kurbo 0.6.0

A 2D curves library
Documentation 
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//! Common mathematical operations

#![allow(missing_docs)]

use arrayvec::ArrayVec;

/// Adds convenience methods to `f32` and `f64`.
pub trait FloatExt<T> {
    /// Rounds to the nearest integer away from zero,
    /// unless the provided value is already an integer.
    ///
    /// It is to `ceil` what `trunc` is to `floor`.
    ///
    /// # Examples
    ///
    /// ```
    /// use kurbo::common::FloatExt;
    ///
    /// let f = 3.7_f64;
    /// let g = 3.0_f64;
    /// let h = -3.7_f64;
    /// let i = -5.1_f32;
    ///
    /// assert_eq!(f.expand(), 4.0);
    /// assert_eq!(g.expand(), 3.0);
    /// assert_eq!(h.expand(), -4.0);
    /// assert_eq!(i.expand(), -6.0);
    /// ```
    fn expand(&self) -> T;
}

impl FloatExt<f64> for f64 {
    #[inline]
    fn expand(&self) -> f64 {
        self.abs().ceil().copysign(*self)
    }
}

impl FloatExt<f32> for f32 {
    #[inline]
    fn expand(&self) -> f32 {
        self.abs().ceil().copysign(*self)
    }
}

/// Find real roots of cubic equation.
///
/// The implementation is not (yet) fully robust, but it does handle the case
/// where `c3` is zero (in that case, solving the quadratic equation).
///
/// See: <http://mathworld.wolfram.com/CubicFormula.html>
///
/// Returns values of x for which c0 + c1 x + c2 x² + c3 x³ = 0.
pub fn solve_cubic(c0: f64, c1: f64, c2: f64, c3: f64) -> ArrayVec<[f64; 3]> {
    let mut result = ArrayVec::new();
    let c3_recip = c3.recip();
    let scaled_c2 = c2 * c3_recip;
    let scaled_c1 = c1 * c3_recip;
    let scaled_c0 = c0 * c3_recip;
    if !(scaled_c0.is_finite() && scaled_c1.is_finite() && scaled_c2.is_finite()) {
        // cubic coefficient is zero or nearly so.
        for root in solve_quadratic(c0, c1, c2) {
            result.push(root);
        }
        return result;
    }
    let (c0, c1, c2) = (scaled_c0, scaled_c1, scaled_c2);
    let q = c1 * (1.0 / 3.0) - c2 * c2 * (1.0 / 9.0); // Q
    let r = (1.0 / 6.0) * c2 * c1 - (1.0 / 27.0) * c2.powi(3) - c0 * 0.5; // R
    let d = q.powi(3) + r * r; // D
    let x0 = c2 * (1.0 / 3.0);
    // TODO: handle the cases where these intermediate results overflow.
    if d > 0.0 {
        let sq = d.sqrt();
        let t1 = (r + sq).cbrt() + (r - sq).cbrt();
        result.push(t1 - x0);
    } else if d == 0.0 {
        let t1 = -r.cbrt();
        let x1 = t1 - x0;
        result.push(x1);
        result.push(-2.0 * t1 - x0);
    } else {
        let sq = (-d).sqrt();
        let rho = r.hypot(sq);
        let th = sq.atan2(r) * (1.0 / 3.0);
        let cbrho = rho.cbrt();
        let c = th.cos();
        let ss3 = th.sin() * 3.0f64.sqrt();
        result.push(2.0 * cbrho * c - x0);
        result.push(-cbrho * (c + ss3) - x0);
        result.push(-cbrho * (c - ss3) - x0);
    }
    result
}

/// Find real roots of quadratic equation.
///
/// Returns values of x for which c0 + c1 x + c2 x² = 0.
///
/// This function tries to be quite numerically robust. If the equation
/// is nearly linear, it will return the root ignoring the quadratic term;
/// the other root might be out of representable range. In the degenerate
/// case where all coefficients are zero, so that all values of x satisfy
/// the equation, a single `0.0` is returned.
pub fn solve_quadratic(c0: f64, c1: f64, c2: f64) -> ArrayVec<[f64; 2]> {
    let mut result = ArrayVec::new();
    let sc0 = c0 * c2.recip();
    let sc1 = c1 * c2.recip();
    if !sc0.is_finite() || !sc1.is_finite() {
        // c2 is zero or very small, treat as linear eqn
        let root = -c0 / c1;
        if root.is_finite() {
            result.push(root);
        } else if c0 == 0.0 && c1 == 0.0 {
            // Degenerate case
            result.push(0.0);
        }
        return result;
    }
    let arg = sc1 * sc1 - 4. * sc0;
    let root1 = if !arg.is_finite() {
        // Likely, calculation of sc1 * sc1 overflowed. Find one root
        // using sc1 x + x² = 0, other root as sc0 / root1.
        -sc1
    } else {
        if arg < 0.0 {
            return result;
        } else if arg == 0.0 {
            result.push(-0.5 * sc1);
            return result;
        }
        // See https://math.stackexchange.com/questions/866331
        -0.5 * (sc1 + arg.sqrt().copysign(sc1))
    };
    let root2 = sc0 / root1;
    if root2.is_finite() {
        // Sort just to be friendly and make results deterministic.
        if root2 > root1 {
            result.push(root1);
            result.push(root2);
        } else {
            result.push(root2);
            result.push(root1);
        }
    } else {
        result.push(root1);
    }
    result
}

