|
||
}
|
||
}
|
||
}
|
||
}
|
||
}
|
||
g
|
||
};
|
||
let grams: [[Vec<f64>; 3]; 2] = [0, 1].map(|d| [0, 1, 2].map(|r| gram(d, r)));
|
||
let g = |d: usize, r: usize, i: usize, k: usize| {
|
||
let c = self.count.get(d).copied().unwrap_or(0);
|
||
grams.get(d).and_then(|m| m.get(r)).and_then(|m| m.get(i * c + k)).copied().unwrap_or(0.0)
|
||
};
|
||
for j in 0..cv {
|
||
for i in 0..cu {
|
||
for l in j.saturating_sub(3)..(j + 4).min(cv) {
|
||
for k in i.saturating_sub(3)..(i + 4).min(cu) {
|
||
let e: f64 = ENERGY.iter().map(|&(ru, rv, w)| w * g(0, ru, i, k) * g(1, rv, j, l)).sum();
|
||
add((i, j), (k, l), lambda * e);
|
||
}
|
||
}
|
||
}
|
||
}
|
||
}
|
||
|
||
/// The surface of the net `heights` over the plane `o`, `x`, `y`, `n`: control point `(i, j)`
|
||
/// at the knot averages `(ξᵢ, ηⱼ)` in the plane, `heights[j·count[0] + i]` along `n`.
|
||
fn surface(&self, heights: &[f64], o: Vec3, x: Vec3, y: Vec3, n: Vec3) -> SplineResult<NurbsSurface> {
|
||
let [cu, cv] = self.count;
|
||
let mut ctrl = Vec::with_capacity(cu * cv);
|
||
for j in 0..cv {
|
||
for i in 0..cu {
|
||
let hij = heights.get(j * cu + i).copied().unwrap_or(f64::NAN);
|
||
ctrl.push(HPoint::weighted(o + x * self.average(0, i) + y * self.average(1, j) + n * hij, 1.0));
|
||
}
|
||
}
|
||
NurbsSurface::from_homogeneous([3, 3], self.knots.clone(), ctrl)
|
||
}
|
||
}
|
||
|
||
/// A symmetric banded matrix: the upper band, row `r` holding columns `r ..= r + bw`.
|
||
#[derive(Clone)]
|
||
pub(crate) struct Band {
|
||
n: usize,
|
||
bw: usize,
|
||
a: Vec<f64>,
|
||
}
|
||
|
||
impl Band {
|
||
pub(crate) fn new(n: usize, bw: usize) -> Band {
|
||
Band { n, bw, a: vec![0.0; n * (bw + 1)] }
|
||
}
|
||
fn index(&self, r: usize, s: usize) -> Option<usize> {
|
||
let (r, s) = if r <= s { (r, s) } else { (s, r) };
|
||
(s - r <= self.bw && s < self.n).then_some(r * (self.bw + 1) + (s - r))
|
||
}
|
||
fn get(&self, r: usize, s: usize) -> f64 {
|
||
self.index(r, s).and_then(|i| self.a.get(i)).copied().unwrap_or(0.0)
|
||
}
|
||
fn set(&mut self, r: usize, s: usize, v: f64) {
|
||
if let Some(x) = self.index(r, s).and_then(|i| self.a.get_mut(i)) {
|
||
*x = v;
|
||
}
|
||
}
|
||
/// Add `v` at `(r, s)` with `r ≤ s` (the lower half is the same by symmetry: callers adding
|
||
/// a full outer product reach each stored entry once through `r ≤ s`).
|
||
pub(crate) fn add(&mut self, r: usize, s: usize, v: f64) {
|
||
if r <= s
|
||
&& let Some(x) = self.index(r, s).and_then(|i| self.a.get_mut(i))
|
||
{
|
||
*x += v;
|
||
}
|
||
}
|
||
/// Solve `A·x = b` by banded Cholesky factorisation `A = Uᵀ·U`; an error when `A` is not
|
||
/// positive definite.
|
||
fn solve(self, b: Vec<f64>) -> SplineResult<Vec<f64>> {
|
||
self.factor()?.substitute(b)
|
||
}
|
||
|
||
/// The Cholesky factor `U` of `A = Uᵀ·U`, in place; an error when `A` is not positive
|
||
/// definite.
