1 /*

     2  * Copyright (c) 2007, 2015, 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

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     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

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    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

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    23  * questions.

    24  */

    25 

    26 package sun.java2d.marlin;

    27 

    28 import java.util.Iterator;

    29 

    30 final class Curve {

    31 

    32     float ax, ay, bx, by, cx, cy, dx, dy;

    33     float dax, day, dbx, dby;

    34     // shared iterator instance

    35     private final BreakPtrIterator iterator = new BreakPtrIterator();

    36 

    37     Curve() {

    38     }

    39 

    40     void set(float[] points, int type) {

    41         switch(type) {

    42         case 8:

    43             set(points[0], points[1],

    44                 points[2], points[3],

    45                 points[4], points[5],

    46                 points[6], points[7]);

    47             return;

    48         case 6:

    49             set(points[0], points[1],

    50                 points[2], points[3],

    51                 points[4], points[5]);

    52             return;

    53         default:

    54             throw new InternalError("Curves can only be cubic or quadratic");

    55         }

    56     }

    57 

    58     void set(float x1, float y1,

    59              float x2, float y2,

    60              float x3, float y3,

    61              float x4, float y4)

    62     {

    63         ax = 3f * (x2 - x3) + x4 - x1;

    64         ay = 3f * (y2 - y3) + y4 - y1;

    65         bx = 3f * (x1 - 2f * x2 + x3);

    66         by = 3f * (y1 - 2f * y2 + y3);

    67         cx = 3f * (x2 - x1);

    68         cy = 3f * (y2 - y1);

    69         dx = x1;

    70         dy = y1;

    71         dax = 3f * ax; day = 3f * ay;

    72         dbx = 2f * bx; dby = 2f * by;

    73     }

    74 

    75     void set(float x1, float y1,

    76              float x2, float y2,

    77              float x3, float y3)

    78     {

    79         ax = 0f; ay = 0f;

    80         bx = x1 - 2f * x2 + x3;

    81         by = y1 - 2f * y2 + y3;

    82         cx = 2f * (x2 - x1);

    83         cy = 2f * (y2 - y1);

    84         dx = x1;

    85         dy = y1;

    86         dax = 0f; day = 0f;

    87         dbx = 2f * bx; dby = 2f * by;

    88     }

    89 

    90     float xat(float t) {

    91         return t * (t * (t * ax + bx) + cx) + dx;

    92     }

    93     float yat(float t) {

    94         return t * (t * (t * ay + by) + cy) + dy;

    95     }

    96 

    97     float dxat(float t) {

    98         return t * (t * dax + dbx) + cx;

    99     }

   100 

   101     float dyat(float t) {

   102         return t * (t * day + dby) + cy;

   103     }

   104 

   105     int dxRoots(float[] roots, int off) {

   106         return Helpers.quadraticRoots(dax, dbx, cx, roots, off);

   107     }

   108 

   109     int dyRoots(float[] roots, int off) {

   110         return Helpers.quadraticRoots(day, dby, cy, roots, off);

   111     }

   112 

   113     int infPoints(float[] pts, int off) {

   114         // inflection point at t if -f'(t)x*f''(t)y + f'(t)y*f''(t)x == 0

   115         // Fortunately, this turns out to be quadratic, so there are at

   116         // most 2 inflection points.

   117         final float a = dax * dby - dbx * day;

   118         final float b = 2f * (cy * dax - day * cx);

   119         final float c = cy * dbx - cx * dby;

   120 

   121         return Helpers.quadraticRoots(a, b, c, pts, off);

   122     }

   123 

   124     // finds points where the first and second derivative are

   125     // perpendicular. This happens when g(t) = f'(t)*f''(t) == 0 (where

   126     // * is a dot product). Unfortunately, we have to solve a cubic.

