-/* FlatteningPathIterator.java -- performs interpolation of curved paths
- Copyright (C) 2002 Free Software Foundation
+/* FlatteningPathIterator.java -- Approximates curves by straight lines
+ Copyright (C) 2003 Free Software Foundation
This file is part of GNU Classpath.
package java.awt.geom;
+import java.util.NoSuchElementException;
+
+
/**
- * This class can be used to perform the flattening required by the Shape
- * interface. It interpolates a curved path segment into a sequence of flat
- * ones within a certain flatness, up to a recursion limit.
+ * A PathIterator for approximating curved path segments by sequences
+ * of straight lines. Instances of this class will only return
+ * segments of type {@link PathIterator#SEG_MOVETO}, {@link
+ * PathIterator#SEG_LINETO}, and {@link PathIterator#SEG_CLOSE}.
+ *
+ * <p>The accuracy of the approximation is determined by two
+ * parameters:
+ *
+ * <ul><li>The <i>flatness</i> is a threshold value for deciding when
+ * a curved segment is consided flat enough for being approximated by
+ * a single straight line. Flatness is defined as the maximal distance
+ * of a curve control point to the straight line that connects the
+ * curve start and end. A lower flatness threshold means a closer
+ * approximation. See {@link QuadCurve2D#getFlatness()} and {@link
+ * CubicCurve2D#getFlatness()} for drawings which illustrate the
+ * meaning of flatness.</li>
+ *
+ * <li>The <i>recursion limit</i> imposes an upper bound for how often
+ * a curved segment gets subdivided. A limit of <i>n</i> means that
+ * for each individual quadratic and cubic Bézier spline
+ * segment, at most 2<sup><small><i>n</i></small></sup> {@link
+ * PathIterator#SEG_LINETO} segments will be created.</li></ul>
+ *
+ * <p><b>Memory Efficiency:</b> The memory consumption grows linearly
+ * with the recursion limit. Neither the <i>flatness</i> parameter nor
+ * the number of segments in the flattened path will affect the memory
+ * consumption.
+ *
+ * <p><b>Thread Safety:</b> Multiple threads can safely work on
+ * separate instances of this class. However, multiple threads should
+ * not concurrently access the same instance, as no synchronization is
+ * performed.
+ *
+ * @see <a href="doc-files/FlatteningPathIterator-1.html"
+ * >Implementation Note</a>
+ *
+ * @author Sascha Brawer (brawer@dandelis.ch)
*
- * @author Eric Blake <ebb9@email.byu.edu>
- * @see Shape
- * @see RectangularShape#getPathIterator(AffineTransform, double)
* @since 1.2
- * @status STUBS ONLY
*/
-public class FlatteningPathIterator implements PathIterator
+public class FlatteningPathIterator
+ implements PathIterator
{
- // The iterator we are applied to.
- private PathIterator subIterator;
- private double flatness;
- private int limit;
+ /**
+ * The PathIterator whose curved segments are being approximated.
+ */
+ private final PathIterator srcIter;
+
+
+ /**
+ * The square of the flatness threshold value, which determines when
+ * a curve segment is considered flat enough that no further
+ * subdivision is needed.
+ *
+ * <p>Calculating flatness actually produces the squared flatness
+ * value. To avoid the relatively expensive calculation of a square
+ * root for each curve segment, we perform all flatness comparisons
+ * on squared values.
+ *
+ * @see QuadCurve2D#getFlatnessSq()
+ * @see CubicCurve2D#getFlatnessSq()
+ */
+ private final double flatnessSq;
+
+
+ /**
+ * The maximal number of subdivions that are performed to
+ * approximate a quadratic or cubic curve segment.
+ */
+ private final int recursionLimit;
+
+
+ /**
+ * A stack for holding the coordinates of subdivided segments.
+ *
+ * @see <a href="doc-files/FlatteningPathIterator-1.html"
+ * >Implementation Note</a>
+ */
+ private double[] stack;
+
+
+ /**
+ * The current stack size.
+ *
+ * @see <a href="doc-files/FlatteningPathIterator-1.html"
+ * >Implementation Note</a>
+ */
+ private int stackSize;
+
+
+ /**
+ * The number of recursions that were performed to arrive at
+ * a segment on the stack.
+ *
+ * @see <a href="doc-files/FlatteningPathIterator-1.html"
+ * >Implementation Note</a>
+ */
+ private int[] recLevel;
+
+
+
+ private final double[] scratch = new double[6];
+
+
+ /**
+ * The segment type of the last segment that was returned by
+ * the source iterator.
+ */
+ private int srcSegType;
+
+ /**
+ * The current <i>x</i> position of the source iterator.
+ */
+ private double srcPosX;
+
+
+ /**
+ * The current <i>y</i> position of the source iterator.
