/* * Copyright (c) 2007, 2016, Oracle and/or its affiliates. All rights reserved. * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. * * This code is free software; you can redistribute it and/or modify it * under the terms of the GNU General Public License version 2 only, as * published by the Free Software Foundation. Oracle designates this * particular file as subject to the "Classpath" exception as provided * by Oracle in the LICENSE file that accompanied this code. * * This code is distributed in the hope that it will be useful, but WITHOUT * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License * version 2 for more details (a copy is included in the LICENSE file that * accompanied this code). * * You should have received a copy of the GNU General Public License version * 2 along with this work; if not, write to the Free Software Foundation, * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. * * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA * or visit www.oracle.com if you need additional information or have any * questions. */ package com.sun.marlin; final class DCurve { double ax, ay, bx, by, cx, cy, dx, dy; double dax, day, dbx, dby; DCurve() { } void set(double[] points, int type) { switch(type) { case 8: set(points[0], points[1], points[2], points[3], points[4], points[5], points[6], points[7]); return; case 6: set(points[0], points[1], points[2], points[3], points[4], points[5]); return; default: throw new InternalError("DCurves can only be cubic or quadratic"); } } void set(double x1, double y1, double x2, double y2, double x3, double y3, double x4, double y4) { ax = 3D * (x2 - x3) + x4 - x1; ay = 3D * (y2 - y3) + y4 - y1; bx = 3D * (x1 - 2D * x2 + x3); by = 3D * (y1 - 2D * y2 + y3); cx = 3D * (x2 - x1); cy = 3D * (y2 - y1); dx = x1; dy = y1; dax = 3D * ax; day = 3D * ay; dbx = 2D * bx; dby = 2D * by; } void set(double x1, double y1, double x2, double y2, double x3, double y3) { ax = 0D; ay = 0D; bx = x1 - 2D * x2 + x3; by = y1 - 2D * y2 + y3; cx = 2D * (x2 - x1); cy = 2D * (y2 - y1); dx = x1; dy = y1; dax = 0D; day = 0D; dbx = 2D * bx; dby = 2D * by; } double xat(double t) { return t * (t * (t * ax + bx) + cx) + dx; } double yat(double t) { return t * (t * (t * ay + by) + cy) + dy; } double dxat(double t) { return t * (t * dax + dbx) + cx; } double dyat(double t) { return t * (t * day + dby) + cy; } int dxRoots(double[] roots, int off) { return DHelpers.quadraticRoots(dax, dbx, cx, roots, off); } int dyRoots(double[] roots, int off) { return DHelpers.quadraticRoots(day, dby, cy, roots, off); } int infPoints(double[] pts, int off) { // inflection point at t if -f'(t)x*f''(t)y + f'(t)y*f''(t)x == 0 // Fortunately, this turns out to be quadratic, so there are at // most 2 inflection points. final double a = dax * dby - dbx * day; final double b = 2D * (cy * dax - day * cx); final double c = cy * dbx - cx * dby; return DHelpers.quadraticRoots(a, b, c, pts, off); } // finds points where the first and second derivative are // perpendicular. This happens when g(t) = f'(t)*f''(t) == 0 (where // * is a dot product). Unfortunately, we have to solve a cubic. private int perpendiculardfddf(double[] pts, int off) { assert pts.length >= off + 4; // these are the coefficients of some multiple of g(t) (not g(t), // because the roots of a polynomial are not changed after multiplication // by a constant, and this way we save a few multiplications). final double a = 2D * (dax*dax + day*day); final double b = 3D * (dax*dbx + day*dby); final double c = 2D * (dax*cx + day*cy) + dbx*dbx + dby*dby; final double d = dbx*cx + dby*cy; return DHelpers.cubicRootsInAB(a, b, c, d, pts, off, 0D, 1D); } // Tries to find the roots of the function ROC(t)-w in [0, 1). It uses // a variant of the false position algorithm to find the roots. False // position requires that 2 initial values x0,x1 be given, and that the // function must have opposite signs at those values. To find such // values, we need the local extrema of the ROC function, for which we // need the roots of its derivative; however, it's harder to find the // roots of the derivative in this case than it is to find the roots // of the original function. So, we find all points where this curve's // first and second derivative are perpendicular, and we pretend these // are our local extrema. There are at most 3 of these, so we will check // at most 4 sub-intervals of (0,1). ROC has asymptotes at inflection // points, so roc-w can have at least 6 roots. This shouldn't be a // problem for what we're trying to do (draw a nice looking curve). int rootsOfROCMinusW(double[] roots, int off, final double w, final double err) { // no OOB exception, because by now off<=6, and roots.length >= 10 assert off <= 6 && roots.length >= 10; int ret = off; int numPerpdfddf = perpendiculardfddf(roots, off); double t0 = 0, ft0 = ROCsq(t0) - w*w; roots[off + numPerpdfddf] = 1D; // always check interval end points numPerpdfddf++; for (int i = off; i < off + numPerpdfddf; i++) { double t1 = roots[i], ft1 = ROCsq(t1) - w*w; if (ft0 == 0D) { roots[ret++] = t0; } else if (ft1 * ft0 < 0D) { // have opposite signs // (ROC(t)^2 == w^2) == (ROC(t) == w) is true because // ROC(t) >= 0 for all t. roots[ret++] = falsePositionROCsqMinusX(t0, t1, w*w, err); } t0 = t1; ft0 = ft1; } return ret - off; } private static double eliminateInf(double x) { return (x == Double.POSITIVE_INFINITY ? Double.MAX_VALUE : (x == Double.NEGATIVE_INFINITY ? Double.MIN_VALUE : x)); } // A slight modification of the false position algorithm on wikipedia. // This only works for the ROCsq-x functions. It might be nice to have // the function as an argument, but that would be awkward in java6. // TODO: It is something to consider for java8 (or whenever lambda // expressions make it into the language), depending on how closures // and turn out. Same goes for the newton's method // algorithm in DHelpers.java private double falsePositionROCsqMinusX(double x0, double x1, final double x, final double err) { final int iterLimit = 100; int side = 0; double t = x1, ft = eliminateInf(ROCsq(t) - x); double s = x0, fs = eliminateInf(ROCsq(s) - x); double r = s, fr; for (int i = 0; i < iterLimit && Math.abs(t - s) > err * Math.abs(t + s); i++) { r = (fs * t - ft * s) / (fs - ft); fr = ROCsq(r) - x; if (sameSign(fr, ft)) { ft = fr; t = r; if (side < 0) { fs /= (1 << (-side)); side--; } else { side = -1; } } else if (fr * fs > 0) { fs = fr; s = r; if (side > 0) { ft /= (1 << side); side++; } else { side = 1; } } else { break; } } return r; } private static boolean sameSign(double x, double y) { // another way is to test if x*y > 0. This is bad for small x, y. return (x < 0D && y < 0D) || (x > 0D && y > 0D); } // returns the radius of curvature squared at t of this curve // see http://en.wikipedia.org/wiki/Radius_of_curvature_(applications) private double ROCsq(final double t) { // dx=xat(t) and dy=yat(t). These calls have been inlined for efficiency final double dx = t * (t * dax + dbx) + cx; final double dy = t * (t * day + dby) + cy; final double ddx = 2D * dax * t + dbx; final double ddy = 2D * day * t + dby; final double dx2dy2 = dx*dx + dy*dy; final double ddx2ddy2 = ddx*ddx + ddy*ddy; final double ddxdxddydy = ddx*dx + ddy*dy; return dx2dy2*((dx2dy2*dx2dy2) / (dx2dy2 * ddx2ddy2 - ddxdxddydy*ddxdxddydy)); } }