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use super::*;
impl<V> NURBSCurve<V> {
/// Constructs the rationalized B-spline curve.
#[inline(always)]
pub const fn new(curve: BSplineCurve<V>) -> Self { NURBSCurve(curve) }
/// Returns the BSpline curve before rationalized.
#[inline(always)]
pub const fn non_rationalized(&self) -> &BSplineCurve<V> { &self.0 }
/// Returns the BSpline curve before rationalized.
#[inline(always)]
pub fn into_non_rationalized(self) -> BSplineCurve<V> { self.0 }
/// Returns the reference of the knot vector.
/// cf.[`BSplineCurve::knot_vec`](./struct.BSplineCurve.html#method.knot_vec)
#[inline(always)]
pub const fn knot_vec(&self) -> &KnotVec { &self.0.knot_vec }
/// Returns the `idx`th knot.
/// cf.[`BSplineCurve::knot`](./struct.BSplineCurve.html#method.knot)
#[inline(always)]
pub fn knot(&self, idx: usize) -> f64 { self.0.knot_vec[idx] }
/// Returns the reference of the control points.
/// cf.[`BSplineCurve::control_points`](./struct.BSplineCurve.html#method.control_points)
#[inline(always)]
pub const fn control_points(&self) -> &Vec<V> { &self.0.control_points }
/// Returns the reference of the control point corresponding to the index `idx`.
/// cf.[`BSplineCurve::control_point`](./struct.BSplineCurve.html#method.control_point)
#[inline(always)]
pub fn control_point(&self, idx: usize) -> &V { &self.0.control_points[idx] }
/// Returns the mutable reference of the control point corresponding to index `idx`.
/// cf.[`BSplineCurve::control_point_mut`](./struct.BSplineCurve.html#method.control_point_mut)
#[inline(always)]
pub fn control_point_mut(&mut self, idx: usize) -> &mut V { &mut self.0.control_points[idx] }
/// Returns the iterator on all control points
/// cf.[`BSplineCurve::control_points_mut`](./struct.BSplineCurve.html#method.control_points_mut)
#[inline(always)]
pub fn control_points_mut(&mut self) -> impl Iterator<Item = &mut V> {
self.0.control_points.iter_mut()
}
/// Apply the given transformation to all control points.
/// cf.[`BSplineCurve::transform_control_points`](./struct.BSplineCurve.html#method.transform_control_points)
#[inline(always)]
pub fn transform_control_points<F: FnMut(&mut V)>(&mut self, f: F) {
self.0.transform_control_points(f)
}
/// Returns the degree of NURBS curve.
/// cf.[`BSplineCurve::degree`](./struct.BSplineCurve.html#method.degree)
#[inline(always)]
pub fn degree(&self) -> usize { self.0.degree() }
/// Inverts a curve.
/// cf.[`BSplineCurve::invert`](./struct.BSplineCurve.html#method.invert)
#[inline(always)]
pub fn invert(&mut self) -> &mut Self {
self.0.invert();
self
}
/// Returns whether the knot vector is clamped or not.
/// cf.[`BSplineCurve::is_clamped`](./struct.BSplineCurve.html#method.is_clamped)
#[inline(always)]
pub fn is_clamped(&self) -> bool { self.0.knot_vec.is_clamped(self.0.degree()) }
/// Normalizes the knot vector.
/// cf.[`BSplineCurve::knot_normalize`](./struct.BSplineCurve.html#method.knot_normalize)
#[inline(always)]
pub fn knot_normalize(&mut self) -> &mut Self {
self.0.knot_vec.try_normalize().unwrap();
self
}
/// Translates the knot vector.
/// cf.[`BSplineCurve::knot_translate`](./struct.BSplineCurve.html#method.knot_translate)
#[inline(always)]
pub fn knot_translate(&mut self, x: f64) -> &mut Self {
self.0.knot_vec.translate(x);
self
}
}
impl<V: Homogeneous<f64>> NURBSCurve<V> {
/// Constructs a rationalization curve from the curves and weights.
/// # Failures
/// the length of `curve.control_points()` and `weights` must be the same.