+ */
+ private double srcPosY;
+
+
+ /**
+ * A flag that indicates when this path iterator has finished its
+ * iteration over path segments.
+ */
+ private boolean done;
+
+
+ /**
+ * Constructs a new PathIterator for approximating an input
+ * PathIterator with straight lines. The approximation works by
+ * recursive subdivisons, until the specified flatness threshold is
+ * not exceeded.
+ *
+ * <p>There will not be more than 10 nested recursion steps, which
+ * means that a single <code>SEG_QUADTO</code> or
+ * <code>SEG_CUBICTO</code> segment is approximated by at most
+ * 2<sup><small>10</small></sup> = 1024 straight lines.
+ */
public FlatteningPathIterator(PathIterator src, double flatness)
{
this(src, flatness, 10);
}
- public FlatteningPathIterator(PathIterator src, double flatness, int limit)
+
+
+ /**
+ * Constructs a new PathIterator for approximating an input
+ * PathIterator with straight lines. The approximation works by
+ * recursive subdivisons, until the specified flatness threshold is
+ * not exceeded. Additionally, the number of recursions is also
+ * bound by the specified recursion limit.
+ */
+ public FlatteningPathIterator(PathIterator src, double flatness,
+ int limit)
{
- subIterator = src;
- this.flatness = flatness;
- this.limit = limit;
if (flatness < 0 || limit < 0)
throw new IllegalArgumentException();
+
+ srcIter = src;
+ flatnessSq = flatness * flatness;
+ recursionLimit = limit;
+ fetchSegment();
}
+
+ /**
+ * Returns the maximally acceptable flatness.
+ *
+ * @see QuadCurve2D#getFlatness()
+ * @see CubicCurve2D#getFlatness()
+ */
public double getFlatness()
{
- return flatness;
+ return Math.sqrt(flatnessSq);
}
+
+ /**
+ * Returns the maximum number of recursive curve subdivisions.
+ */
public int getRecursionLimit()
{
- return limit;
+ return recursionLimit;
}
+
+ // Documentation will be copied from PathIterator.
public int getWindingRule()
{
- return subIterator.getWindingRule();
+ return srcIter.getWindingRule();
}
+
+ // Documentation will be copied from PathIterator.
public boolean isDone()
{
- return subIterator.isDone();
+ return done;
}
+
+ // Documentation will be copied from PathIterator.
public void next()
{
- throw new Error("not implemented");
+ if (stackSize > 0)
+ {
+ --stackSize;
+ if (stackSize > 0)
+ {
+ switch (srcSegType)
+ {
+ case PathIterator.SEG_QUADTO:
+ subdivideQuadratic();
+ return;
+
+ case PathIterator.SEG_CUBICTO:
+ subdivideCubic();
+ return;
+
+ default:
+ throw new IllegalStateException();
+ }
+ }
+ }
+
+ srcIter.next();
+ fetchSegment();
}
+
+ // Documentation will be copied from PathIterator.
public int currentSegment(double[] coords)
{
- throw new Error("not implemented");
+ if (done)
+ throw new NoSuchElementException();
+
+ switch (srcSegType)
+ {
+ case PathIterator.SEG_CLOSE:
+ return srcSegType;
+
+ case PathIterator.SEG_MOVETO:
+ case PathIterator.SEG_LINETO:
+ coords[0] = srcPosX;
+ coords[1] = srcPosY;
+ return srcSegType;
+
+ case PathIterator.SEG_QUADTO:
+ if (stackSize == 0)
+ {
+ coords[0] = srcPosX;
+ coords[1] = srcPosY;
+ }
+ else
+ {
+ int sp = stack.length - 4 * stackSize;
+ coords[0] = stack[sp + 2];
+ coords[1] = stack[sp + 3];
+ }
+ return PathIterator.SEG_LINETO;
+
+ case PathIterator.SEG_CUBICTO:
+ if (stackSize == 0)
+ {
+ coords[0] = srcPosX;
+ coords[1] = srcPosY;
+ }
+ else
+ {
+ int sp = stack.length - 6 * stackSize;
+ coords[0] = stack[sp + 4];
+ coords[1] = stack[sp + 5];
+ }
+ return PathIterator.SEG_LINETO;
+ }
+
+ throw new IllegalStateException();
}
+
+
+ // Documentation will be copied from PathIterator.
