/* * Paper.js * * This file is part of Paper.js, a JavaScript Vector Graphics Library, * based on Scriptographer.org and designed to be largely API compatible. * http://paperjs.org/ * http://scriptographer.org/ * * Copyright (c) 2011, 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, cos = Math.cos, PI = Math.PI; return { TOLERANCE: 10e-6, /** * 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, tolerance) { // After Numerical Recipes in C, 2nd edition, Press et al., // 5.6, Quadratic and Cubic Equations // If problem is actually linear, return 0 or 1 easy roots if (abs(a) < tolerance) { if (abs(b) >= tolerance) return [ -c / b ]; // If all the coefficients are 0, infinite values are // possible! if (abs(c) < tolerance) return Infinity; // Infinite solutions return []; // 0 solutions } var q = b * b - 4 * a * c; if (q < 0) return []; // 0 solutions q = sqrt(q); if (b < 0) q = -q; q = (b + q) * -0.5; var roots = []; if (abs(q) >= tolerance) roots.push(c / q); if (abs(a) >= tolerance) roots.push(q / a); return roots; // 0, 1 or 2 solutions }, /** * 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, tolerance) { // After Numerical Recipes in C, 2nd edition, Press et al., // 5.6, Quadratic and Cubic Equations if (abs(a) < tolerance) return Numerical.solveQuadratic(b, c, d, tolerance); // Normalize b /= a; c /= a; d /= a; // Compute discriminants var Q = (b * b - 3 * c) / 9, R = (2 * b * b * b - 9 * b * c + 27 * d) / 54, Q3 = Q * Q * Q, R2 = R * R; b /= 3; // Divide by 3 as that's required below if (R2 < Q3) { // Three real roots // This sqrt and division is safe, since R2 >= 0, so Q3 > R2, // so Q3 > 0. The acos is also safe, since R2/Q3 < 1, and // thus R/sqrt(Q3) < 1. var theta = Math.acos(R / sqrt(Q3)), // This sqrt is safe, since Q3 >= 0, and thus Q >= 0 q = -2 * sqrt(Q); return [ q * cos(theta / 3) - b, q * cos((theta + 2 * PI) / 3) - b, q * cos((theta - 2 * PI) / 3) - b ]; } else { // One real root var A = -Math.pow(abs(R) + sqrt(R2 - Q3), 1 / 3); if (R < 0) A = -A; var B = (abs(A) < tolerance) ? 0 : Q / A; return [ (A + B) - b ]; } return []; } }; };