1 /*
   2  * Copyright (c) 2007, 2016, 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.
  24  */
  25 
  26 package sun.java2d.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(float[] points, int type) {
  37         switch(type) {
  38         case 8:
  39             set(points[0], points[1],
  40                 points[2], points[3],
  41                 points[4], points[5],
  42                 points[6], points[7]);
  43             return;
  44         case 6:
  45             set(points[0], points[1],
  46                 points[2], points[3],
  47                 points[4], points[5]);
  48             return;
  49         default:
  50             throw new InternalError("Curves can only be cubic or quadratic");
  51         }
  52     }
  53 
  54     void set(float x1, float y1,
  55              float x2, float y2,
  56              float x3, float y3,
  57              float x4, float y4)
  58     {
  59         ax = 3.0f * (x2 - x3) + x4 - x1;
  60         ay = 3.0f * (y2 - y3) + y4 - y1;
  61         bx = 3.0f * (x1 - 2.0f * x2 + x3);
  62         by = 3.0f * (y1 - 2.0f * y2 + y3);
  63         cx = 3.0f * (x2 - x1);
  64         cy = 3.0f * (y2 - y1);
  65         dx = x1;
  66         dy = y1;
  67         dax = 3.0f * ax; day = 3.0f * ay;
  68         dbx = 2.0f * bx; dby = 2.0f * by;
  69     }
  70 
  71     void set(float x1, float y1,
  72              float x2, float y2,
  73              float x3, float y3)
  74     {
  75         ax = 0.0f; ay = 0.0f;
  76         bx = x1 - 2.0f * x2 + x3;
  77         by = y1 - 2.0f * y2 + y3;
  78         cx = 2.0f * (x2 - x1);
  79         cy = 2.0f * (y2 - y1);
  80         dx = x1;
  81         dy = y1;
  82         dax = 0.0f; day = 0.0f;
  83         dbx = 2.0f * bx; dby = 2.0f * by;
  84     }
  85 
  86     float xat(float t) {
  87         return t * (t * (t * ax + bx) + cx) + dx;
  88     }
  89     float yat(float t) {
  90         return t * (t * (t * ay + by) + cy) + dy;
  91     }
  92 
  93     float dxat(float t) {
  94         return t * (t * dax + dbx) + cx;
  95     }
  96 
  97     float dyat(float t) {
  98         return t * (t * day + dby) + cy;
  99     }
 100 
 101     int dxRoots(float[] roots, int off) {
 102         return Helpers.quadraticRoots(dax, dbx, cx, roots, off);
 103     }
 104 
 105     int dyRoots(float[] roots, int off) {
 106         return Helpers.quadraticRoots(day, dby, cy, roots, off);
 107     }
 108 
 109     int infPoints(float[] pts, int off) {
 110         // inflection point at t if -f'(t)x*f''(t)y + f'(t)y*f''(t)x == 0
 111         // Fortunately, this turns out to be quadratic, so there are at
 112         // most 2 inflection points.
 113         final float a = dax * dby - dbx * day;
 114         final float b = 2.0f * (cy * dax - day * cx);
 115         final float c = cy * dbx - cx * dby;
 116 
 117         return Helpers.quadraticRoots(a, b, c, pts, off);
 118     }
 119 
 120     // finds points where the first and second derivative are
 121     // perpendicular. This happens when g(t) = f'(t)*f''(t) == 0 (where
 122     // * is a dot product). Unfortunately, we have to solve a cubic.
 123     private int perpendiculardfddf(float[] pts, int off) {
 124         assert pts.length >= off + 4;
 125 
 126         // these are the coefficients of some multiple of g(t) (not g(t),
 127         // because the roots of a polynomial are not changed after multiplication
 128         // by a constant, and this way we save a few multiplications).
 129         final float a = 2.0f * (dax*dax + day*day);
 130         final float b = 3.0f * (dax*dbx + day*dby);
 131         final float c = 2.0f * (dax*cx + day*cy) + dbx*dbx + dby*dby;
 132         final float d = dbx*cx + dby*cy;
 133         return Helpers.cubicRootsInAB(a, b, c, d, pts, off, 0.0f, 1.0f);
 134     }
 135 
 136     // Tries to find the roots of the function ROC(t)-w in [0, 1). It uses
 137     // a variant of the false position algorithm to find the roots. False
 138     // position requires that 2 initial values x0,x1 be given, and that the
 139     // function must have opposite signs at those values. To find such
 140     // values, we need the local extrema of the ROC function, for which we
 141     // need the roots of its derivative; however, it's harder to find the
 142     // roots of the derivative in this case than it is to find the roots
 143     // of the original function. So, we find all points where this curve's
 144     // first and second derivative are perpendicular, and we pretend these
 145     // are our local extrema. There are at most 3 of these, so we will check
 146     // at most 4 sub-intervals of (0,1). ROC has asymptotes at inflection
 147     // points, so roc-w can have at least 6 roots. This shouldn't be a
 148     // problem for what we're trying to do (draw a nice looking curve).