public int currentSegment(float[] coords)
{
- throw new Error("not implemented");
+ if (done)
+ throw new NoSuchElementException();
+
+ switch (srcSegType)
+ {
+ case PathIterator.SEG_CLOSE:
+ return srcSegType;
+
+ case PathIterator.SEG_MOVETO:
+ case PathIterator.SEG_LINETO:
+ coords[0] = (float) srcPosX;
+ coords[1] = (float) srcPosY;
+ return srcSegType;
+
+ case PathIterator.SEG_QUADTO:
+ if (stackSize == 0)
+ {
+ coords[0] = (float) srcPosX;
+ coords[1] = (float) srcPosY;
+ }
+ else
+ {
+ int sp = stack.length - 4 * stackSize;
+ coords[0] = (float) stack[sp + 2];
+ coords[1] = (float) stack[sp + 3];
+ }
+ return PathIterator.SEG_LINETO;
+
+ case PathIterator.SEG_CUBICTO:
+ if (stackSize == 0)
+ {
+ coords[0] = (float) srcPosX;
+ coords[1] = (float) srcPosY;
+ }
+ else
+ {
+ int sp = stack.length - 6 * stackSize;
+ coords[0] = (float) stack[sp + 4];
+ coords[1] = (float) stack[sp + 5];
+ }
+ return PathIterator.SEG_LINETO;
+ }
+
+ throw new IllegalStateException();
+ }
+
+
+ /**
+ * Fetches the next segment from the source iterator.
+ */
+ private void fetchSegment()
+ {
+ int sp;
+
+ if (srcIter.isDone())
+ {
+ done = true;
+ return;
+ }
+
+ srcSegType = srcIter.currentSegment(scratch);
+
+ switch (srcSegType)
+ {
+ case PathIterator.SEG_CLOSE:
+ return;
+
+ case PathIterator.SEG_MOVETO:
+ case PathIterator.SEG_LINETO:
+ srcPosX = scratch[0];
+ srcPosY = scratch[1];
+ return;
+
+ case PathIterator.SEG_QUADTO:
+ if (recursionLimit == 0)
+ {
+ srcPosX = scratch[2];
+ srcPosY = scratch[3];
+ stackSize = 0;
+ return;
+ }
+ sp = 4 * recursionLimit;
+ stackSize = 1;
+ if (stack == null)
+ {
+ stack = new double[sp + /* 4 + 2 */ 6];
+ recLevel = new int[recursionLimit + 1];
+ }
+ recLevel[0] = 0;
+ stack[sp] = srcPosX; // P1.x
+ stack[sp + 1] = srcPosY; // P1.y
+ stack[sp + 2] = scratch[0]; // C.x
+ stack[sp + 3] = scratch[1]; // C.y
+ srcPosX = stack[sp + 4] = scratch[2]; // P2.x
+ srcPosY = stack[sp + 5] = scratch[3]; // P2.y
+ subdivideQuadratic();
+ break;
+
+ case PathIterator.SEG_CUBICTO:
+ if (recursionLimit == 0)
+ {
+ srcPosX = scratch[4];
+ srcPosY = scratch[5];
+ stackSize = 0;
+ return;
+ }
+ sp = 6 * recursionLimit;
+ stackSize = 1;
+ if ((stack == null) || (stack.length < sp + 8))
+ {
+ stack = new double[sp + /* 6 + 2 */ 8];
+ recLevel = new int[recursionLimit + 1];
+ }
+ recLevel[0] = 0;
+ stack[sp] = srcPosX; // P1.x
+ stack[sp + 1] = srcPosY; // P1.y
+ stack[sp + 2] = scratch[0]; // C1.x
+ stack[sp + 3] = scratch[1]; // C1.y
+ stack[sp + 4] = scratch[2]; // C2.x
+ stack[sp + 5] = scratch[3]; // C2.y
+ srcPosX = stack[sp + 6] = scratch[4]; // P2.x
+ srcPosY = stack[sp + 7] = scratch[5]; // P2.y
+ subdivideCubic();
+ return;
+ }
+ }
+
+
+ /**
+ * Repeatedly subdivides the quadratic curve segment that is on top
+ * of the stack. The iteration terminates when the recursion limit
+ * has been reached, or when the resulting segment is flat enough.
+ */
+ private void subdivideQuadratic()
+ {
+ int sp;
+ int level;
+
+ sp = stack.length - 4 * stackSize - 2;
+ level = recLevel[stackSize - 1];
+ while ((level < recursionLimit)
+ && (QuadCurve2D.getFlatnessSq(stack, sp) >= flatnessSq))
+ {
+ recLevel[stackSize] = recLevel[stackSize - 1] = ++level;
+ QuadCurve2D.subdivide(stack, sp, stack, sp - 4, stack, sp);
+ ++stackSize;
+ sp -= 4;
+ }
+ }
+
+
+ /**
+ * Repeatedly subdivides the cubic curve segment that is on top
+ * of the stack. The iteration terminates when the recursion limit
+ * has been reached, or when the resulting segment is flat enough.