|
||
pub(crate) fn factor(mut self) -> SplineResult<Band> {
|
||
let (n, bw) = (self.n, self.bw);
|
||
for r in 0..n {
|
||
let mut d = self.get(r, r);
|
||
for k in r.saturating_sub(bw)..r {
|
||
let u = self.get(k, r);
|
||
d -= u * u;
|
||
}
|
||
if !(d > 0.0) {
|
||
return invalid("the fit's equations are singular");
|
||
}
|
||
let d = d.sqrt();
|
||
self.set(r, r, d);
|
||
for s in r + 1..(r + bw + 1).min(n) {
|
||
let mut v = self.get(r, s);
|
||
for k in s.saturating_sub(bw)..r {
|
||
v -= self.get(k, r) * self.get(k, s);
|
||
}
|
||
self.set(r, s, v / d);
|
||
}
|
||
}
|
||
Ok(self)
|
||
}
|
||
|
||
/// [`Band::factor`] by rows: once row `r` of `U` is known, the rows below it within the band
|
||
/// are updated from it at once (right-looking), which walks memory row by row instead of down
|
||
/// columns. The same factor, to rounding; what the general fill, whose bands are wide, uses.
|
||
pub(crate) fn factor_rows(mut self) -> SplineResult<Band> {
|
||
let (n, bw) = (self.n, self.bw);
|
||
let w = bw + 1;
|
||
let mut row = vec![0.0; w];
|
||
for r in 0..n {
|
||
let Some(here) = self.a.get_mut(r * w..r * w + w) else { return invalid("the fit's equations are out of shape") };
|
||
let d = here.first().copied().unwrap_or(0.0);
|
||
if !(d > 0.0) {
|
||
return invalid("the fit's equations are singular");
|
||
}
|
||
let d = d.sqrt();
|
||
for (k, x) in here.iter_mut().enumerate() {
|
||
*x = if k == 0 { d } else { *x / d };
|
||
}
|
||
row.copy_from_slice(here);
|
||
let end = (r + bw + 1).min(n);
|
||
for s in r + 1..end {
|
||
let us = row.get(s - r).copied().unwrap_or(0.0);
|
||
if us == 0.0 {
|
||
continue;
|
||
}
|
||
let len = end - s;
|
||
let (Some(dst), Some(src)) = (self.a.get_mut(s * w..s * w + len), row.get(s - r..s - r + len)) else {
|
||
return invalid("the fit's equations are out of shape");
|
||
};
|
||
for (x, y) in dst.iter_mut().zip(src) {
|
||
*x -= us * y;
|
||
}
|
||
}
|
||
}
|
||
Ok(self)
|
||
}
|
||
|
||
/// Solve `Uᵀ·U·x = b` with a factor from [`Band::factor`].
|
||
pub(crate) fn substitute(&self, mut b: Vec<f64>) -> SplineResult<Vec<f64>> {
|
||
let (n, bw) = (self.n, self.bw);
|
||
// Uᵀ·y = b, then U·x = y.
|
||
for r in 0..n {
|
||
let mut v = b.get(r).copied().unwrap_or(0.0);
|
||
for k in r.saturating_sub(bw)..r {
|
||
v -= self.get(k, r) * b.get(k).copied().unwrap_or(0.0);
|
||
}
|
||
if let Some(x) = b.get_mut(r) {
|
||
*x = v / self.get(r, r);
|
||
}
|
||
}
|
||
for r in (0..n).rev() {
|
||
let mut v = b.get(r).copied().unwrap_or(0.0);
|
||
for s in r + 1..(r + bw + 1).min(n) {
|
||
v -= self.get(r, s) * b.get(s).copied().unwrap_or(0.0);
|
||
}
|
||
if let Some(x) = b.get_mut(r) {
|
||
*x = v / self.get(r, r);
|
||
}
|
||
}
|
||
if b.iter().any(|x| !x.is_finite()) {
|
||
return invalid("the fit is not finite");
|
||
}
|
||
Ok(b)
|
||
}
|
||
}
|
||
|
||
#[cfg(test)]
|
||
mod tests {
|
||
use super::*;
|
||
|
||
/// The closed polygon through `corners` in space, side `k` from corner `k` to `k+1`.
|
||
fn polygon3(corners: &[Vec3]) -> impl FnMut(usize, f64) -> Option<Vec3> + '_ {
|
||
move |k, t| {
|
||
let n = corners.len();
|
||
Some(corners.get(k)?.lerp(*corners.get((k + 1) % n)?, t))
|
||
}
|
||
}
|
||
|
||
/// The polygon `corners` in the plane, lifted by `z(x, y)` all along.
