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* Paper.js - The Swiss Army Knife of Vector Graphics Scripting.
* http://paperjs.org/
*
* Copyright (c) 2011 - 2020, Jürg Lehni & Jonathan Puckey
* http://juerglehni.com/ & https://puckey.studio/
*
* Distributed under the MIT license. See LICENSE file for details.
*
* All rights reserved.
*/
// TODO: remove eslint-disable comment and deal with errors over time
/* eslint-disable */
/**
* @name Numerical
* @namespace
* @private
*/
// @ts-expect-error = 'new' expression, whose target lacks a construct signature
export const Numerical = new (function () {
// Lookup tables for abscissas and weights with values for n = 2 .. 16.
// As values are symmetric, only store half of them and adapt algorithm
// to factor in symmetry.
var abscissas = [
[0.5773502691896257645091488],
[0, 0.7745966692414833770358531],
[0.3399810435848562648026658, 0.8611363115940525752239465],
[0, 0.5384693101056830910363144, 0.9061798459386639927976269],
[0.2386191860831969086305017, 0.6612093864662645136613996, 0.9324695142031520278123016],
[0, 0.4058451513773971669066064, 0.7415311855993944398638648, 0.9491079123427585245261897],
[0.1834346424956498049394761, 0.525532409916328985817739, 0.7966664774136267395915539, 0.9602898564975362316835609],
[
0, 0.324253423403808929038538, 0.613371432700590397308702, 0.8360311073266357942994298,
0.9681602395076260898355762,
],
[
0.148874338981631210884826, 0.4333953941292471907992659, 0.6794095682990244062343274, 0.8650633666889845107320967,
0.973906528517171720077964,
],
[
0, 0.269543155952344972331532, 0.5190961292068118159257257, 0.7301520055740493240934163,
0.8870625997680952990751578, 0.978228658146056992803938,
],
[
0.1252334085114689154724414, 0.3678314989981801937526915, 0.5873179542866174472967024,
0.7699026741943046870368938, 0.9041172563704748566784659, 0.9815606342467192506905491,
],
[
0, 0.2304583159551347940655281, 0.4484927510364468528779129, 0.6423493394403402206439846,
0.8015780907333099127942065, 0.9175983992229779652065478, 0.9841830547185881494728294,
],
[
0.1080549487073436620662447, 0.3191123689278897604356718, 0.5152486363581540919652907,
0.6872929048116854701480198, 0.8272013150697649931897947, 0.9284348836635735173363911,
0.9862838086968123388415973,
],
[
0, 0.2011940939974345223006283, 0.3941513470775633698972074, 0.5709721726085388475372267,
0.7244177313601700474161861, 0.8482065834104272162006483, 0.9372733924007059043077589,
0.9879925180204854284895657,
],
[
0.0950125098376374401853193, 0.2816035507792589132304605, 0.4580167776572273863424194,
0.6178762444026437484466718, 0.7554044083550030338951012, 0.8656312023878317438804679,
0.9445750230732325760779884, 0.9894009349916499325961542,
],
];
var weights = [
[1],
[0.8888888888888888888888889, 0.5555555555555555555555556],
[0.6521451548625461426269361, 0.3478548451374538573730639],
[0.5688888888888888888888889, 0.4786286704993664680412915, 0.236926885056189087514264],
[0.4679139345726910473898703, 0.3607615730481386075698335, 0.1713244923791703450402961],
[0.417959183673469387755102, 0.3818300505051189449503698, 0.2797053914892766679014678, 0.1294849661688696932706114],
[0.3626837833783619829651504, 0.3137066458778872873379622, 0.222381034453374470544356, 0.1012285362903762591525314],
[
0.3302393550012597631645251, 0.3123470770400028400686304, 0.2606106964029354623187429, 0.180648160694857404058472,
0.0812743883615744119718922,
],
[
0.295524224714752870173893, 0.2692667193099963550912269, 0.2190863625159820439955349, 0.1494513491505805931457763,
0.0666713443086881375935688,
],
[
0.2729250867779006307144835, 0.2628045445102466621806889, 0.2331937645919904799185237,
0.1862902109277342514260976, 0.1255803694649046246346943, 0.0556685671161736664827537,
],
[
0.2491470458134027850005624, 0.2334925365383548087608499, 0.2031674267230659217490645,
0.1600783285433462263346525, 0.1069393259953184309602547, 0.047175336386511827194616,
],
[
0.2325515532308739101945895, 0.2262831802628972384120902, 0.2078160475368885023125232,