+ */
+ private void subdivideCubic()
+ {
+ int sp;
+ int level;
+
+ sp = stack.length - 6 * stackSize - 2;
+ level = recLevel[stackSize - 1];
+ while ((level < recursionLimit)
+ && (CubicCurve2D.getFlatnessSq(stack, sp) >= flatnessSq))
+ {
+ recLevel[stackSize] = recLevel[stackSize - 1] = ++level;
+
+ CubicCurve2D.subdivide(stack, sp, stack, sp - 6, stack, sp);
+ ++stackSize;
+ sp -= 6;
+ }
}
-} // class FlatteningPathIterator
+
+
+ /* These routines were useful for debugging. Since they would
+ * just bloat the implementation, they are commented out.
+ *
+ *
+
+ private static String segToString(int segType, double[] d, int offset)
+ {
+ String s;
+
+ switch (segType)
+ {
+ case PathIterator.SEG_CLOSE:
+ return "SEG_CLOSE";
+
+ case PathIterator.SEG_MOVETO:
+ return "SEG_MOVETO (" + d[offset] + ", " + d[offset + 1] + ")";
+
+ case PathIterator.SEG_LINETO:
+ return "SEG_LINETO (" + d[offset] + ", " + d[offset + 1] + ")";
+
+ case PathIterator.SEG_QUADTO:
+ return "SEG_QUADTO (" + d[offset] + ", " + d[offset + 1]
+ + ") (" + d[offset + 2] + ", " + d[offset + 3] + ")";
+
+ case PathIterator.SEG_CUBICTO:
+ return "SEG_CUBICTO (" + d[offset] + ", " + d[offset + 1]
+ + ") (" + d[offset + 2] + ", " + d[offset + 3]
+ + ") (" + d[offset + 4] + ", " + d[offset + 5] + ")";
+ }
+
+ throw new IllegalStateException();
+ }
+
+
+ private void dumpQuadraticStack(String msg)
+ {
+ int sp = stack.length - 4 * stackSize - 2;
+ int i = 0;
+ System.err.print(" " + msg + ":");
+ while (sp < stack.length)
+ {
+ System.err.print(" (" + stack[sp] + ", " + stack[sp+1] + ")");
+ if (i < recLevel.length)
+ System.out.print("/" + recLevel[i++]);
+ if (sp + 3 < stack.length)
+ System.err.print(" [" + stack[sp+2] + ", " + stack[sp+3] + "]");
+ sp += 4;
+ }
+ System.err.println();
+ }
+
+
+ private void dumpCubicStack(String msg)
+ {
+ int sp = stack.length - 6 * stackSize - 2;
+ int i = 0;
+ System.err.print(" " + msg + ":");
+ while (sp < stack.length)
+ {
+ System.err.print(" (" + stack[sp] + ", " + stack[sp+1] + ")");
+ if (i < recLevel.length)
+ System.out.print("/" + recLevel[i++]);
+ if (sp + 3 < stack.length)
+ {
+ System.err.print(" [" + stack[sp+2] + ", " + stack[sp+3] + "]");
+ System.err.print(" [" + stack[sp+4] + ", " + stack[sp+5] + "]");
+ }
+ sp += 6;
+ }
+ System.err.println();
+ }
+
+ *
+ *
+ */
+}
--- /dev/null
+<?xml version="1.0" encoding="US-ASCII"?>
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en">
+<head>
+ <title>The GNU Implementation of java.awt.geom.FlatteningPathIterator</title>
+ <meta name="author" content="Sascha Brawer" />
+ <style type="text/css"><!--
+ td { white-space: nowrap; }
+ li { margin: 2mm 0; }
+ --></style>
+</head>
+<body>
+
+<h1>The GNU Implementation of FlatteningPathIterator</h1>
+
+<p><i><a href="http://www.dandelis.ch/people/brawer/">Sascha
+Brawer</a>, November 2003</i></p>
+
+<p>This document describes the GNU implementation of the class
+<code>java.awt.geom.FlatteningPathIterator</code>. It does
+<em>not</em> describe how a programmer should use this class; please
+refer to the generated API documentation for this purpose. Instead, it
+is intended for maintenance programmers who want to understand the
+implementation, for example because they want to extend the class or
+fix a bug.</p>
+
+
+<h2>Data Structures</h2>
+
+<p>The algorithm uses a stack. Its allocation is delayed to the time
+when the source path iterator actually returns the first curved
+segment (either <code>SEG_QUADTO</code> or <code>SEG_CUBICTO</code>).