 149     int rootsOfROCMinusW(float[] roots, int off, final float w, final float err) {
 150         // no OOB exception, because by now off<=6, and roots.length >= 10
 151         assert off <= 6 && roots.length >= 10;
 152         int ret = off;
 153         int numPerpdfddf = perpendiculardfddf(roots, off);
 154         float t0 = 0.0f, ft0 = ROCsq(t0) - w*w;
 155         roots[off + numPerpdfddf] = 1.0f; // always check interval end points
 156         numPerpdfddf++;
 157         for (int i = off; i < off + numPerpdfddf; i++) {
 158             float t1 = roots[i], ft1 = ROCsq(t1) - w*w;
 159             if (ft0 == 0.0f) {
 160                 roots[ret++] = t0;
 161             } else if (ft1 * ft0 < 0.0f) { // have opposite signs
 162                 // (ROC(t)^2 == w^2) == (ROC(t) == w) is true because
 163                 // ROC(t) >= 0 for all t.
 164                 roots[ret++] = falsePositionROCsqMinusX(t0, t1, w*w, err);
 165             }
 166             t0 = t1;
 167             ft0 = ft1;
 168         }
 169 
 170         return ret - off;
 171     }
 172 
 173     private static float eliminateInf(float x) {
 174         return (x == Float.POSITIVE_INFINITY ? Float.MAX_VALUE :
 175             (x == Float.NEGATIVE_INFINITY ? Float.MIN_VALUE : x));
 176     }
 177 
 178     // A slight modification of the false position algorithm on wikipedia.
 179     // This only works for the ROCsq-x functions. It might be nice to have
 180     // the function as an argument, but that would be awkward in java6.
 181     // TODO: It is something to consider for java8 (or whenever lambda
 182     // expressions make it into the language), depending on how closures
 183     // and turn out. Same goes for the newton's method
 184     // algorithm in Helpers.java
 185     private float falsePositionROCsqMinusX(float x0, float x1,
 186                                            final float x, final float err)
 187     {
 188         final int iterLimit = 100;
 189         int side = 0;
 190         float t = x1, ft = eliminateInf(ROCsq(t) - x);
 191         float s = x0, fs = eliminateInf(ROCsq(s) - x);
 192         float r = s, fr;
 193         for (int i = 0; i < iterLimit && Math.abs(t - s) > err * Math.abs(t + s); i++) {
 194             r = (fs * t - ft * s) / (fs - ft);
 195             fr = ROCsq(r) - x;
 196             if (sameSign(fr, ft)) {
 197                 ft = fr; t = r;
 198                 if (side < 0) {
 199                     fs /= (1 << (-side));
 200                     side--;
 201                 } else {
 202                     side = -1;
 203                 }
 204             } else if (fr * fs > 0) {
 205                 fs = fr; s = r;
 206                 if (side > 0) {
 207                     ft /= (1 << side);
 208                     side++;
 209                 } else {
 210                     side = 1;
 211                 }
 212             } else {
 213                 break;
 214             }
 215         }
 216         return r;
 217     }
 218 
 219     private static boolean sameSign(float x, float y) {
 220         // another way is to test if x*y > 0. This is bad for small x, y.
 221         return (x < 0.0f && y < 0.0f) || (x > 0.0f && y > 0.0f);
 222     }
 223 
 224     // returns the radius of curvature squared at t of this curve
 225     // see http://en.wikipedia.org/wiki/Radius_of_curvature_(applications)
 226     private float ROCsq(final float t) {
 227         // dx=xat(t) and dy=yat(t). These calls have been inlined for efficiency
 228         final float dx = t * (t * dax + dbx) + cx;
 229         final float dy = t * (t * day + dby) + cy;
 230         final float ddx = 2.0f * dax * t + dbx;
 231         final float ddy = 2.0f * day * t + dby;
 232         final float dx2dy2 = dx*dx + dy*dy;
 233         final float ddx2ddy2 = ddx*ddx + ddy*ddy;
 234         final float ddxdxddydy = ddx*dx + ddy*dy;
 235         return dx2dy2*((dx2dy2*dx2dy2) / (dx2dy2 * ddx2ddy2 - ddxdxddydy*ddxdxddydy));
































































 236     }
 237 }
 238 
--- EOF ---