   127     private int perpendiculardfddf(float[] pts, int off) {

   128         assert pts.length >= off + 4;

   129 

   130         // these are the coefficients of some multiple of g(t) (not g(t),

   131         // because the roots of a polynomial are not changed after multiplication

   132         // by a constant, and this way we save a few multiplications).

   133         final float a = 2f * (dax*dax + day*day);

   134         final float b = 3f * (dax*dbx + day*dby);

   135         final float c = 2f * (dax*cx + day*cy) + dbx*dbx + dby*dby;

   136         final float d = dbx*cx + dby*cy;

   137         return Helpers.cubicRootsInAB(a, b, c, d, pts, off, 0f, 1f);

   138     }

   139 

   140     // Tries to find the roots of the function ROC(t)-w in [0, 1). It uses

   141     // a variant of the false position algorithm to find the roots. False

   142     // position requires that 2 initial values x0,x1 be given, and that the

   143     // function must have opposite signs at those values. To find such

   144     // values, we need the local extrema of the ROC function, for which we

   145     // need the roots of its derivative; however, it's harder to find the

   146     // roots of the derivative in this case than it is to find the roots

   147     // of the original function. So, we find all points where this curve's

   148     // first and second derivative are perpendicular, and we pretend these

   149     // are our local extrema. There are at most 3 of these, so we will check

   150     // at most 4 sub-intervals of (0,1). ROC has asymptotes at inflection

   151     // points, so roc-w can have at least 6 roots. This shouldn't be a

   152     // problem for what we're trying to do (draw a nice looking curve).

   153     int rootsOfROCMinusW(float[] roots, int off, final float w, final float err) {

   154         // no OOB exception, because by now off<=6, and roots.length >= 10

   155         assert off <= 6 && roots.length >= 10;

   156         int ret = off;

   157         int numPerpdfddf = perpendiculardfddf(roots, off);

   158         float t0 = 0, ft0 = ROCsq(t0) - w*w;

   159         roots[off + numPerpdfddf] = 1f; // always check interval end points

   160         numPerpdfddf++;

   161         for (int i = off; i < off + numPerpdfddf; i++) {

   162             float t1 = roots[i], ft1 = ROCsq(t1) - w*w;

   163             if (ft0 == 0f) {

   164                 roots[ret++] = t0;

   165             } else if (ft1 * ft0 < 0f) { // have opposite signs

   166                 // (ROC(t)^2 == w^2) == (ROC(t) == w) is true because

   167                 // ROC(t) >= 0 for all t.

   168                 roots[ret++] = falsePositionROCsqMinusX(t0, t1, w*w, err);

   169             }

   170             t0 = t1;

   171             ft0 = ft1;

   172         }

   173 

   174         return ret - off;

   175     }

   176 

   177     private static float eliminateInf(float x) {

   178         return (x == Float.POSITIVE_INFINITY ? Float.MAX_VALUE :

   179             (x == Float.NEGATIVE_INFINITY ? Float.MIN_VALUE : x));

   180     }

   181 

   182     // A slight modification of the false position algorithm on wikipedia.

   183     // This only works for the ROCsq-x functions. It might be nice to have

   184     // the function as an argument, but that would be awkward in java6.

   185     // TODO: It is something to consider for java8 (or whenever lambda

   186     // expressions make it into the language), depending on how closures

   187     // and turn out. Same goes for the newton's method

   188     // algorithm in Helpers.java

   189     private float falsePositionROCsqMinusX(float x0, float x1,

   190                                            final float x, final float err)

   191     {

   192         final int iterLimit = 100;

   193         int side = 0;

   194         float t = x1, ft = eliminateInf(ROCsq(t) - x);

   195         float s = x0, fs = eliminateInf(ROCsq(s) - x);

   196         float r = s, fr;

   197         for (int i = 0; i < iterLimit && Math.abs(t - s) > err * Math.abs(t + s); i++) {

   198             r = (fs * t - ft * s) / (fs - ft);

   199             fr = ROCsq(r) - x;