+If the input path does not contain any curved segments, the value of
+the <code>stack</code> variable stays <code>null</code>. In this quite
+common case, the memory consumption is minimal.</p>
+
+<dl><dt><code>stack</code></dt><dd>The variable <code>stack</code> is
+a <code>double</code> array that holds the start, control and end
+points of individual sub-segments.</dd>
+
+<dt><code>recLevel</code></dt><dd>The variable <code>recLevel</code>
+holds how many recursive sub-divisions were needed to calculate a
+segment. The original curve has recursion level 0. For each
+sub-division, the corresponding recursion level is increased by
+one.</dd>
+
+<dt><code>stackSize</code></dt><dd>Finally, the variable
+<code>stackSize</code> indicates how many sub-segments are stored on
+the stack.</dd></dl>
+
+<h2>Algorithm</h2>
+
+<p>The implementation separately processes each segment that the
+base iterator returns.</p>
+
+<p>In the case of <code>SEG_CLOSE</code>,
+<code>SEG_MOVETO</code> and <code>SEG_LINETO</code> segments, the
+implementation simply hands the segment to the consumer, without actually
+doing anything.</p>
+
+<p>Any <code>SEG_QUADTO</code> and <code>SEG_CUBICTO</code> segments
+need to be flattened. Flattening is performed with a fixed-sized
+stack, holding the coordinates of subdivided segments. When the base
+iterator returns a <code>SEG_QUADTO</code> and
+<code>SEG_CUBICTO</code> segments, it is recursively flattened as
+follows:</p>
+
+<ol><li>Intialization: Allocate memory for the stack (unless a
+sufficiently large stack has been allocated previously). Push the
+original quadratic or cubic curve onto the stack. Mark that segment as
+having a <code>recLevel</code> of zero.</li>
+
+<li>If the stack is empty, flattening the segment is complete,
+and the next segment is fetched from the base iterator.</li>
+
+<li>If the stack is not empty, pop a curve segment from the
+stack.
+
+ <ul><li>If its <code>recLevel</code> exceeds the recursion limit,
+ hand the current segment to the consumer.</li>
+
+ <li>Calculate the squared flatness of the segment. If it smaller
+ than <code>flatnessSq</code>, hand the current segment to the
+ consumer.</li>
+
+ <li>Otherwise, split the segment in two halves. Push the right
+ half onto the stack. Then, push the left half onto the stack.
+ Continue with step two.</li></ul></li>
+</ol>
+
+<p>The implementation is slightly complicated by the fact that
+consumers <em>pull</em> the flattened segments from the
+<code>FlatteningPathIterator</code>. This means that we actually
+cannot “hand the curent segment over to the consumer.”
+But the algorithm is easier to understand if one assumes a
+<em>push</em> paradigm.</p>
+
+
+<h2>Example</h2>
+
+<p>The following example shows how a
+<code>FlatteningPathIterator</code> processes a
+<code>SEG_QUADTO</code> segment. It is (arbitrarily) assumed that the
+recursion limit was set to 2.</p>
+
+<blockquote>
+<table border="1" cellspacing="0" cellpadding="8">
+ <tr align="center" valign="baseline">
+ <th></th><th>A</th><th>B</th><th>C</th>
+ <th>D</th><th>E</th><th>F</th><th>G</th><th>H</th>
+ </tr>
+ <tr align="center" valign="baseline">
+ <th><code>stack[0]</code></th>
+ <td>—</td>
+ <td>—</td>
+ <td><i>S<sub>ll</sub>.x</i></td>
+ <td>—</td>
+ <td>—</td>
+ <td>—</td>
+ <td>—</td>
+ <td>—</td>
+ </tr>
+ <tr align="center" valign="baseline">
+ <th><code>stack[1]</code></th>
+ <td>—</td>
