1 /*
   2  * Copyright (c) 2007, 2018, Oracle and/or its affiliates. All rights reserved.
   3  * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
   4  *
   5  * This code is free software; you can redistribute it and/or modify it
   6  * under the terms of the GNU General Public License version 2 only, as
   7  * published by the Free Software Foundation.  Oracle designates this
   8  * particular file as subject to the "Classpath" exception as provided
   9  * by Oracle in the LICENSE file that accompanied this code.
  10  *
  11  * This code is distributed in the hope that it will be useful, but WITHOUT
  12  * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
  13  * FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
  14  * version 2 for more details (a copy is included in the LICENSE file that
  15  * accompanied this code).
  16  *
  17  * You should have received a copy of the GNU General Public License version
  18  * 2 along with this work; if not, write to the Free Software Foundation,
  19  * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
  20  *
  21  * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
  22  * or visit www.oracle.com if you need additional information or have any
  23  * questions.
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  25 
  26 package com.sun.marlin;
  27 
  28 final class Curve {
  29 
  30     float ax, ay, bx, by, cx, cy, dx, dy;
  31     float dax, day, dbx, dby;
  32 
  33     Curve() {
  34     }
  35 
  36     void set(final float[] points, final int type) {
  37         // if instead of switch (perf + most probable cases first)
  38         if (type == 8) {
  39             set(points[0], points[1],
  40                 points[2], points[3],
  41                 points[4], points[5],
  42                 points[6], points[7]);
  43         } else if (type == 4) {
  44             set(points[0], points[1],
  45                 points[2], points[3]);
  46         } else {
  47             set(points[0], points[1],
  48                 points[2], points[3],
  49                 points[4], points[5]);
  50         }
  51     }
  52 
  53     void set(final float x1, final float y1,
  54              final float x2, final float y2,
  55              final float x3, final float y3,
  56              final float x4, final float y4)
  57     {
  58         final float dx32 = 3.0f * (x3 - x2);
  59         final float dy32 = 3.0f * (y3 - y2);
  60         final float dx21 = 3.0f * (x2 - x1);
  61         final float dy21 = 3.0f * (y2 - y1);
  62         ax = (x4 - x1) - dx32;  // A = P3 - P0 - 3 (P2 - P1) = (P3 - P0) + 3 (P1 - P2)
  63         ay = (y4 - y1) - dy32;
  64         bx = (dx32 - dx21);     // B = 3 (P2 - P1) - 3(P1 - P0) = 3 (P2 + P0) - 6 P1
  65         by = (dy32 - dy21);
  66         cx = dx21;              // C = 3 (P1 - P0)
  67         cy = dy21;
  68         dx = x1;                // D = P0
  69         dy = y1;
  70         dax = 3.0f * ax;
  71         day = 3.0f * ay;
  72         dbx = 2.0f * bx;
  73         dby = 2.0f * by;
  74     }
  75 
  76     void set(final float x1, final float y1,
  77              final float x2, final float y2,
  78              final float x3, final float y3)
  79     {
  80         final float dx21 = (x2 - x1);
  81         final float dy21 = (y2 - y1);
  82         ax = 0.0f;              // A = 0
  83         ay = 0.0f;
  84         bx = (x3 - x2) - dx21;  // B = P3 - P0 - 2 P2
  85         by = (y3 - y2) - dy21;
  86         cx = 2.0f * dx21;       // C = 2 (P2 - P1)
  87         cy = 2.0f * dy21;
  88         dx = x1;                // D = P1
  89         dy = y1;
  90         dax = 0.0f;
  91         day = 0.0f;
  92         dbx = 2.0f * bx;
  93         dby = 2.0f * by;
  94     }
  95 
  96     void set(final float x1, final float y1,
  97              final float x2, final float y2)
  98     {
  99         final float dx21 = (x2 - x1);
 100         final float dy21 = (y2 - y1);
 101         ax = 0.0f;              // A = 0
 102         ay = 0.0f;
 103         bx = 0.0f;              // B = 0
 104         by = 0.0f;
 105         cx = dx21;              // C = (P2 - P1)
 106         cy = dy21;
 107         dx = x1;                // D = P1
 108         dy = y1;
 109         dax = 0.0f;
 110         day = 0.0f;
 111         dbx = 0.0f;
 112         dby = 0.0f;
 113     }
 114 
 115     int dxRoots(final float[] roots, final int off) {
 116         return Helpers.quadraticRoots(dax, dbx, cx, roots, off);
 117     }
 118 
 119     int dyRoots(final float[] roots, final int off) {
 120         return Helpers.quadraticRoots(day, dby, cy, roots, off);
 121     }
 122 
 123     int infPoints(final float[] pts, final int off) {
 124         // inflection point at t if -f'(t)x*f''(t)y + f'(t)y*f''(t)x == 0
 125         // Fortunately, this turns out to be quadratic, so there are at
 126         // most 2 inflection points.
 127         final float a = dax * dby - dbx * day;
 128         final float b = 2.0f * (cy * dax - day * cx);
 129         final float c = cy * dbx - cx * dby;
 130 
 131         return Helpers.quadraticRoots(a, b, c, pts, off);
 132     }
 133 
 134     int xPoints(final float[] ts, final int off, final float x)
 135     {
 136         return Helpers.cubicRootsInAB(ax, bx, cx, dx - x, ts, off, 0.0f, 1.0f);
 137     }
 138 
 139     int yPoints(final float[] ts, final int off, final float y)
 140     {
 141         return Helpers.cubicRootsInAB(ay, by, cy, dy - y, ts, off, 0.0f, 1.0f);
 142     }
 143 
 144     // finds points where the first and second derivative are
 145     // perpendicular. This happens when g(t) = f'(t)*f''(t) == 0 (where
 146     // * is a dot product). Unfortunately, we have to solve a cubic.