#[inline(always)]
pub fn try_from_bspline_and_weights(
curve: BSplineCurve<V::Point>,
weights: Vec<f64>,
) -> Result<Self> {
let BSplineCurve {
knot_vec,
control_points,
} = curve;
if control_points.len() != weights.len() {
return Err(Error::DifferentLength);
}
let control_points = control_points
.into_iter()
.zip(weights)
.map(|(pt, w)| V::from_point_weight(pt, w))
.collect();
Ok(Self(BSplineCurve::new_unchecked(knot_vec, control_points)))
}
}
impl<V: Homogeneous<f64> + ControlPoint<f64, Diff = V>> NURBSCurve<V> {
/// Returns the closure of substitution.
#[inline(always)]
pub fn get_closure(&self) -> impl Fn(f64) -> V::Point + '_ { move |t| self.subs(t) }
}
impl<V: Homogeneous<f64> + ControlPoint<f64, Diff = V>> NURBSCurve<V>
where V::Point: Tolerance
{
/// Returns whether all control points are the same or not.
/// If the knot vector is clamped, it means whether the curve is constant or not.
/// # Examples
/// ```
/// use truck_geometry::*;
///
/// let knot_vec = KnotVec::bezier_knot(2);
/// let pt = Vector3::new(1.0, 2.0, 1.0);
/// // allows differences upto scalars
/// let mut ctrl_pts = vec![pt.clone(), pt.clone() * 2.0, pt.clone() * 3.0];
/// let bspcurve = BSplineCurve::new(knot_vec.clone(), ctrl_pts.clone());
/// assert!(!bspcurve.is_const());
/// let const_curve = NURBSCurve::new(bspcurve);
/// assert!(const_curve.is_const());
///
/// ctrl_pts.push(Vector3::new(2.0, 3.0, 1.0));
/// let curve = NURBSCurve::new(BSplineCurve::new(knot_vec.clone(), ctrl_pts.clone()));
/// assert!(!curve.is_const());
/// ```
/// # Remarks
/// If the knot vector is not clamped and the BSpline basis function is not partition of unity,
/// then perhaps returns true even if the curve is not constant.
/// ```
/// use truck_geometry::*;
/// let knot_vec = KnotVec::uniform_knot(1, 5);
/// let ctrl_pts = vec![Vector2::new(1.0, 2.0), Vector2::new(1.0, 2.0)];
/// let bspcurve = BSplineCurve::new(knot_vec, ctrl_pts);
///
/// // bspcurve is not constant.
/// assert_eq!(bspcurve.subs(0.0), Vector2::new(0.0, 0.0));
/// assert_ne!(bspcurve.subs(0.5), Vector2::new(0.0, 0.0));
///
/// // bspcurve.is_const() is true
/// assert!(bspcurve.is_const());
/// ```
pub fn is_const(&self) -> bool {
let pt = self.0.control_points[0].to_point();
self.0
.control_points
.iter()
.all(move |vec| vec.to_point().near(&pt))
}
/// Determine whether `self` and `other` is near as the B-spline curves or not.
///
/// Divides each knot interval into the number of degree equal parts,
/// and check `|self(t) - other(t)| < TOLERANCE` for each end points `t`.
/// # Examples
/// ```
/// use truck_geometry::*;
/// let knot_vec = KnotVec::from(
/// vec![0.0, 0.0, 0.0, 1.0, 2.0, 3.0, 4.0, 4.0, 4.0]
/// );
/// let ctrl_pts = vec![
/// Vector3::new(1.0, 1.0, 1.0),
/// Vector3::new(3.0, 2.0, 2.0),
/// Vector3::new(2.0, 3.0, 1.0),
/// Vector3::new(4.0, 5.0, 2.0),
/// Vector3::new(5.0, 4.0, 1.0),
/// Vector3::new(1.0, 1.0, 2.0),
/// ];
/// let curve0 = NURBSCurve::new(BSplineCurve::new(knot_vec, ctrl_pts));
/// let mut curve1 = curve0.clone();
/// assert!(curve0.near_as_curve(&curve1));
/// *curve1.control_point_mut(1) += Vector3::new(0.01, 0.0002, 0.0);
/// assert!(!curve0.near_as_curve(&curve1));
/// ```
#[inline(always)]
pub fn near_as_curve(&self, other: &Self) -> bool {
self.0
.sub_near_as_curve(&other.0, 2, move |x, y| x.to_point().near(&y.to_point()))
}
/// Determines `self` and `other` is near in square order as the B-spline curves or not.
///
/// Divide each knot interval into the number of degree equal parts,
/// and check `|self(t) - other(t)| < TOLERANCE`for each end points `t`.