+ <td>—</td>
+ <td><i>S<sub>ll</sub>.y</i></td>
+ <td>—</td>
+ <td>—</td>
+ <td>—</td>
+ <td>—</td>
+ <td>—</td>
+ </tr>
+ <tr align="center" valign="baseline">
+ <th><code>stack[2]</code></th>
+ <td>—</td>
+ <td>—</td>
+ <td><i>C<sub>ll</sub>.x</i></td>
+ <td>—</td>
+ <td>—</td>
+ <td>—</td>
+ <td>—</td>
+ <td>—</td>
+ </tr>
+ <tr align="center" valign="baseline">
+ <th><code>stack[3]</code></th>
+ <td>—</td>
+ <td>—</td>
+ <td><i>C<sub>ll</sub>.y</i></td>
+ <td>—</td>
+ <td>—</td>
+ <td>—</td>
+ <td>—</td>
+ <td>—</td>
+ </tr>
+ <tr align="center" valign="baseline">
+ <th><code>stack[4]</code></th>
+ <td>—</td>
+ <td><i>S<sub>l</sub>.x</i></td>
+ <td><i>E<sub>ll</sub>.x</i>
+ = <i>S<sub>lr</sub>.x</i></td>
+ <td><i>S<sub>lr</sub>.x</i></td>
+ <td>—</td>
+ <td><i>S<sub>rl</sub>.x</i></td>
+ <td>—</td>
+ <td>—</td>
+ </tr>
+ <tr align="center" valign="baseline">
+ <th><code>stack[5]</code></th>
+ <td>—</td>
+ <td><i>S<sub>l</sub>.y</i></td>
+ <td><i>E<sub>ll</sub>.x</i>
+ = <i>S<sub>lr</sub>.y</i></td>
+ <td><i>S<sub>lr</sub>.y</i></td>
+ <td>—</td>
+ <td><i>S<sub>rl</sub>.y</i></td>
+ <td>—</td>
+ <td>—</td>
+ </tr>
+ <tr align="center" valign="baseline">
+ <th><code>stack[6]</code></th>
+ <td>—</td>
+ <td><i>C<sub>l</sub>.x</i></td>
+ <td><i>C<sub>lr</sub>.x</i></td>
+ <td><i>C<sub>lr</sub>.x</i></td>
+ <td>—</td>
+ <td><i>C<sub>rl</sub>.x</i></td>
+ <td>—</td>
+ <td>—</td>
+ </tr>
+ <tr align="center" valign="baseline">
+ <th><code>stack[7]</code></th>
+ <td>—</td>
+ <td><i>C<sub>l</sub>.y</i></td>
+ <td><i>C<sub>lr</sub>.y</i></td>
+ <td><i>C<sub>lr</sub>.y</i></td>
+ <td>—</td>
+ <td><i>C<sub>rl</sub>.y</i></td>
+ <td>—</td>
+ <td>—</td>
+ </tr>
+ <tr align="center" valign="baseline">
+ <th><code>stack[8]</code></th>
+ <td><i>S.x</i></td>
+ <td><i>E<sub>l</sub>.x</i>
+ = <i>S<sub>r</sub>.x</i></td>
+ <td><i>E<sub>lr</sub>.x</i>
+ = <i>S<sub>r</sub>.x</i></td>
+ <td><i>E<sub>lr</sub>.x</i>
+ = <i>S<sub>r</sub>.x</i></td>
+ <td><i>S<sub>r</sub>.x</i></td>
+ <td><i>E<sub>rl</sub>.x</i>
+ = <i>S<sub>rr</sub>.x</i></td>
+ <td><i>S<sub>rr</sub>.x</i></td>
+ <td>—</td>
+ </tr>
+ <tr align="center" valign="baseline">
+ <th><code>stack[9]</code></th>
+ <td><i>S.y</i></td>
+ <td><i>E<sub>l</sub>.y</i>
+ = <i>S<sub>r</sub>.y</i></td>
+ <td><i>E<sub>lr</sub>.y</i>
+ = <i>S<sub>r</sub>.y</i></td>
+ <td><i>E<sub>lr</sub>.y</i>
+ = <i>S<sub>r</sub>.y</i></td>
+ <td><i>S<sub>r</sub>.y</i></td>
+ <td><i>E<sub>rl</sub>.y</i>
+ = <i>S<sub>rr</sub>.y</i></td>
+ <td><i>S<sub>rr</sub>.y</i></td>
+ <td>—</td>
+ </tr>
+ <tr align="center" valign="baseline">
+ <th><code>stack[10]</code></th>
+ <td><i>C.x</i></td>
+ <td><i>C<sub>r</sub>.x</i></td>
+ <td><i>C<sub>r</sub>.x</i></td>
+ <td><i>C<sub>r</sub>.x</i></td>
+ <td><i>C<sub>r</sub>.x</i></td>
+ <td><i>C<sub>rr</sub>.x</i></td>
+ <td><i>C<sub>rr</sub>.x</i></td>
+ <td>—</td>
+ </tr>
+ <tr align="center" valign="baseline">
+ <th><code>stack[11]</code></th>
+ <td><i>C.y</i></td>
+ <td><i>C<sub>r</sub>.y</i></td>
+ <td><i>C<sub>r</sub>.y</i></td>
+ <td><i>C<sub>r</sub>.y</i></td>
+ <td><i>C<sub>r</sub>.y</i></td>
+ <td><i>C<sub>rr</sub>.y</i></td>
+ <td><i>C<sub>rr</sub>.y</i></td>
+ <td>—</td>
+ </tr>
+ <tr align="center" valign="baseline">
+ <th><code>stack[12]</code></th>
+ <td><i>E.x</i></td>
+ <td><i>E<sub>r</sub>.x</i></td>
+ <td><i>E<sub>r</sub>.x</i></td>
+ <td><i>E<sub>r</sub>.x</i></td>
+ <td><i>E<sub>r</sub>.x</i></td>
