diff options
| author | Sascha Brawer <brawer@dandelis.ch> | 2003-11-19 13:02:11 +0100 |
|---|---|---|
| committer | Michael Koch <mkoch@gcc.gnu.org> | 2003-11-19 12:02:11 +0000 |
| commit | b6b8f690470abd887a1f4de734548edc510f9290 (patch) | |
| tree | cef7d50742f7b5bcf27c9b376c2e4b564d0d1d04 /libjava/java/awt | |
| parent | 1f33554abb31ee66b0b61be508d91d1cafdc4b0b (diff) | |
| download | gcc-b6b8f690470abd887a1f4de734548edc510f9290.zip gcc-b6b8f690470abd887a1f4de734548edc510f9290.tar.gz gcc-b6b8f690470abd887a1f4de734548edc510f9290.tar.bz2 | |
FlatteningPathIterator.java: Entirely re-written.
2003-11-19 Sascha Brawer <brawer@dandelis.ch>
* java/awt/geom/FlatteningPathIterator.java: Entirely re-written.
* java/awt/geom/doc-files/FlatteningPathIterator-1.html:
Describe how the implementation works.
From-SVN: r73734
Diffstat (limited to 'libjava/java/awt')
| -rw-r--r-- | libjava/java/awt/geom/FlatteningPathIterator.java | 526 | ||||
| -rw-r--r-- | libjava/java/awt/geom/doc-files/FlatteningPathIterator-1.html | 481 |
2 files changed, 981 insertions, 26 deletions
diff --git a/libjava/java/awt/geom/FlatteningPathIterator.java b/libjava/java/awt/geom/FlatteningPathIterator.java index a7a57ef..94ff145 100644 --- a/libjava/java/awt/geom/FlatteningPathIterator.java +++ b/libjava/java/awt/geom/FlatteningPathIterator.java @@ -1,5 +1,5 @@ -/* 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. @@ -38,68 +38,542 @@ exception statement from your version. */ 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(); + } + + * + * + */ +} diff --git a/libjava/java/awt/geom/doc-files/FlatteningPathIterator-1.html b/libjava/java/awt/geom/doc-files/FlatteningPathIterator-1.html new file mode 100644 index 0000000..5a52d69 --- /dev/null +++ b/libjava/java/awt/geom/doc-files/FlatteningPathIterator-1.html @@ -0,0 +1,481 @@ +<?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> |