0.1781459807619457382800467, 0.1388735102197872384636018, 0.0921214998377284479144218,
0.0404840047653158795200216,
],
[
0.2152638534631577901958764, 0.2051984637212956039659241, 0.1855383974779378137417166,
0.1572031671581935345696019, 0.1215185706879031846894148, 0.0801580871597602098056333,
0.0351194603317518630318329,
],
[
0.2025782419255612728806202, 0.1984314853271115764561183, 0.1861610000155622110268006,
0.1662692058169939335532009, 0.1395706779261543144478048, 0.1071592204671719350118695,
0.0703660474881081247092674, 0.0307532419961172683546284,
],
[
0.1894506104550684962853967, 0.1826034150449235888667637, 0.1691565193950025381893121,
0.1495959888165767320815017, 0.1246289712555338720524763, 0.0951585116824927848099251,
0.0622535239386478928628438, 0.0271524594117540948517806,
],
];
// Math short-cuts for often used methods and values
var abs = Math.abs,
sqrt = Math.sqrt,
pow = Math.pow,
// Fallback to polyfill:
log2 =
Math.log2 ||
function (x) {
return Math.log(x) * Math.LOG2E;
},
// Constants
EPSILON = 1e-12,
MACHINE_EPSILON = 1.12e-16;
function clamp(value, min, max) {
return value < min ? min : value > max ? max : value;
}
function getDiscriminant(a, b, c) {
// d = b^2 - a * c computed accurately enough by a tricky scheme.
// Ported from @hkrish's polysolve.c
function split(v) {
var x = v * 134217729,
y = v - x,
hi = y + x, // Don't optimize y away!
lo = v - hi;
return [hi, lo];
}
var D = b * b - a * c,
E = b * b + a * c;
if (abs(D) * 3 < E) {
var ad = split(a),
bd = split(b),
cd = split(c),
p = b * b,
dp = bd[0] * bd[0] - p + 2 * bd[0] * bd[1] + bd[1] * bd[1],
q = a * c,
dq = ad[0] * cd[0] - q + ad[0] * cd[1] + ad[1] * cd[0] + ad[1] * cd[1];
D = p - q + (dp - dq); // Don’t omit parentheses!
}
return D;
}
function getNormalizationFactor() {
// Normalize coefficients à la Jenkins & Traub's RPOLY.
// Normalization is done by scaling coefficients with a power of 2, so
// that all the bits in the mantissa remain unchanged.
// Use the infinity norm (max(sum(abs(a)…)) to determine the appropriate
// scale factor. See @hkrish in #1087#issuecomment-231526156
var norm = Math.max.apply(Math, arguments);
return norm && (norm < 1e-8 || norm > 1e8) ? pow(2, -Math.round(log2(norm))) : 0;
}
return /** @lends Numerical */ {
/**
* A very small absolute value used to check if a value is very close to
* zero. The value should be large enough to offset any floating point
* noise, but small enough to be meaningful in computation in a nominal
* range (see MACHINE_EPSILON).
*
* http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html
* http://www.cs.berkeley.edu/~wkahan/Math128/Cubic.pdf
*/
EPSILON: EPSILON,
/**
* The machine epsilon for a double precision (Javascript Number) is
* 2.220446049250313e-16. (try this in the js console:
* (function(){ for (var e = 1; 1 < 1+e/2;) e/=2; return e }())
*
* The constant MACHINE_EPSILON here refers to the constants δ and ε
* such that, the error introduced by addition, multiplication on a
* 64bit float (js Number) will be less than δ and ε. That is to say,
* for all X and Y representable by a js Number object, S and P be their
* 'exact' sum and product respectively, then
* |S - (x+y)| <= δ|S|, and |P - (x*y)| <= ε|P|.
* This amounts to about half of the actual machine epsilon.
*/
MACHINE_EPSILON: MACHINE_EPSILON,
/**
* The epsilon to be used when handling curve-time parameters. This
* cannot be smaller, because errors add up to around 2e-7 in the bezier
* fat-line clipping code as a result of recursive sub-division.
*/
CURVETIME_EPSILON: 1e-8,
/**
* The epsilon to be used when performing "geometric" checks, such as
* distances between points and lines.
*/
GEOMETRIC_EPSILON: 1e-7,
/**
* The epsilon to be used when performing "trigonometric" checks, such
* as examining cross products to check for collinearity.
*/
TRIGONOMETRIC_EPSILON: 1e-8,
/**
* The epsilon to be used when performing angular checks in degrees,
* e.g. in `arcTo()`.