 147     private int perpendiculardfddf(final float[] pts, final int off) {
 148         assert pts.length >= off + 4;
 149 
 150         // these are the coefficients of some multiple of g(t) (not g(t),
 151         // because the roots of a polynomial are not changed after multiplication
 152         // by a constant, and this way we save a few multiplications).
 153         final float a = 2.0f * (dax * dax + day * day);
 154         final float b = 3.0f * (dax * dbx + day * dby);
 155         final float c = 2.0f * (dax * cx  + day * cy) + dbx * dbx + dby * dby;
 156         final float d = dbx * cx + dby * cy;
 157 
 158         return Helpers.cubicRootsInAB(a, b, c, d, pts, off, 0.0f, 1.0f);
 159     }
 160 
 161     // Tries to find the roots of the function ROC(t)-w in [0, 1). It uses
 162     // a variant of the false position algorithm to find the roots. False
 163     // position requires that 2 initial values x0,x1 be given, and that the
 164     // function must have opposite signs at those values. To find such
 165     // values, we need the local extrema of the ROC function, for which we
 166     // need the roots of its derivative; however, it's harder to find the
 167     // roots of the derivative in this case than it is to find the roots
 168     // of the original function. So, we find all points where this curve's
 169     // first and second derivative are perpendicular, and we pretend these
 170     // are our local extrema. There are at most 3 of these, so we will check
 171     // at most 4 sub-intervals of (0,1). ROC has asymptotes at inflection
 172     // points, so roc-w can have at least 6 roots. This shouldn't be a
 173     // problem for what we're trying to do (draw a nice looking curve).
 174     int rootsOfROCMinusW(final float[] roots, final int off, final float w2, final float err) {
 175         // no OOB exception, because by now off<=6, and roots.length >= 10
 176         assert off <= 6 && roots.length >= 10;
 177 
 178         int ret = off;
 179         final int end = off + perpendiculardfddf(roots, off);
 180         roots[end] = 1.0f; // always check interval end points
 181 
 182         float t0 = 0.0f, ft0 = ROCsq(t0) - w2;
 183 
 184         for (int i = off; i <= end; i++) {
 185             float t1 = roots[i], ft1 = ROCsq(t1) - w2;
 186             if (ft0 == 0.0f) {
 187                 roots[ret++] = t0;
 188             } else if (ft1 * ft0 < 0.0f) { // have opposite signs
 189                 // (ROC(t)^2 == w^2) == (ROC(t) == w) is true because
 190                 // ROC(t) >= 0 for all t.
 191                 roots[ret++] = falsePositionROCsqMinusX(t0, t1, w2, err);
 192             }
 193             t0 = t1;
 194             ft0 = ft1;
 195         }
 196 
 197         return ret - off;
 198     }
 199 
 200     private static float eliminateInf(final float x) {
 201         return (x == Float.POSITIVE_INFINITY ? Float.MAX_VALUE :
 202                (x == Float.NEGATIVE_INFINITY ? Float.MIN_VALUE : x));
 203     }
 204 
 205     // A slight modification of the false position algorithm on wikipedia.
 206     // This only works for the ROCsq-x functions. It might be nice to have
 207     // the function as an argument, but that would be awkward in java6.
 208     // TODO: It is something to consider for java8 (or whenever lambda
 209     // expressions make it into the language), depending on how closures
 210     // and turn out. Same goes for the newton's method
 211     // algorithm in Helpers.java
 212     private float falsePositionROCsqMinusX(final float t0, final float t1,
 213                                            final float w2, final float err)
 214     {
 215         final int iterLimit = 100;
 216         int side = 0;
 217         float t = t1, ft = eliminateInf(ROCsq(t) - w2);
 218         float s = t0, fs = eliminateInf(ROCsq(s) - w2);
 219         float r = s, fr;
 220 
 221         for (int i = 0; i < iterLimit && Math.abs(t - s) > err * Math.abs(t + s); i++) {
 222             r = (fs * t - ft * s) / (fs - ft);
 223             fr = ROCsq(r) - w2;
 224             if (sameSign(fr, ft)) {
 225                 ft = fr; t = r;
 226                 if (side < 0) {
 227                     fs /= (1 << (-side));
 228                     side--;
 229                 } else {
 230                     side = -1;
 231                 }
 232             } else if (fr * fs > 0.0f) {
 233                 fs = fr; s = r;
 234                 if (side > 0) {
 235                     ft /= (1 << side);
 236                     side++;
 237                 } else {
 238                     side = 1;
 239                 }
 240             } else {
 241                 break;
 242             }
 243         }
 244         return r;
 245     }
 246 
 247     private static boolean sameSign(final float x, final float y) {
 248         // another way is to test if x*y > 0. This is bad for small x, y.
 249         return (x < 0.0f && y < 0.0f) || (x > 0.0f && y > 0.0f);
 250     }
 251 
 252     // returns the radius of curvature squared at t of this curve
 253     // see http://en.wikipedia.org/wiki/Radius_of_curvature_(applications)
 254     private float ROCsq(final float t) {
 255         final float dx = t * (t * dax + dbx) + cx;
 256         final float dy = t * (t * day + dby) + cy;
 257         final float ddx = 2.0f * dax * t + dbx;
 258         final float ddy = 2.0f * day * t + dby;
 259         final float dx2dy2 = dx * dx + dy * dy;
 260         final float ddx2ddy2 = ddx * ddx + ddy * ddy;
 261         final float ddxdxddydy = ddx * dx + ddy * dy;
 262         return dx2dy2 * ((dx2dy2 * dx2dy2) / (dx2dy2 * ddx2ddy2 - ddxdxddydy * ddxdxddydy));
 263     }
 264 }