mirror of
https://github.com/scratchfoundation/paper.js.git
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206 lines
8.6 KiB
JavaScript
206 lines
8.6 KiB
JavaScript
/*
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* Paper.js - The Swiss Army Knife of Vector Graphics Scripting.
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* http://paperjs.org/
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*
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* Copyright (c) 2011 - 2013, Juerg Lehni & Jonathan Puckey
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* http://lehni.org/ & http://jonathanpuckey.com/
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*
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* Distributed under the MIT license. See LICENSE file for details.
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*
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* All rights reserved.
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*/
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var Numerical = new function() {
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// Lookup tables for abscissas and weights with values for n = 2 .. 16.
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// As values are symetric, only store half of them and addapt algorithm
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// to factor in symetry.
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var abscissas = [
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[ 0.5773502691896257645091488],
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[0,0.7745966692414833770358531],
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[ 0.3399810435848562648026658,0.8611363115940525752239465],
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[0,0.5384693101056830910363144,0.9061798459386639927976269],
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[ 0.2386191860831969086305017,0.6612093864662645136613996,0.9324695142031520278123016],
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[0,0.4058451513773971669066064,0.7415311855993944398638648,0.9491079123427585245261897],
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[ 0.1834346424956498049394761,0.5255324099163289858177390,0.7966664774136267395915539,0.9602898564975362316835609],
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[0,0.3242534234038089290385380,0.6133714327005903973087020,0.8360311073266357942994298,0.9681602395076260898355762],
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[ 0.1488743389816312108848260,0.4333953941292471907992659,0.6794095682990244062343274,0.8650633666889845107320967,0.9739065285171717200779640],
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[0,0.2695431559523449723315320,0.5190961292068118159257257,0.7301520055740493240934163,0.8870625997680952990751578,0.9782286581460569928039380],
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[ 0.1252334085114689154724414,0.3678314989981801937526915,0.5873179542866174472967024,0.7699026741943046870368938,0.9041172563704748566784659,0.9815606342467192506905491],
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[0,0.2304583159551347940655281,0.4484927510364468528779129,0.6423493394403402206439846,0.8015780907333099127942065,0.9175983992229779652065478,0.9841830547185881494728294],
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[ 0.1080549487073436620662447,0.3191123689278897604356718,0.5152486363581540919652907,0.6872929048116854701480198,0.8272013150697649931897947,0.9284348836635735173363911,0.9862838086968123388415973],
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[0,0.2011940939974345223006283,0.3941513470775633698972074,0.5709721726085388475372267,0.7244177313601700474161861,0.8482065834104272162006483,0.9372733924007059043077589,0.9879925180204854284895657],
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[ 0.0950125098376374401853193,0.2816035507792589132304605,0.4580167776572273863424194,0.6178762444026437484466718,0.7554044083550030338951012,0.8656312023878317438804679,0.9445750230732325760779884,0.9894009349916499325961542]
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];
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var weights = [
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[1],
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[0.8888888888888888888888889,0.5555555555555555555555556],
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[0.6521451548625461426269361,0.3478548451374538573730639],
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[0.5688888888888888888888889,0.4786286704993664680412915,0.2369268850561890875142640],
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[0.4679139345726910473898703,0.3607615730481386075698335,0.1713244923791703450402961],
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[0.4179591836734693877551020,0.3818300505051189449503698,0.2797053914892766679014678,0.1294849661688696932706114],
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[0.3626837833783619829651504,0.3137066458778872873379622,0.2223810344533744705443560,0.1012285362903762591525314],
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[0.3302393550012597631645251,0.3123470770400028400686304,0.2606106964029354623187429,0.1806481606948574040584720,0.0812743883615744119718922],
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[0.2955242247147528701738930,0.2692667193099963550912269,0.2190863625159820439955349,0.1494513491505805931457763,0.0666713443086881375935688],
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[0.2729250867779006307144835,0.2628045445102466621806889,0.2331937645919904799185237,0.1862902109277342514260976,0.1255803694649046246346943,0.0556685671161736664827537],
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[0.2491470458134027850005624,0.2334925365383548087608499,0.2031674267230659217490645,0.1600783285433462263346525,0.1069393259953184309602547,0.0471753363865118271946160],
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[0.2325515532308739101945895,0.2262831802628972384120902,0.2078160475368885023125232,0.1781459807619457382800467,0.1388735102197872384636018,0.0921214998377284479144218,0.0404840047653158795200216],
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[0.2152638534631577901958764,0.2051984637212956039659241,0.1855383974779378137417166,0.1572031671581935345696019,0.1215185706879031846894148,0.0801580871597602098056333,0.0351194603317518630318329],
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[0.2025782419255612728806202,0.1984314853271115764561183,0.1861610000155622110268006,0.1662692058169939335532009,0.1395706779261543144478048,0.1071592204671719350118695,0.0703660474881081247092674,0.0307532419961172683546284],
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[0.1894506104550684962853967,0.1826034150449235888667637,0.1691565193950025381893121,0.1495959888165767320815017,0.1246289712555338720524763,0.0951585116824927848099251,0.0622535239386478928628438,0.0271524594117540948517806]
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];
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// Math short-cuts for often used methods and values
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var abs = Math.abs,
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sqrt = Math.sqrt,
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pow = Math.pow,
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cos = Math.cos,
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PI = Math.PI;
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return {
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TOLERANCE: 10e-6,
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// Precision when comparing against 0
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EPSILON: 10e-12,
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// Kappa, see: http://www.whizkidtech.redprince.net/bezier/circle/kappa/
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KAPPA: 4 * (sqrt(2) - 1) / 3,
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/**
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* Check if the value is 0, within a tolerance defined by
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* Numerical.EPSILON.