+ <td><i>E<sub>rr</sub>.x</i></td>
+ <td><i>E<sub>rr</sub>.x</i></td>
+ <td>—</td>
+ </tr>
+ <tr align="center" valign="baseline">
+ <th><code>stack[13]</code></th>
+ <td><i>E.y</i></td>
+ <td><i>E<sub>r</sub>.y</i></td>
+ <td><i>E<sub>r</sub>.y</i></td>
+ <td><i>E<sub>r</sub>.y</i></td>
+ <td><i>E<sub>r</sub>.y</i></td>
+ <td><i>E<sub>rr</sub>.y</i></td>
+ <td><i>E<sub>rr</sub>.x</i></td>
+ <td>—</td>
+ </tr>
+ <tr align="center" valign="baseline">
+ <th><code>stackSize</code></th>
+ <td>1</td>
+ <td>2</td>
+ <td>3</td>
+ <td>2</td>
+ <td>1</td>
+ <td>2</td>
+ <td>1</td>
+ <td>0</td>
+ </tr>
+ <tr align="center" valign="baseline">
+ <th><code>recLevel[2]</code></th>
+ <td>—</td>
+ <td>—</td>
+ <td>2</td>
+ <td>—</td>
+ <td>—</td>
+ <td>—</td>
+ <td>—</td>
+ <td>—</td>
+ </tr>
+ <tr align="center" valign="baseline">
+ <th><code>recLevel[1]</code></th>
+ <td>—</td>
+ <td>1</td>
+ <td>2</td>
+ <td>2</td>
+ <td>—</td>
+ <td>2</td>
+ <td>—</td>
+ <td>—</td>
+ </tr>
+ <tr align="center" valign="baseline">
+ <th><code>recLevel[0]</code></th>
+ <td>0</td>
+ <td>1</td>
+ <td>1</td>
+ <td>1</td>
+ <td>1</td>
+ <td>2</td>
+ <td>2</td>
+ <td>—</td>
+ </tr>
+ </table>
+</blockquote>
+
+<ol>
+
+<li>The data structures are initialized as follows.
+
+<ul><li>The segment’s end point <i>E</i>, control point
+<i>C</i>, and start point <i>S</i> are pushed onto the stack.</li>
+
+ <li>Currently, the curve in the stack would be approximated by one
+ single straight line segment (<i>S</i> – <i>E</i>).
+ Therefore, <code>stackSize</code> is set to 1.</li>
+
+ <li>This single straight line segment is approximating the original
+ curve, which can be seen as the result of zero recursive
+ splits. Therefore, <code>recLevel[0]</code> is set to
+ zero.</li></ul>
+
+Column A shows the state after the initialization step.</li>
+
+<li>The algorithm proceeds by taking the topmost curve segment
+(<i>S</i> – <i>C</i> – <i>E</i>) from the stack.
+
+ <ul><li>The recursion level of this segment (stored in
+ <code>recLevel[0]</code>) is zero, which is smaller than
+ the limit 2.</li>
+
+ <li>The method <code>java.awt.geom.QuadCurve2D.getFlatnessSq</code>
+ is called to calculate the squared flatness.</li>
+
+ <li>For the sake of argument, we assume that the squared flatness is
+ exceeding the threshold stored in <code>flatnessSq</code>. Thus, the
+ curve segment <i>S</i> – <i>C</i> – <i>E</i> gets
+ subdivided into a left and a right half, namely
+ <i>S<sub>l</sub></i> – <i>C<sub>l</sub></i> –
+ <i>E<sub>l</sub></i> and <i>S<sub>r</sub></i> –
+ <i>C<sub>r</sub></i> – <i>E<sub>r</sub></i>. Both halves are
+ pushed onto the stack, so the left half is now on top.
+
+ <br /> <br />The left half starts at the same point
+ as the original curve, so <i>S<sub>l</sub></i> has the same
+ coordinates as <i>S</i>. Similarly, the end point of the right
+ half and of the original curve are identical
+ (<i>E<sub>r</sub></i> = <i>E</i>). More interestingly, the left
+ half ends where the right half starts. Because
+ <i>E<sub>l</sub></i> = <i>S<sub>r</sub></i>, their coordinates need
+ to be stored only once, which amounts to saving 16 bytes (two
+ <code>double</code> values) for each iteration.</li></ul>
+
+Column B shows the state after the first iteration.</li>
+
+<li>Again, the topmost curve segment (<i>S<sub>l</sub></i>
+– <i>C<sub>l</sub></i> – <i>E<sub>l</sub></i>) is
+taken from the stack.