   200             if (sameSign(fr, ft)) {

   201                 ft = fr; t = r;

   202                 if (side < 0) {

   203                     fs /= (1 << (-side));

   204                     side--;

   205                 } else {

   206                     side = -1;

   207                 }

   208             } else if (fr * fs > 0) {

   209                 fs = fr; s = r;

   210                 if (side > 0) {

   211                     ft /= (1 << side);

   212                     side++;

   213                 } else {

   214                     side = 1;

   215                 }

   216             } else {

   217                 break;

   218             }

   219         }

   220         return r;

   221     }

   222 

   223     private static boolean sameSign(float x, float y) {

   224         // another way is to test if x*y > 0. This is bad for small x, y.

   225         return (x < 0f && y < 0f) || (x > 0f && y > 0f);

   226     }

   227 

   228     // returns the radius of curvature squared at t of this curve

   229     // see http://en.wikipedia.org/wiki/Radius_of_curvature_(applications)

   230     private float ROCsq(final float t) {

   231         // dx=xat(t) and dy=yat(t). These calls have been inlined for efficiency

   232         final float dx = t * (t * dax + dbx) + cx;

   233         final float dy = t * (t * day + dby) + cy;

   234         final float ddx = 2f * dax * t + dbx;

   235         final float ddy = 2f * day * t + dby;

   236         final float dx2dy2 = dx*dx + dy*dy;

   237         final float ddx2ddy2 = ddx*ddx + ddy*ddy;

   238         final float ddxdxddydy = ddx*dx + ddy*dy;

   239         return dx2dy2*((dx2dy2*dx2dy2) / (dx2dy2 * ddx2ddy2 - ddxdxddydy*ddxdxddydy));

   240     }

   241 

   242     // curve to be broken should be in pts

   243     // this will change the contents of pts but not Ts

   244     // TODO: There's no reason for Ts to be an array. All we need is a sequence

   245     // of t values at which to subdivide. An array statisfies this condition,

   246     // but is unnecessarily restrictive. Ts should be an Iterator<Float> instead.

   247     // Doing this will also make dashing easier, since we could easily make

   248     // LengthIterator an Iterator<Float> and feed it to this function to simplify

   249     // the loop in Dasher.somethingTo.

   250     BreakPtrIterator breakPtsAtTs(final float[] pts, final int type,

   251                                   final float[] Ts, final int numTs)

   252     {

   253         assert pts.length >= 2*type && numTs <= Ts.length;

   254 

   255         // initialize shared iterator:

   256         iterator.init(pts, type, Ts, numTs);

   257 

   258         return iterator;

   259     }

   260 

   261     static final class BreakPtrIterator {

   262         private int nextCurveIdx;

   263         private int curCurveOff;

   264         private float prevT;

   265         private float[] pts;

   266         private int type;

   267         private float[] ts;

   268         private int numTs;

   269 

   270         void init(final float[] pts, final int type,

   271                   final float[] ts, final int numTs) {

   272             this.pts = pts;

   273             this.type = type;

   274             this.ts = ts;

   275             this.numTs = numTs;

   276 

   277             nextCurveIdx = 0;

   278             curCurveOff = 0;

   279             prevT = 0f;

   280         }

   281 

   282         public boolean hasNext() {

   283             return nextCurveIdx <= numTs;

   284         }

   285 

   286         public int next() {

   287             int ret;

   288             if (nextCurveIdx < numTs) {

   289                 float curT = ts[nextCurveIdx];

   290                 float splitT = (curT - prevT) / (1f - prevT);

   291                 Helpers.subdivideAt(splitT,

   292                                     pts, curCurveOff,

   293                                     pts, 0,

   294                                     pts, type, type);

   295                 prevT = curT;

   296                 ret = 0;

   297                 curCurveOff = type;

   298             } else {

   299                 ret = curCurveOff;

   300             }

   301             nextCurveIdx++;

   302             return ret;

   303         }

   304     }

   305 }

   306