|
||
fn lifted<'a>(corners: &'a [[f64; 2]], z: &'a dyn Fn(f64, f64) -> f64) -> impl FnMut(usize, f64) -> Option<Vec3> + 'a {
|
||
move |k, t| {
|
||
let n = corners.len();
|
||
let (a, b) = (corners.get(k)?, corners.get((k + 1) % n)?);
|
||
let (x, y) = (a[0] + (b[0] - a[0]) * t, a[1] + (b[1] - a[1]) * t);
|
||
Some(Vec3::new(x, y, z(x, y)))
|
||
}
|
||
}
|
||
|
||
fn ring(n: usize, r: f64) -> Vec<[f64; 2]> {
|
||
(0..n).map(|k| k as f64 * std::f64::consts::TAU / n as f64).map(|a| [r * a.cos(), r * a.sin()]).collect()
|
||
}
|
||
|
||
/// The farthest the surface is from `z` over the polygon's inside (sampled on a grid).
|
||
fn off_inside(f: &LoopFill, corners: &[[f64; 2]], z: &dyn Fn(f64, f64) -> f64) -> f64 {
|
||
let inside = |x: f64, y: f64| {
|
||
let n = corners.len();
|
||
let mut w = false;
|
||
for k in 0..n {
|
||
let (a, b) = (corners[k], corners[(k + 1) % n]);
|
||
if (a[1] > y) != (b[1] > y) && x < a[0] + (y - a[1]) / (b[1] - a[1]) * (b[0] - a[0]) {
|
||
w = !w;
|
||
}
|
||
}
|
||
w
|
||
};
|
||
let s = &f.surface;
|
||
let [du, dv] = s.domain();
|
||
let mut worst = 0.0f64;
|
||
for i in 0..=40 {
|
||
for j in 0..=40 {
|
||
let (u, v) = ((du[0] + (du[1] - du[0]) * i as f64 / 40.0).min(du[1]), (dv[0] + (dv[1] - dv[0]) * j as f64 / 40.0).min(dv[1]));
|
||
let p = s.eval(u, v).unwrap();
|
||
if inside(p.x, p.y) {
|
||
worst = worst.max((p.z - z(p.x, p.y)).abs());
|
||
}
|
||
}
|
||
}
|
||
worst
|
||
}
|
||
|
||
#[test]
|
||
fn planes_are_reproduced_and_rims_of_quadrics_are_filled_closely() {
|
||
// A tilted plane over a hexagon: the smoothness energy vanishes on it, so the fill is it.
|
||
let hex = ring(6, 10.0);
|
||
let plane = |x: f64, y: f64| 0.3 * x - 0.2 * y + 1.0;
|
||
let f = fill_loop(6, &mut lifted(&hex, &plane), 1e-6, 0.0).unwrap();
|
||
let inside = off_inside(&f, &hex, &plane);
|
||
eprintln!("plane: deviation {:e}, inside off by {inside:e}, {} evaluations", f.deviation, f.evaluations);
|
||
assert!(f.deviation <= 1e-6 && inside < 1e-9);
|
||
// A saddle and a dome: the least-bending fill through the rim is not the quadric (only
|
||
// planes are reproduced), but stays near it (the plate is free to turn at the rim).
|
||
let saddle = |x: f64, y: f64| 0.05 * x * y;
|
||
let dome = |x: f64, y: f64| 3.0 - 0.02 * (x * x + y * y);
|
||
for (name, z, bound) in [("saddle", &saddle as &dyn Fn(f64, f64) -> f64, 0.1), ("dome", &dome, 1.0)] {
|
||
let f = fill_loop(6, &mut lifted(&hex, z), 2e-5, 0.0).unwrap();
|
||
let inside = off_inside(&f, &hex, z);
|
||
eprintln!("{name}: deviation {:e}, inside off by {inside:e}", f.deviation);
|
||
assert!(f.deviation <= 2e-5 && inside < bound, "{name}: {inside}");
|
||
}
|
||
}
|
||
|
||
#[test]
|
||
fn concave_loops_fill_and_the_result_does_not_depend_on_where_the_loop_starts() {
|
||
// A dart, an arrow-head pentagon and a plus, on a cylinder.
