paper.js/src/util/Numerical.js

/*
 * 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/
 *
 * Distributed under the MIT license. See LICENSE file for details.
 *
 * Copyright (c) 2011, Juerg Lehni & Jonathan Puckey
 * http://lehni.org/ & http://jonathanpuckey.com/
 *
 * All rights reserved.
 */

var Numerical = new function() {

	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]
	],

	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]
	];

	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;
		},

		/**
		 * Newton-Raphson Method Using Derivative. This is a special version
		 * that clips results to 0 .. 1, as required by Paper.js iterative
		 * approach for curve time parametrization: Ending up far outside
		 * the bezier curve boundaries resulted in inprecision of added up
		 * curve lengths.
		 */
		findRootNewton: function(f, fd, a, b, n, tol) {
			var x = 0.5 * (a + b);
			for (var i = 0; i < n; i++) {
				var dx = f(x) / fd(x);
				x -= dx;
				// Clip to 0 .. t .. 1. See comment above
				if (x < 0) x = 0;
				else if (x > 1) x = 1;
				if (Math.abs(dx) < tol)
					return x;
			}
			// If we did not succeed, fall back on False Position method for
			// accurate results.
			return Numerical.findRootFalsePosition(f, a, b, n, tol);
		},

		findRootFalsePosition: function(f, a, b, n, tol) {
			var fa = f(a),
				fb = f(b),
				dx = b - a,
				del, x;
			for (var i = 0; i < n; i++) {
				x = a + dx * fa / (fa - fb);
				var fx = f(x);
				if (fx < 0) {
					del = a - x;
					a = x;
					fa = fx;
				} else {
					del = b - x;
					b = x;
					fb = fx;
				}
				dx = b - a;
				if (Math.abs(del) < tol || fx == 0)
					return x;
			}
			return x;
		},
	};
};