/*

* Paper.js - The Swiss Army Knife of Vector Graphics Scripting.

* http://paperjs.org/

*

* Copyright (c) 2011 - 2013, Juerg Lehni & Jonathan Puckey

* http://lehni.org/ & http://jonathanpuckey.com/

*

* Distributed under the MIT license. See LICENSE file for details.

*

* All rights reserved.

*/

var Numerical = new function() {

// Lookup tables for abscissas and weights with values for n = 2 .. 16.

// As values are symetric, only store half of them and addapt algorithm

// to factor in symetry.

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],

[ 0.1834346424956498049394761,0.5255324099163289858177390,0.7966664774136267395915539,0.9602898564975362316835609],

[0,0.3242534234038089290385380,0.6133714327005903973087020,0.8360311073266357942994298,0.9681602395076260898355762],

[ 0.1488743389816312108848260,0.4333953941292471907992659,0.6794095682990244062343274,0.8650633666889845107320967,0.9739065285171717200779640],

[0,0.2695431559523449723315320,0.5190961292068118159257257,0.7301520055740493240934163,0.8870625997680952990751578,0.9782286581460569928039380],

[ 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],

[0.5688888888888888888888889,0.4786286704993664680412915,0.2369268850561890875142640],

[0.4679139345726910473898703,0.3607615730481386075698335,0.1713244923791703450402961],

[0.4179591836734693877551020,0.3818300505051189449503698,0.2797053914892766679014678,0.1294849661688696932706114],

[0.3626837833783619829651504,0.3137066458778872873379622,0.2223810344533744705443560,0.1012285362903762591525314],

[0.3302393550012597631645251,0.3123470770400028400686304,0.2606106964029354623187429,0.1806481606948574040584720,0.0812743883615744119718922],

[0.2955242247147528701738930,0.2692667193099963550912269,0.2190863625159820439955349,0.1494513491505805931457763,0.0666713443086881375935688],

[0.2729250867779006307144835,0.2628045445102466621806889,0.2331937645919904799185237,0.1862902109277342514260976,0.1255803694649046246346943,0.0556685671161736664827537],

[0.2491470458134027850005624,0.2334925365383548087608499,0.2031674267230659217490645,0.1600783285433462263346525,0.1069393259953184309602547,0.0471753363865118271946160],

[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,

cos = Math.cos,

PI = Math.PI;

return {

TOLERANCE: 10e-6,

// Precision when comparing against 0

EPSILON: 10e-12,

// Kappa, see: http://www.whizkidtech.redprince.net/bezier/circle/kappa/

KAPPA: 4 * (sqrt(2) - 1) / 3,

/**

* Check if the value is 0, within a tolerance defined by

* Numerical.EPSILON.

*/

isZero: function(val) {

return abs(val) <= Numerical.EPSILON;

},

/**

* Gauss-Legendre Numerical Integration.

*/

integrate: function(f, a, b, n) {

var x = abscissas[n - 2],

w = weights[n - 2],

A = 0.5 * (b - a),

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),

dx = fx / df(x);

// See if we can trust the Newton-Raphson result. If not we use

// bisection to find another candiate for Newton's method.

if (abs(dx) < tolerance)

return x;

// Generate a candidate for Newton's method.

var nx = x - dx;

// 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 ? 0.5 * (a + b) : nx;

} else {

a = x;

x = nx >= b ? 0.5 * (a + b) : nx;

}

}

},

/**

* Solves the quadratic polynomial with coefficients a, b, c for roots

* (zero crossings) and and returns the solutions in an array.

*

* a*x^2 + b*x + c = 0

*/

solveQuadratic: function(a, b, c, roots, min, max) {

var unbound = min === undefined,

count = 0;

function add(root) {

if (unbound || root >= min && root <= max)

roots[count++] = root;

return count;

}

// Code ported over and adapted from Uintah library (MIT license).

var epsilon = this.EPSILON;

// If a is 0, equation is actually linear, return 0 or 1 easy roots.

if (abs(a) < epsilon) {

if (abs(b) >= epsilon)

return add(-c / b);

// If all the coefficients are 0, we have infinite solutions!

return abs(c) < epsilon ? -1 : 0; // Infinite or 0 solutions

}

// Convert to normal form: x^2 + px + q = 0

var p = b / (2 * a);

var q = c / a;

var p2 = p * p;

if (p2 < q - epsilon)

return 0;

var s = p2 > q ? sqrt(p2 - q) : 0;

add (s - p);

if (s > 0)

add(-s - p);

return count;

},

/**

* Solves the cubic polynomial with coefficients a, b, c, d for roots

* (zero crossings) and and returns the solutions in an array.

*

* a*x^3 + b*x^2 + c*x + d = 0

*/

solveCubic: function(a, b, c, d, roots, min, max) {

var epsilon = this.EPSILON;

// If a is 0, equation is actually quadratic.

if (abs(a) < epsilon)

return Numerical.solveQuadratic(b, c, d, roots, min, max);

var unbound = min === undefined,

count = 0;

function add(root) {

if (unbound || root >= min && root <= max)

roots[count++] = root;

return count;

}

// Code ported over and adapted from Uintah library (MIT license).

// Normalize to form: x^3 + b x^2 + c x + d = 0:

b /= a;

c /= a;

d /= a;

// Compute discriminants

var bb = b * b,

p = (bb - 3 * c) / 9,

q = (2 * bb * b - 9 * b * c + 27 * d) / 54,

// Use Cardano's formula

ppp = p * p * p,

D = q * q - ppp;

// Substitute x = y - b/3 to eliminate quadric term: x^3 +px + q = 0

b /= 3;

if (abs(D) < epsilon) {

if (abs(q) < epsilon) // One triple solution.

return add(-b);

// One single and one double solution.

var sqp = sqrt(p),

snq = q > 0 ? 1 : -1;

add(-snq * 2 * sqp - b);

return add(snq * sqp - b);

}

if (D < 0) { // Casus irreducibilis: three real solutions

var sqp = sqrt(p),

phi = Math.acos(q / (sqp * sqp * sqp)) / 3,

t = -2 * sqp,

o = 2 * PI / 3;

add(t * cos(phi) - b);

add(t * cos(phi + o) - b);

return add(t * cos(phi - o) - b);

}

// One real solution

var A = (q > 0 ? -1 : 1) * pow(abs(q) + sqrt(D), 1 / 3);

return add(A + p / A - b);

}

};

};