+
+ <ul><li>The recursion level of this segment (stored in
+ <code>recLevel[1]</code>) is 1, which is smaller than
+ the limit 2.</li>
+
+ <li>The method <code>java.awt.geom.QuadCurve2D.getFlatnessSq</code>
+ is called to calculate the squared flatness.</li>
+
+ <li>Assuming that the segment is still not considered
+ flat enough, it gets subdivided into a left
+ (<i>S<sub>ll</sub></i> – <i>C<sub>ll</sub></i> –
+ <i>E<sub>ll</sub></i>) and a right (<i>S<sub>lr</sub></i>
+ – <i>C<sub>lr</sub></i> – <i>E<sub>lr</sub></i>)
+ half.</li></ul>
+
+Column C shows the state after the second iteration.</li>
+
+<li>The topmost curve segment (<i>S<sub>ll</sub></i> –
+<i>C<sub>ll</sub></i> – <i>E<sub>ll</sub></i>) is popped from
+the stack.
+
+ <ul><li>The recursion level of this segment (stored in
+ <code>recLevel[2]</code>) is 2, which is <em>not</em> smaller than
+ the limit 2. Therefore, a <code>SEG_LINETO</code> (from
+ <i>S<sub>ll</sub></i> to <i>E<sub>ll</sub></i>) is passed to the
+ consumer.</li></ul>
+
+ The new state is shown in column D.</li>
+
+
+<li>The topmost curve segment (<i>S<sub>lr</sub></i> –
+<i>C<sub>lr</sub></i> – <i>E<sub>lr</sub></i>) is popped from
+the stack.
+
+ <ul><li>The recursion level of this segment (stored in
+ <code>recLevel[1]</code>) is 2, which is <em>not</em> smaller than
+ the limit 2. Therefore, a <code>SEG_LINETO</code> (from
+ <i>S<sub>lr</sub></i> to <i>E<sub>lr</sub></i>) is passed to the
+ consumer.</li></ul>
+
+ The new state is shown in column E.</li>
+
+<li>The algorithm proceeds by taking the topmost curve segment
+(<i>S<sub>r</sub></i> – <i>C<sub>r</sub></i> –
+<i>E<sub>r</sub></i>) from the stack.
+
+ <ul><li>The recursion level of this segment (stored in
+ <code>recLevel[0]</code>) is 1, which is smaller than
+ the limit 2.</li>
+
+ <li>The method <code>java.awt.geom.QuadCurve2D.getFlatnessSq</code>
+ is called to calculate the squared flatness.</li>
+
+ <li>For the sake of argument, we again assume that the squared
+ flatness is exceeding the threshold stored in
+ <code>flatnessSq</code>. Thus, the curve segment
+ (<i>S<sub>r</sub></i> – <i>C<sub>r</sub></i> –
+ <i>E<sub>r</sub></i>) is subdivided into a left and a right half,
+ namely
+ <i>S<sub>rl</sub></i> – <i>C<sub>rl</sub></i> –
+ <i>E<sub>rl</sub></i> and <i>S<sub>rr</sub></i> –
+ <i>C<sub>rr</sub></i> – <i>E<sub>rr</sub></i>. Both halves
+ are pushed onto the stack.</li></ul>
+
+ The new state is shown in column F.</li>
+
+<li>The topmost curve segment (<i>S<sub>rl</sub></i> –
+<i>C<sub>rl</sub></i> – <i>E<sub>rl</sub></i>) is popped from
+the stack.
+
+ <ul><li>The recursion level of this segment (stored in
+ <code>recLevel[2]</code>) is 2, which is <em>not</em> smaller than
+ the limit 2. Therefore, a <code>SEG_LINETO</code> (from
+ <i>S<sub>rl</sub></i> to <i>E<sub>rl</sub></i>) is passed to the
+ consumer.</li></ul>
+
+ The new state is shown in column G.</li>
+
+<li>The topmost curve segment (<i>S<sub>rr</sub></i> –
+<i>C<sub>rr</sub></i> – <i>E<sub>rr</sub></i>) is popped from
+the stack.
+
+ <ul><li>The recursion level of this segment (stored in
+ <code>recLevel[2]</code>) is 2, which is <em>not</em> smaller than
+ the limit 2. Therefore, a <code>SEG_LINETO</code> (from
+ <i>S<sub>rr</sub></i> to <i>E<sub>rr</sub></i>) is passed to the
+ consumer.</li></ul>
+
+ The new state is shown in column H.</li>
+
+<li>The stack is now empty. The FlatteningPathIterator will fetch the
+next segment from the base iterator, and process it.</li>
+
+</ol>
+
+<p>In order to split the most recently pushed segment, the
+<code>subdivideQuadratic()</code> method passes <code>stack</code>
+directly to
+<code>QuadCurve2D.subdivide(double[],int,double[],int,double[],int)</code>.
+Because the stack grows towards the beginning of the array, no data
+needs to be copied around: <code>subdivide</code> will directly store
+the result into the stack, which will have the contents shown to the
+right.</p>
+
+</body>
+</html>
OSDN Git Service