/// # Examples
/// ```
/// use truck_geometry::*;
/// let knot_vec = KnotVec::from(
/// vec![0.0, 0.0, 0.0, 1.0, 2.0, 3.0, 4.0, 4.0, 4.0]
/// );
/// let ctrl_pts = vec![
/// Vector3::new(1.0, 1.0, 1.0),
/// Vector3::new(3.0, 2.0, 2.0),
/// Vector3::new(2.0, 3.0, 1.0),
/// Vector3::new(4.0, 5.0, 2.0),
/// Vector3::new(5.0, 4.0, 1.0),
/// Vector3::new(1.0, 1.0, 2.0),
/// ];
/// let curve0 = NURBSCurve::new(BSplineCurve::new(knot_vec, ctrl_pts));
/// let mut curve1 = curve0.clone();
/// assert!(curve0.near_as_curve(&curve1));
/// *curve1.control_point_mut(1) += Vector3::new(0.01, TOLERANCE, 0.0);
/// assert!(!curve0.near2_as_curve(&curve1));
/// ```
#[inline(always)]
pub fn near2_as_curve(&self, other: &Self) -> bool {
self.0
.sub_near_as_curve(&other.0, 2, move |x, y| x.to_point().near2(&y.to_point()))
}
}
impl<V: Homogeneous<f64> + ControlPoint<f64, Diff = V> + Tolerance> NURBSCurve<V> {
/// Adds a knot `x`, and do not change `self` as a curve.
/// cf.[`BSplineCurve::add_knot`](./struct.BSplineCurve.html#method.add_knot)
pub fn add_knot(&mut self, x: f64) -> &mut Self {
self.0.add_knot(x);
self
}
/// Removes a knot corresponding to the indices `idx`, and do not change `self` as a curve.
/// If cannot remove the knot, do not change `self` and return `self`.
/// cf.[`BSplineCurve::remove_knot`](./struct.BSplineCurve.html#method.remove_knot)
pub fn remove_knot(&mut self, idx: usize) -> &mut Self {
let _ = self.try_remove_knot(idx);
self
}
/// Removes a knot corresponding to the indice `idx`, and do not change `self` as a curve.
/// If the knot cannot be removed, returns
/// [`Error::CannotRemoveKnot`](./errors/enum.Error.html#variant.CannotRemoveKnot).
/// cf.[`BSplineCurve::try_remove_knot`](./struct.BSplineCurve.html#method.try_remove_knot)
pub fn try_remove_knot(&mut self, idx: usize) -> Result<&mut Self> {
self.0.try_remove_knot(idx)?;
Ok(self)
}
/// Elevates 1 degree.
/// cf.[`BSplineCurve::elevate_degree`](./struct.BSplineCurve.html#method.elevate_degree)
pub fn elevate_degree(&mut self) -> &mut Self {
self.0.elevate_degree();
self
}
/// Makes the NURBS curve clamped
/// cf.[`BSplineCurve::clamp`](./struct.BSplineCurve.html#method.clamp)
#[inline(always)]
pub fn clamp(&mut self) -> &mut Self {
self.0.clamp();
self
}
/// Repeats `Self::try_remove_knot()` from the back knot in turn until the knot cannot be removed.
/// cf.[`BSplineCurve::optimize`](./struct.BSplineCurve.html#method.optimize)
pub fn optimize(&mut self) -> &mut Self {
self.0.optimize();
self
}
/// Makes two splines having the same degrees.
/// cf.[`BSplineCurve::syncro_degree`](./struct.BSplineCurve.html#method.syncro_degree)
pub fn syncro_degree(&mut self, other: &mut Self) {
let (degree0, degree1) = (self.degree(), other.degree());
for _ in degree0..degree1 {
self.elevate_degree();
}
for _ in degree1..degree0 {
other.elevate_degree();
}
}
/// Makes two splines having the same normalized knot vectors.