|
||
let z = |x: f64, _: f64| 50.0 - (1600.0 - x * x).sqrt();
|
||
let dart = vec![[0.0, 0.0], [20.0, -10.0], [8.0, 0.0], [20.0, 10.0]];
|
||
let arrow = vec![[0.0, 0.0], [10.0, -8.0], [10.0, -3.0], [20.0, 0.0], [10.0, 3.0]];
|
||
let plus = vec![
|
||
[-3.0, -9.0],
|
||
[3.0, -9.0],
|
||
[3.0, -3.0],
|
||
[9.0, -3.0],
|
||
[9.0, 3.0],
|
||
[3.0, 3.0],
|
||
[3.0, 9.0],
|
||
[-3.0, 9.0],
|
||
[-3.0, 3.0],
|
||
[-9.0, 3.0],
|
||
[-9.0, -3.0],
|
||
[-3.0, -3.0],
|
||
];
|
||
let flat_corner = vec![[-10.0, -10.0], [0.0, -10.0], [10.0, -10.0], [10.0, 10.0], [-10.0, 10.0]];
|
||
for (name, c) in [("dart", dart), ("arrow", arrow), ("plus", plus), ("flat corner", flat_corner)] {
|
||
let n = c.len();
|
||
let f = fill_loop(n, &mut lifted(&c, &z), 1e-4, 0.0).unwrap();
|
||
let inside = off_inside(&f, &c, &z);
|
||
eprintln!("{name}: deviation {:e}, inside off the cylinder by {inside:e}, {} evaluations", f.deviation, f.evaluations);
|
||
// The least-bending fill, not the cylinder: within a few tenths of a millimetre of it.
|
||
assert!(f.deviation <= 1e-4 && inside < 0.2, "{name}: {inside}");
|
||
// The same loop from another corner and the other way round.
|
||
let mut back = c.clone();
|
||
back.rotate_left(2);
|
||
back.reverse();
|
||
let g = fill_loop(n, &mut lifted(&back, &z), 1e-4, 0.0).unwrap();
|
||
assert_ne!(f.counter_clockwise, g.counter_clockwise);
|
||
for (x, y) in [(4.0, 0.0), (6.0, 1.0), (5.0, -1.0)] {
|
||
let p = Vec3::new(x, y, z(x, y));
|
||
let gap = f.point_over(p).unwrap().dist(g.point_over(p).unwrap());
|
||
assert!(gap < 1e-7, "{name}: {gap}");
|
||
}
|
||
}
|
||
}
|
||
|
||
#[test]
|
||
fn the_fill_is_the_same_at_any_size_and_place() {
|
||
// The arrow on the cylinder, scaled by 100 and by 1/100, turned and moved: the same fill,
|
||
// scaled and moved alike.
|
||
let z = |x: f64, _: f64| 50.0 - (1600.0 - x * x).sqrt();
|
||
let arrow = [[0.0, 0.0], [10.0, -8.0], [10.0, -3.0], [20.0, 0.0], [10.0, 3.0]];
|
||
let f = fill_loop(5, &mut lifted(&arrow, &z), 1e-4, 0.0).unwrap();
|
||
let axis = Vec3::new(1.0, -2.0, 0.5).normalized().unwrap();
|
||
let shift = Vec3::new(30.0, -12.0, 7.0);
|
||
for scale in [100.0, 0.01] {
|
||
let place = |p: Vec3| {
|
||
// Scaled, turned by 0.9 about `axis` (Rodrigues), moved.
|
||
let p = p * scale;
|
||
let (c, s) = (0.9f64.cos(), 0.9f64.sin());
|
||
p * c + axis.cross(p) * s + axis * (axis.dot(p) * (1.0 - c)) + shift * scale
|
||
};
|
||
let mut moved = |k: usize, t: f64| lifted(&arrow, &z)(k, t).map(place);
|
||
let g = fill_loop(5, &mut moved, 1e-4 * scale, 0.0).unwrap();
|
||
let mut worst = 0.0f64;
|
||
for (x, y) in [(4.0, 0.0), (6.0, 1.0), (5.0, -1.0), (9.0, -2.0), (12.0, 0.5)] {
|
||
let p = Vec3::new(x, y, z(x, y));
|
||
worst = worst.max(g.point_over(place(p)).unwrap().dist(place(f.point_over(p).unwrap())) / scale);
|
||
}
|
||
eprintln!("arrow scaled by {scale}, turned and moved: {} evaluations, surfaces apart by {worst:e} (relative)", g.evaluations);
|
||
assert!(worst < 1e-7, "scale {scale}: {worst}");
|
||
}
|
||
}
|
||
|
||
#[test]
|
||
fn a_long_narrow_slot_fills() {
|
||
// A 200 × 1 mm slot on a cylinder of radius 300 running along it.