// Tables of Legendre-Gauss quadrature coefficients, adapted from:
// <https://pomax.github.io/bezierinfo/legendre-gauss.html>

pub const GAUSS_LEGENDRE_COEFFS_3: &[(f64, f64)] = &[
    (0.8888888888888888, 0.0000000000000000),
    (0.5555555555555556, -0.7745966692414834),
    (0.5555555555555556, 0.7745966692414834),
];

pub const GAUSS_LEGENDRE_COEFFS_5: &[(f64, f64)] = &[
    (0.5688888888888889, 0.0000000000000000),
    (0.4786286704993665, -0.5384693101056831),
    (0.4786286704993665, 0.5384693101056831),
    (0.2369268850561891, -0.9061798459386640),
    (0.2369268850561891, 0.9061798459386640),
];

pub const GAUSS_LEGENDRE_COEFFS_7: &[(f64, f64)] = &[
    (0.4179591836734694, 0.0000000000000000),
    (0.3818300505051189, 0.4058451513773972),
    (0.3818300505051189, -0.4058451513773972),
    (0.2797053914892766, -0.7415311855993945),
    (0.2797053914892766, 0.7415311855993945),
    (0.1294849661688697, -0.9491079123427585),
    (0.1294849661688697, 0.9491079123427585),
];

pub const GAUSS_LEGENDRE_COEFFS_9: &[(f64, f64)] = &[
    (0.3302393550012598, 0.0000000000000000),
    (0.1806481606948574, -0.8360311073266358),
    (0.1806481606948574, 0.8360311073266358),
    (0.0812743883615744, -0.9681602395076261),
    (0.0812743883615744, 0.9681602395076261),
    (0.3123470770400029, -0.3242534234038089),
    (0.3123470770400029, 0.3242534234038089),
    (0.2606106964029354, -0.6133714327005904),
    (0.2606106964029354, 0.6133714327005904),
];

pub const GAUSS_LEGENDRE_COEFFS_11: &[(f64, f64)] = &[
    (0.2729250867779006, 0.0000000000000000),
    (0.2628045445102467, -0.2695431559523450),
    (0.2628045445102467, 0.2695431559523450),
    (0.2331937645919905, -0.5190961292068118),
    (0.2331937645919905, 0.5190961292068118),
    (0.1862902109277343, -0.7301520055740494),
    (0.1862902109277343, 0.7301520055740494),
    (0.1255803694649046, -0.8870625997680953),
    (0.1255803694649046, 0.8870625997680953),
    (0.0556685671161737, -0.9782286581460570),
    (0.0556685671161737, 0.9782286581460570),
];

pub const GAUSS_LEGENDRE_COEFFS_24: &[(f64, f64)] = &[
    (0.1279381953467522, -0.0640568928626056),
    (0.1279381953467522, 0.0640568928626056),
    (0.1258374563468283, -0.1911188674736163),
    (0.1258374563468283, 0.1911188674736163),
    (0.1216704729278034, -0.3150426796961634),
    (0.1216704729278034, 0.3150426796961634),
    (0.1155056680537256, -0.4337935076260451),
    (0.1155056680537256, 0.4337935076260451),
    (0.1074442701159656, -0.5454214713888396),
    (0.1074442701159656, 0.5454214713888396),
    (0.0976186521041139, -0.6480936519369755),
    (0.0976186521041139, 0.6480936519369755),
    (0.0861901615319533, -0.7401241915785544),
    (0.0861901615319533, 0.7401241915785544),
    (0.0733464814110803, -0.8200019859739029),
    (0.0733464814110803, 0.8200019859739029),
    (0.0592985849154368, -0.8864155270044011),
    (0.0592985849154368, 0.8864155270044011),
    (0.0442774388174198, -0.9382745520027328),
    (0.0442774388174198, 0.9382745520027328),
    (0.0285313886289337, -0.9747285559713095),
    (0.0285313886289337, 0.9747285559713095),
    (0.0123412297999872, -0.9951872199970213),
    (0.0123412297999872, 0.9951872199970213),
];

#[cfg(test)]
mod tests {
    use crate::common::*;
    use arrayvec::{Array, ArrayVec};

    fn verify<T: Array<Item = f64>>(mut roots: ArrayVec<T>, expected: &[f64]) {
        assert!(expected.len() == roots.len());
        let epsilon = 1e-6;
        roots.sort_by(|a, b| a.partial_cmp(b).unwrap());
        for i in 0..expected.len() {
            assert!((roots[i] - expected[i]).abs() < epsilon);
        }
    }

    #[test]
    fn test_solve_cubic() {
        verify(solve_cubic(-5.0, 0.0, 0.0, 1.0), &[5.0f64.cbrt()]);
        verify(solve_cubic(-5.0, -1.0, 0.0, 1.0), &[1.90416085913492]);
        verify(solve_cubic(0.0, -1.0, 0.0, 1.0), &[-1.0, 0.0, 1.0]);
        verify(solve_cubic(-2.0, -3.0, 0.0, 1.0), &[-1.0, 2.0]);
        verify(solve_cubic(2.0, -3.0, 0.0, 1.0), &[-2.0, 1.0]);
        verify(solve_cubic(2.0 - 1e-12, 5.0, 4.0, 1.0), &[-2.0, -1.0, -1.0]);
        verify(solve_cubic(2.0 + 1e-12, 5.0, 4.0, 1.0), &[-2.0]);
    }

    #[test]
    fn test_solve_quadratic() {
        verify(
            solve_quadratic(-5.0, 0.0, 1.0),
            &[-5.0f64.sqrt(), 5.0f64.sqrt()],
        );
        verify(solve_quadratic(5.0, 0.0, 1.0), &[]);
        verify(solve_quadratic(5.0, 1.0, 0.0), &[-5.0]);
        verify(solve_quadratic(1.0, 2.0, 1.0), &[-1.0]);
    }
}