/// cf.[`BSplineCurve::syncro_knots`](./struct.BSplineCurve.html#method.syncro_knots)
pub fn syncro_knots(&mut self, other: &mut Self) { self.0.syncro_knots(&mut other.0) }
}
impl<V: Homogeneous<f64> + ControlPoint<f64, Diff = V> + Tolerance> ParameterTransform
for NURBSCurve<V>
{
#[inline(always)]
fn parameter_transform(&mut self, scalar: f64, r#move: f64) -> &mut Self {
self.0.parameter_transform(scalar, r#move);
self
}
}
impl<V: Homogeneous<f64> + ControlPoint<f64, Diff = V> + Tolerance> Cut for NURBSCurve<V> {
#[inline(always)]
fn cut(&mut self, t: f64) -> Self { NURBSCurve(self.0.cut(t)) }
}
impl<V: Homogeneous<f64> + ControlPoint<f64, Diff = V> + Tolerance> Concat<NURBSCurve<V>>
for NURBSCurve<V>
where <V as Homogeneous<f64>>::Point: Debug
{
type Output = NURBSCurve<V>;
fn try_concat(
&self,
other: &Self,
) -> std::result::Result<Self, ConcatError<<V as Homogeneous<f64>>::Point>> {
let curve0 = self.clone();
let mut curve1 = other.clone();
let w0 = curve0.0.control_points.last().unwrap().weight();
let w1 = curve1.0.control_points[0].weight();
curve1.transform_control_points(|pt| *pt *= w0 / w1);
match curve0.0.try_concat(&curve1.0) {
Ok(curve) => Ok(NURBSCurve::new(curve)),
Err(err) => Err(err.point_map(|v| v.to_point())),
}
}
}
#[test]
fn concat_positive_test() {
let mut part0 = NURBSCurve::new(BSplineCurve::new(
KnotVec::uniform_knot(4, 4),
(0..8)
.map(|_| {
Vector4::new(
rand::random::<f64>(),
rand::random::<f64>(),
rand::random::<f64>(),
rand::random::<f64>() + 0.5,
)
})
.collect(),
));
let mut part1 = part0.cut(0.56);
let w = 20.0 * rand::random::<f64>() - 10.0;
part1.transform_control_points(|vec| *vec *= w);
assert_near!(part0.back(), part1.front());
concat_random_test(&part0, &part1, 10);
}
impl<V: Homogeneous<f64> + ControlPoint<f64, Diff = V> + Tolerance> NURBSCurve<V>
where V::Point: Tolerance
{
/// Makes the rational curve locally injective.
/// # Example
/// ```
/// use truck_geometry::*;
/// const N : usize = 100; // sample size for test
///
/// let knot_vec = KnotVec::from(
/// vec![0.0, 0.0, 0.0, 1.0, 3.0, 4.0, 4.0, 4.0]
/// );
/// let control_points = vec![
/// Vector4::new(1.0, 0.0, 0.0, 1.0),
/// Vector4::new(0.0, 1.0, 0.0, 1.0),
/// Vector4::new(0.0, 2.0, 0.0, 2.0),
/// Vector4::new(0.0, 3.0, 0.0, 3.0),
/// Vector4::new(0.0, 0.0, 3.0, 3.0),
/// ];
///
/// let mut curve = NURBSCurve::new(BSplineCurve::new(knot_vec, control_points));
/// let mut flag = false;
/// for i in 0..N {
/// let t = 4.0 * (i as f64) / (N as f64);
/// let pt0 = curve.subs(t);
/// let pt1 = curve.subs(t + 1.0 / (N as f64));
/// flag = flag || pt0.near(&pt1);
/// }
/// // There exists t such that bspcurve(t) == bspcurve(t + 0.01).
/// assert!(flag);
///
/// curve.make_locally_injective().knot_normalize();
/// let mut flag = false;
/// for i in 0..N {
/// let t = 1.0 * (i as f64) / (N as f64);
/// let pt0 = curve.subs(t);
/// let pt1 = curve.subs(t + 1.0 / (N as f64));
/// flag = flag || pt0.near(&pt1);
/// }
/// // There does not exist t such that bspcurve(t) == bspcurve(t + 0.01).
/// assert!(!flag);
/// ```
/// # Remarks
/// If `self` is a constant curve, then does nothing.