|
||
let z = |x: f64, _: f64| 300.0 - (90_000.0 - (x - 100.0) * (x - 100.0)).sqrt();
|
||
let slot = [[0.0, 0.0], [200.0, 0.0], [200.0, 1.0], [0.0, 1.0]];
|
||
let f = fill_loop(4, &mut lifted(&slot, &z), 2e-4, 0.0).unwrap();
|
||
let inside = off_inside(&f, &slot, &z);
|
||
eprintln!("slot: deviation {:e}, inside off the cylinder by {inside:e}, {} evaluations", f.deviation, f.evaluations);
|
||
assert!(f.deviation <= 2e-4 && inside < 1e-2);
|
||
}
|
||
|
||
#[test]
|
||
fn loops_that_do_not_bound_a_height_field_are_refused() {
|
||
let unsupported = |r: SplineResult<LoopFill>| matches!(r, Err(SplineError::Unsupported(_)));
|
||
let flat = |_: f64, _: f64| 0.0;
|
||
// A figure eight (lobes of different sizes).
|
||
let eight = [[0.0, 0.0], [10.0, 10.0], [10.0, 0.0], [0.0, 6.0]];
|
||
assert!(unsupported(fill_loop(4, &mut lifted(&eight, &flat), 1e-4, 0.0)));
|
||
// A loop with a vertical spike in it (up along the normal and down again).
|
||
let spike = [
|
||
Vec3::ZERO,
|
||
Vec3::new(10.0, 0.0, 0.0),
|
||
Vec3::new(10.0, 10.0, 0.0),
|
||
Vec3::new(5.0, 10.0, 0.0),
|
||
Vec3::new(5.0, 10.0, 3.0),
|
||
Vec3::new(5.0, 10.0, 0.0),
|
||
Vec3::new(0.0, 10.0, 0.0),
|
||
];
|
||
assert!(unsupported(fill_loop(7, &mut polygon3(&spike), 1e-4, 0.0)));
|
||
// A loop that runs back over itself.
|
||
let back = [[0.0, 0.0], [10.0, 0.0], [4.0, 0.0], [10.0, 10.0], [0.0, 10.0]];
|
||
assert!(unsupported(fill_loop(5, &mut lifted(&back, &flat), 1e-4, 0.0)));
|
||
// A loop too wavy for one surface.
|
||
let wavy = |x: f64, y: f64| (x * 9.0).sin() * (y * 7.0).cos();
|
||
assert!(unsupported(fill_loop(64, &mut lifted(&ring(64, 10.0), &wavy), 1e-6, 0.0)));
|
||
}
|
||
|
||
#[test]
|
||
fn hostile_loops_are_refused() {
|
||
let flat = |_: f64, _: f64| 0.0;
|
||
let five = ring(5, 10.0);
|
||
assert!(fill_loop(0, &mut lifted(&five, &flat), 1e-4, 0.0).is_err());
|
||
assert!(fill_loop(5, &mut |_, _| Some(Vec3::X), 1e-4, 0.0).is_err(), "no area");
|
||
assert!(fill_loop(5, &mut |_, _| None, 1e-4, 0.0).is_err());
|
||
assert!(fill_loop(5, &mut |k, t| Some(Vec3::new(k as f64, t, f64::NAN)), 1e-4, 0.0).is_err());
|
||
assert!(fill_loop(5, &mut lifted(&five, &flat), 0.0, 0.0).is_err() && fill_loop(5, &mut lifted(&five, &flat), f64::NAN, 0.0).is_err());
|
||
assert!(fill_loop(10_000, &mut lifted(&five, &flat), 1e-4, 0.0).is_err());
|
||
let huge = |x: f64, y: f64| (x * y) * 1e300;
|
||
let _ = fill_loop(5, &mut lifted(&five, &huge), 1e-4, 0.0);
|
||
let tiny: Vec<[f64; 2]> = five.iter().map(|p| [p[0] * 1e-300, p[1] * 1e-300]).collect();
|
||
let _ = fill_loop(5, &mut lifted(&tiny, &flat), 1e-4, 0.0);
|
||
let big: Vec<[f64; 2]> = five.iter().map(|p| [p[0] * 1e300, p[1] * 1e300]).collect();
|
||
let _ = fill_loop(5, &mut lifted(&big, &flat), 1e-4, 0.0);
|
||
}
|
||
|
||
/// A patch whose rim is a dome: the crown fills its middle by `crown` times the loop's size
|
||
/// and leaves the loop — the body's own edges — exactly where it was.