*/
ANGULAR_EPSILON: 1e-5,
/**
* Kappa is the value which which to scale the curve handles when
* drawing a circle with bezier curves.
*
* http://whizkidtech.redprince.net/bezier/circle/kappa/
*/
KAPPA: (4 * (sqrt(2) - 1)) / 3,
/**
* Checks if the value is 0, within a tolerance defined by
* Numerical.EPSILON.
*/
isZero: function (val) {
return val >= -EPSILON && val <= EPSILON;
},
isMachineZero: function (val) {
return val >= -MACHINE_EPSILON && val <= MACHINE_EPSILON;
},
/**
* Returns a number whose value is clamped by the given range.
*
* @param {Number} value the value to be clamped
* @param {Number} min the lower boundary of the range
* @param {Number} max the upper boundary of the range
* @return {Number} a number in the range of [min, max]
*/
clamp: clamp,
/**
* Gauss-Legendre Numerical Integration.
*/
integrate: function (f, a, b, n) {
var x = abscissas[n - 2],
w = weights[n - 2],
A = (b - a) * 0.5,
B = A + a,
i = 0,
m = (n + 1) >> 1,
sum = n & 1 ? w[i++] * f(B) : 0; // Handle odd n
while (i < m) {
var Ax = A * x[i];
sum += w[i++] * (f(B + Ax) + f(B - Ax));
}
return A * sum;
},
/**
* Root finding using Newton-Raphson Method combined with Bisection.
*/
findRoot: function (f, df, x, a, b, n, tolerance) {
for (var i = 0; i < n; i++) {
var fx = f(x),
// Calculate a new candidate with the Newton-Raphson method.
dx = fx / df(x),
nx = x - dx;
// See if we can trust the Newton-Raphson result. If not we use
// bisection to find another candidate for Newton's method.
if (abs(dx) < tolerance) {
x = nx;
break;
}
// Update the root-bounding interval and test for containment of
// the candidate. If candidate is outside the root-bounding
// interval, use bisection instead.
// There is no need to compare to lower / upper because the
// tangent line has positive slope, guaranteeing that the x-axis
// intercept is larger than lower / smaller than upper.
if (fx > 0) {
b = x;
x = nx <= a ? (a + b) * 0.5 : nx;
} else {
a = x;
x = nx >= b ? (a + b) * 0.5 : nx;
}
}
// Return the best result even though we haven't gotten close
// enough to the root... (In paper.js this never seems to happen).
// But make sure, that it actually is within the given range [a, b]
return clamp(x, a, b);
},
/**
* Solve a quadratic equation in a numerically robust manner;
* given a quadratic equation ax² + bx + c = 0, find the values of x.
*
* References:
* Kahan W. - "To Solve a Real Cubic Equation"
* http://www.cs.berkeley.edu/~wkahan/Math128/Cubic.pdf
* Blinn J. - "How to solve a Quadratic Equation"
* Harikrishnan G.
* https://gist.github.com/hkrish/9e0de1f121971ee0fbab281f5c986de9
*
* @param {Number} a the quadratic term
* @param {Number} b the linear term
* @param {Number} c the constant term
* @param {Number[]} roots the array to store the roots in
* @param {Number} [min] the lower bound of the allowed roots
* @param {Number} [max] the upper bound of the allowed roots
* @return {Number} the number of real roots found, or -1 if there are
* infinite solutions
*
* @author Harikrishnan Gopalakrishnan <hari.exeption@gmail.com>
*/
solveQuadratic: function (a, b, c, roots, min, max) {
var x1,
x2 = Infinity;
if (abs(a) < EPSILON) {
// This could just be a linear equation
if (abs(b) < EPSILON) return abs(c) < EPSILON ? -1 : 0;
x1 = -c / b;
} else {
// a, b, c are expected to be the coefficients of the equation:
// Ax² - 2Bx + C == 0, so we take b = -b/2:
b *= -0.5;
var D = getDiscriminant(a, b, c);
// If the discriminant is very small, we can try to normalize
// the coefficients, so that we may get better accuracy.
if (D && abs(D) < MACHINE_EPSILON) {
// @ts-expect-error = Expected 0 arguments, but got 3
var f = getNormalizationFactor(abs(a), abs(b), abs(c));
if (f) {
a *= f;
b *= f;
c *= f;
D = getDiscriminant(a, b, c);
}
}
if (D >= -MACHINE_EPSILON) {
// No real roots if D < 0
var Q = D < 0 ? 0 : sqrt(D),
R = b + (b < 0 ? -Q : Q);
// Try to minimize floating point noise.