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*/
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isZero: function(val) {
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return abs(val) <= this.EPSILON;
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},
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/**
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* Gauss-Legendre Numerical Integration.
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*/
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integrate: function(f, a, b, n) {
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var x = abscissas[n - 2],
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w = weights[n - 2],
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A = 0.5 * (b - a),
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B = A + a,
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i = 0,
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m = (n + 1) >> 1,
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sum = n & 1 ? w[i++] * f(B) : 0; // Handle odd n
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while (i < m) {
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var Ax = A * x[i];
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sum += w[i++] * (f(B + Ax) + f(B - Ax));
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}
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return A * sum;
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},
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/**
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* Root finding using Newton-Raphson Method combined with Bisection.
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*/
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findRoot: function(f, df, x, a, b, n, tolerance) {
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for (var i = 0; i < n; i++) {
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var fx = f(x),
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dx = fx / df(x);
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// See if we can trust the Newton-Raphson result. If not we use
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// bisection to find another candiate for Newton's method.
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if (abs(dx) < tolerance)
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return x;
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// Generate a candidate for Newton's method.
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var nx = x - dx;
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// Update the root-bounding interval and test for containment of
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// the candidate. If candidate is outside the root-bounding
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// interval, use bisection instead.
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// There is no need to compare to lower / upper because the
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// tangent line has positive slope, guaranteeing that the x-axis
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// intercept is larger than lower / smaller than upper.
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if (fx > 0) {
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b = x;
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x = nx <= a ? 0.5 * (a + b) : nx;
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} else {
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a = x;
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x = nx >= b ? 0.5 * (a + b) : nx;
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}
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}
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},
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/**
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* Solves the quadratic polynomial with coefficients a, b, c for roots
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* (zero crossings) and and returns the solutions in an array.
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*
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* a*x^2 + b*x + c = 0
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*/
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solveQuadratic: function(a, b, c, roots) {
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// Code ported over and adapted from Uintah library (MIT license).
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var epsilon = this.EPSILON;
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// If a is 0, equation is actually linear, return 0 or 1 easy roots.
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if (abs(a) < epsilon) {
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if (abs(b) >= epsilon) {
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roots[0] = -c / b;
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return 1;
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}
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// If all the coefficients are 0, we have infinite solutions!
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return abs(c) < epsilon ? -1 : 0; // Infinite or 0 solutions
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}
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var q = b * b - 4 * a * c;
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if (q < 0)
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return 0; // 0 solutions
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q = sqrt(q);
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a *= 2; // Prepare division by (2 * a)
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var n = 0;
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roots[n++] = (-b - q) / a;
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if (q > 0)
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roots[n++] = (-b + q) / a;
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return n; // 1 or 2 solutions
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},
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/**
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* Solves the cubic polynomial with coefficients a, b, c, d for roots
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* (zero crossings) and and returns the solutions in an array.
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*
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* a*x^3 + b*x^2 + c*x + d = 0
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*/
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solveCubic: function(a, b, c, d, roots) {
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// Code ported over and adapted from Uintah library (MIT license).
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var epsilon = this.EPSILON;
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// If a is 0, equation is actually quadratic.
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if (abs(a) < epsilon)
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return Numerical.solveQuadratic(b, c, d, roots);
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// Normalize to form: x^3 + b x^2 + c x + d = 0:
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b /= a;
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c /= a;
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d /= a;
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// Compute discriminants
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var bb = b * b,
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p = (bb - 3 * c) / 9,
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q = (2 * bb * b - 9 * b * c + 27 * d) / 54,
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// Use Cardano's formula
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ppp = p * p * p,
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D = q * q - ppp;
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// Substitute x = y - b/3 to eliminate quadric term: x^3 +px + q = 0
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b /= 3;
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if (abs(D) < epsilon) {
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if (abs(q) < epsilon) { // One triple solution.
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roots[0] = - b;
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return 1;
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}
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// One single and one double solution.
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var sqp = sqrt(p),
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snq = q > 0 ? 1 : -1;
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roots[0] = -snq * 2 * sqp - b;
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roots[1] = snq * sqp - b;
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return 2;
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}
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if (D < 0) { // Casus irreducibilis: three real solutions
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var sqp = sqrt(p),
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phi = Math.acos(q / (sqp * sqp * sqp)) / 3,
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t = -2 * sqp,
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o = 2 * PI / 3;
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roots[0] = t * cos(phi) - b;
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roots[1] = t * cos(phi + o) - b;
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roots[2] = t * cos(phi - o) - b;
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return 3;
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}
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// One real solution
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var A = (q > 0 ? -1 : 1) * pow(abs(q) + sqrt(D), 1 / 3);
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roots[0] = A + p / A - b;
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return 1;
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}
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};
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};
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