|
||
#[test]
|
||
fn a_crown_fills_the_middle_of_a_patch_and_leaves_its_loop_alone() {
|
||
let hex = ring(6, 10.0);
|
||
let dome = |x: f64, y: f64| 3.0 - 0.02 * (x * x + y * y);
|
||
let plain = fill_loop(6, &mut lifted(&hex, &dome), 2e-5, 0.0).unwrap();
|
||
// Crown 0 is the fit as it was: no refinement, no bump, nothing moved.
|
||
assert_eq!((plain.crown, plain.crowned, plain.crown_drift), (0.0, 0, 0.0));
|
||
assert_eq!(plain.surface.count(), [19, 17], "crown 0 keeps the fitted net (16 × 14 spans)");
|
||
let crown = 0.25;
|
||
let full = fill_loop(6, &mut lifted(&hex, &dome), 2e-5, crown).unwrap();
|
||
assert!(full.crowned > 0, "the crown needs room inside the loop");
|
||
assert_eq!(full.crown_drift, 0.0, "the bump must not move the loop at all");
|
||
assert_eq!(full.deviation, plain.deviation, "the crown must not touch the fit");
|
||
// The middle of the box rises by `crown` times the loop's size: the bubble is 1 there.
|
||
let [du, dv] = plain.surface.domain();
|
||
let (uc, vc) = ((du[0] + du[1]) / 2.0, (dv[0] + dv[1]) / 2.0);
|
||
let size = (du[1] - du[0]).hypot(dv[1] - dv[0]);
|
||
let mid = |f: &LoopFill| f.surface.eval(uc, vc).unwrap_or(Vec3::ZERO);
|
||
let bump = (mid(&full) - mid(&plain)).dot(plain.normal);
|
||
eprintln!("crown {crown}: {} points took the bump, the middle rose by {bump} of {}", full.crowned, crown * size);
|
||
assert!((bump - crown * size).abs() < 1e-9 * size, "the middle rose by {bump}, wanted {}", crown * size);
|
||
// A flatter crown lowers the middle by the same.
|
||
let flat = fill_loop(6, &mut lifted(&hex, &dome), 2e-5, -crown).unwrap();
|
||
let dip = (mid(&flat) - mid(&plain)).dot(plain.normal);
|
||
assert!((dip + crown * size).abs() < 1e-9 * size, "the middle fell by {dip}");
|
||
// The loop's own points are still on the surface, as closely as they were.
|
||
let off = |f: &LoopFill| {
|
||
hex.iter()
|
||
.map(|p| f.point_over(Vec3::new(p[0], p[1], dome(p[0], p[1]))).map_or(f64::INFINITY, |q| (q.z - dome(p[0], p[1])).abs()))
|
||
.fold(0.0f64, f64::max)
|
||
};
|
||
assert!(off(&full) <= off(&plain) + 1e-9, "the crowned fill left the loop: {:e} against {:e}", off(&full), off(&plain));
|
||
}
|
||
|
||
#[test]
|
||
fn hostile_crowns_are_refused() {
|
||
let hex = ring(6, 10.0);
|
||
let dome = |x: f64, y: f64| 3.0 - 0.02 * (x * x + y * y);
|
||
for crown in [f64::NAN, f64::INFINITY, f64::NEG_INFINITY, 3.01, -0.91, 1e300] {
|
||
assert!(fill_loop(6, &mut lifted(&hex, &dome), 2e-5, crown).is_err(), "crown {crown}");
|
||
}
|
||
assert!(fill_loop(6, &mut lifted(&hex, &dome), 2e-5, MAX_CROWN).is_ok());
|
||
assert!(fill_loop(6, &mut lifted(&hex, &dome), 2e-5, MIN_CROWN).is_ok());
|
||
}
|
||
}
|