if (R === 0) {
x1 = c / a;
x2 = -x1;
} else {
x1 = R / a;
x2 = c / R;
}
}
}
var count = 0,
boundless = min == null,
minB = min - EPSILON,
maxB = max + EPSILON;
// We need to include EPSILON in the comparisons with min / max,
// as some solutions are ever so lightly out of bounds.
if (isFinite(x1) && (boundless || (x1 > minB && x1 < maxB)))
roots[count++] = boundless ? x1 : clamp(x1, min, max);
if (x2 !== x1 && isFinite(x2) && (boundless || (x2 > minB && x2 < maxB)))
roots[count++] = boundless ? x2 : clamp(x2, min, max);
return count;
},
/**
* Solve a cubic equation, using numerically stable methods,
* given an equation of the form ax³ + bx² + cx + d = 0.
*
* This algorithm avoids the trigonometric/inverse trigonometric
* calculations required by the "Italins"' formula. Cardano's method
* works well enough for exact computations, this method takes a
* numerical approach where the double precision error bound is kept
* very low.
*
* References:
* Kahan W. - "To Solve a Real Cubic Equation"
* http://www.cs.berkeley.edu/~wkahan/Math128/Cubic.pdf
* Harikrishnan G.
* https://gist.github.com/hkrish/9e0de1f121971ee0fbab281f5c986de9
*
* W. Kahan's paper contains inferences on accuracy of cubic
* zero-finding methods. Also testing methods for robustness.
*
* @param {Number} a the cubic term (x³ term)
* @param {Number} b the quadratic term (x² term)
* @param {Number} c the linear term (x term)
* @param {Number} d the constant term
* @param {Number[]} roots the array to store the roots in
* @param {Number} [min] the lower bound of the allowed roots
* @param {Number} [max] the upper bound of the allowed roots
* @return {Number} the number of real roots found, or -1 if there are
* infinite solutions
*
* @author Harikrishnan Gopalakrishnan <hari.exeption@gmail.com>
*/
solveCubic: function (a, b, c, d, roots, min, max) {
// @ts-expect-error = Expected 0 arguments, but got 4
var f = getNormalizationFactor(abs(a), abs(b), abs(c), abs(d)),
x,
b1,
c2,
qd,
q;
if (f) {
a *= f;
b *= f;
c *= f;
d *= f;
}
function evaluate(x0) {
x = x0;
// Evaluate q, q', b1 and c2 at x
var tmp = a * x;
b1 = tmp + b;
c2 = b1 * x + c;
qd = (tmp + b1) * x + c2;
q = c2 * x + d;
}
// If a or d is zero, we only need to solve a quadratic, so we set
// the coefficients appropriately.
if (abs(a) < EPSILON) {
a = b;
b1 = c;
c2 = d;
x = Infinity;
} else if (abs(d) < EPSILON) {
b1 = b;
c2 = c;
x = 0;
} else {
// Here onwards we iterate for the leftmost root. Proceed to
// deflate the cubic into a quadratic (as a side effect to the
// iteration) and solve the quadratic.
evaluate(-(b / a) / 3);
// Get a good initial approximation.
var t = q / a,
r = pow(abs(t), 1 / 3),
s = t < 0 ? -1 : 1,
td = -qd / a,
// See Kahan's notes on why 1.324718*... works.
rd = td > 0 ? 1.324717957244746 * Math.max(r, sqrt(td)) : r,
x0 = x - s * rd;
if (x0 !== x) {
do {
evaluate(x0);
// Newton's. Divide by 1 + MACHINE_EPSILON (1.000...002)
// to avoid x0 crossing over a root.
x0 = qd === 0 ? x : x - q / qd / (1 + MACHINE_EPSILON);
} while (s * x0 > s * x);
// Adjust the coefficients for the quadratic.
if (abs(a) * x * x > abs(d / x)) {
c2 = -d / x;
b1 = (c2 - c) / x;
}
}
}
// The cubic has been deflated to a quadratic.
var count = Numerical.solveQuadratic(a, b1, c2, roots, min, max),
boundless = min == null;
if (
isFinite(x) &&
(count === 0 || (count > 0 && x !== roots[0] && x !== roots[1])) &&
(boundless || (x > min - EPSILON && x < max + EPSILON))
)
roots[count++] = boundless ? x : clamp(x, min, max);
return count;
},
};
})();
|