/// ```
/// use truck_geometry::*;
/// let knot_vec = KnotVec::from(vec![0.0, 0.0, 0.0, 1.0, 2.0, 2.0, 2.0]);
/// let ctrl_pts = vec![
/// Vector3::new(1.0, 1.0, 1.0),
/// Vector3::new(2.0, 2.0, 2.0),
/// Vector3::new(3.0, 3.0, 3.0),
/// Vector3::new(4.0, 4.0, 4.0),
/// ];
/// let mut curve = NURBSCurve::new(BSplineCurve::new(knot_vec, ctrl_pts));
/// let org_curve = curve.clone();
/// curve.make_locally_injective();
/// assert_eq!(curve, org_curve);
/// ```
pub fn make_locally_injective(&mut self) -> &mut Self {
let mut iter = self.0.bezier_decomposition().into_iter();
while let Some(bezier) = iter.next().map(NURBSCurve::new) {
if !bezier.is_const() {
*self = bezier;
break;
}
}
let mut x = 0.0;
for mut bezier in iter.map(NURBSCurve::new) {
if bezier.is_const() {
x += bezier.0.knot_vec.range_length();
} else {
let s0 = self.0.control_points.last().unwrap().weight();
let s1 = bezier.0.control_points[0].weight();
bezier
.0
.control_points
.iter_mut()
.for_each(move |vec| *vec *= s0 / s1);
self.concat(bezier.knot_translate(-x));
}
}
self
}
}
impl<V: Homogeneous<f64> + ControlPoint<f64, Diff = V>> ParameterDivision1D for NURBSCurve<V>
where V::Point: MetricSpace<Metric = f64> + HashGen<f64>
{
type Point = V::Point;
#[inline(always)]
fn parameter_division(&self, range: (f64, f64), tol: f64) -> (Vec<f64>, Vec<V::Point>) {
algo::curve::parameter_division(self, range, tol)
}
}
impl<V: Homogeneous<f64> + ControlPoint<f64, Diff = V>> SearchNearestParameter<D1> for NURBSCurve<V>
where
V::Point: MetricSpace<Metric = f64>,
<V::Point as EuclideanSpace>::Diff: InnerSpace + Tolerance,
{
type Point = V::Point;
/// Searches the parameter `t` which minimize |self(t) - point| by Newton's method with initial guess `hint`.
/// Returns `None` if the number of attempts exceeds `trial` i.e. if `trial == 0`, then the trial is only one time.
/// # Examples
/// ```
/// use truck_geometry::*;
///
/// // Defines the half unit circle in x > 0.
/// let knot_vec = KnotVec::bezier_knot(2);
/// let ctrl_pts = vec![Vector3::new(0.0, -1.0, 1.0), Vector3::new(1.0, 0.0, 0.0), Vector3::new(0.0, 1.0, 1.0)];
/// let curve = NURBSCurve::new(BSplineCurve::new(knot_vec, ctrl_pts));
///
/// // search rational nearest parameter
/// let pt = Point2::new(1.0, 2.0);
/// let hint = 0.8;
/// let t = curve.search_nearest_parameter(pt, Some(hint), 100).unwrap();
///
/// // check the answer
/// let res = curve.subs(t);
/// let ans = Point2::from_vec(pt.to_vec().normalize());
/// assert_near!(res, ans);
/// ```
/// # Remarks
/// It may converge to a local solution depending on the hint.
/// ```
/// use truck_geometry::*;
///
/// // Same curve and point as above example
/// let knot_vec = KnotVec::bezier_knot(2);
/// let ctrl_pts = vec![Vector3::new(0.0, -1.0, 1.0), Vector3::new(1.0, 0.0, 0.0), Vector3::new(0.0, 1.0, 1.0)];
/// let curve = NURBSCurve::new(BSplineCurve::new(knot_vec, ctrl_pts));
/// let pt = Point2::new(1.0, 2.0);
///
/// // another hint
/// let hint = 0.5;
///
/// // Newton's method is vibration divergent.
/// assert!(curve.search_nearest_parameter(pt, Some(hint), 100).is_none());
/// ```
#[inline(always)]
fn search_nearest_parameter<H: Into<SPHint1D>>(
&self,
point: V::Point,
hint: H,
trial: usize,
) -> Option<f64> {
let hint = match hint.into() {
SPHint1D::Parameter(hint) => hint,
SPHint1D::Range(x, y) => {
algo::curve::presearch(self, point, (x, y), PRESEARCH_DIVISION)
}
SPHint1D::None => {
algo::curve::presearch(self, point, self.parameter_range(), PRESEARCH_DIVISION)
}
};
algo::curve::search_nearest_parameter(self, point, hint, trial)
}
}
impl<V: Homogeneous<f64> + ControlPoint<f64, Diff = V>> SearchParameter<D1> for NURBSCurve<V>
where
V::Point: MetricSpace<Metric = f64>,
<V::Point as EuclideanSpace>::Diff: InnerSpace + Tolerance,
{
type Point = V::Point;
#[inline(always)]
fn search_parameter<H: Into<SPHint1D>>(
&self,
point: V::Point,
hint: H,
trial: usize,
) -> Option<f64> {
let hint = match hint.into() {
SPHint1D::Parameter(hint) => hint,
SPHint1D::Range(x, y) => {
algo::curve::presearch(self, point, (x, y), PRESEARCH_DIVISION)
}
SPHint1D::None => {
algo::curve::presearch(self, point, self.parameter_range(), PRESEARCH_DIVISION)
}
};
algo::curve::search_parameter(self, point, hint, trial)
}
}
impl<V: Homogeneous<f64>> NURBSCurve<V>
where V::Point:
MetricSpace<Metric = f64> + std::ops::Index<usize, Output = f64> + Bounded<f64> + Copy
{
/// Returns the bounding box including all control points.
#[inline(always)]
pub fn roughly_bounding_box(&self) -> BoundingBox<V::Point> {
self.0.control_points.iter().map(|p| p.to_point()).collect()
}
}
impl<V: Homogeneous<f64> + ControlPoint<f64, Diff = V>> ParametricCurve for NURBSCurve<V> {
type Point = V::Point;
type Vector = <V::Point as EuclideanSpace>::Diff;
#[inline(always)]
fn subs(&self, t: f64) -> Self::Point { self.0.subs(t).to_point() }
#[inline(always)]
fn der(&self, t: f64) -> Self::Vector {
let pt = self.0.subs(t);
let der = self.0.der(t);
pt.rat_der(der)
}
#[inline(always)]
fn der2(&self, t: f64) -> Self::Vector {
let pt = self.0.subs(t);
let der = self.0.der(t);
let der2 = self.0.der2(t);
pt.rat_der2(der, der2)
}
}
impl<V: Homogeneous<f64> + ControlPoint<f64, Diff = V>> BoundedCurve for NURBSCurve<V> {
#[inline(always)]
fn parameter_range(&self) -> (f64, f64) {
(
self.0.knot_vec[0],
self.0.knot_vec[self.0.knot_vec.len() - 1],
)
}
}
impl<V: Clone> Invertible for NURBSCurve<V> {
#[inline(always)]
fn invert(&mut self) { self.invert(); }
#[inline(always)]
fn inverse(&self) -> Self {
let mut curve = self.0.clone();
curve.invert();
NURBSCurve(curve)
}
}
impl<M, V: Copy> Transformed<M> for NURBSCurve<V>
where M: Copy + std::ops::Mul<V, Output = V>
{
#[inline(always)]
fn transform_by(&mut self, trans: M) {
self.0
.control_points
.iter_mut()
.for_each(move |v| *v = trans * *v)
}
}
impl<V: Homogeneous<f64>> From<BSplineCurve<V::Point>> for NURBSCurve<V> {
fn from(bspcurve: BSplineCurve<V::Point>) -> NURBSCurve<V> {
NURBSCurve::new(BSplineCurve::new_unchecked(
bspcurve.knot_vec,
bspcurve
.control_points
.into_iter()
.map(V::from_point)
.collect(),
))
}
}
#[test]
fn test_parameter_division() {
let knot_vec = KnotVec::uniform_knot(2, 3);
let ctrl_pts = vec![
Vector4::new(0.0, 0.0, 0.0, 1.0),
Vector4::new(2.0, 0.0, 0.0, 2.0),
Vector4::new(0.0, 3.0, 0.0, 3.0),
Vector4::new(0.0, 0.0, 2.0, 2.0),
Vector4::new(1.0, 1.0, 1.0, 1.0),
];
let curve = NURBSCurve::new(BSplineCurve::new(knot_vec, ctrl_pts));
let tol = 0.01;
let (div, pts) = curve.parameter_division(curve.parameter_range(), tol * 0.5);
let knot_vec = curve.knot_vec();
assert_eq!(knot_vec[0], div[0]);
assert_eq!(knot_vec.range_length(), div.last().unwrap() - div[0]);
for i in 1..div.len() {
let pt0 = curve.subs(div[i - 1]);
assert_eq!(pt0, pts[i - 1]);
let pt1 = curve.subs(div[i]);
assert_eq!(pt1, pts[i]);
let value_middle = pt0.midpoint(pt1);
let param_middle = curve.subs((div[i - 1] + div[i]) / 2.0);
let dist = value_middle.distance(param_middle);
assert!(dist < tol, "large distance: {}", dist);
}
}