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<head>
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<title>The GNU Implementation of java.awt.geom.FlatteningPathIterator</title>
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<meta name="author" content="Sascha Brawer" />
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</head>
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<body>
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<h1>The GNU Implementation of FlatteningPathIterator</h1>
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<p><i><a href="http://www.dandelis.ch/people/brawer/">Sascha
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Brawer</a>, November 2003</i></p>
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<p>This document describes the GNU implementation of the class
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<code>java.awt.geom.FlatteningPathIterator</code>. It does
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<em>not</em> describe how a programmer should use this class; please
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refer to the generated API documentation for this purpose. Instead, it
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is intended for maintenance programmers who want to understand the
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implementation, for example because they want to extend the class or
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fix a bug.</p>
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<h2>Data Structures</h2>
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<p>The algorithm uses a stack. Its allocation is delayed to the time
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when the source path iterator actually returns the first curved
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segment (either <code>SEG_QUADTO</code> or <code>SEG_CUBICTO</code>).
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If the input path does not contain any curved segments, the value of
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the <code>stack</code> variable stays <code>null</code>. In this quite
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common case, the memory consumption is minimal.</p>
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<dl><dt><code>stack</code></dt><dd>The variable <code>stack</code> is
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a <code>double</code> array that holds the start, control and end
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points of individual sub-segments.</dd>
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<dt><code>recLevel</code></dt><dd>The variable <code>recLevel</code>
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holds how many recursive sub-divisions were needed to calculate a
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segment. The original curve has recursion level 0. For each
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sub-division, the corresponding recursion level is increased by
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one.</dd>
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<dt><code>stackSize</code></dt><dd>Finally, the variable
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<code>stackSize</code> indicates how many sub-segments are stored on
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the stack.</dd></dl>
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<h2>Algorithm</h2>
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<p>The implementation separately processes each segment that the
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base iterator returns.</p>
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<p>In the case of <code>SEG_CLOSE</code>,
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<code>SEG_MOVETO</code> and <code>SEG_LINETO</code> segments, the
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implementation simply hands the segment to the consumer, without actually
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doing anything.</p>
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<p>Any <code>SEG_QUADTO</code> and <code>SEG_CUBICTO</code> segments
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need to be flattened. Flattening is performed with a fixed-sized
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stack, holding the coordinates of subdivided segments. When the base
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iterator returns a <code>SEG_QUADTO</code> and
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<code>SEG_CUBICTO</code> segments, it is recursively flattened as
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follows:</p>
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<ol><li>Intialization: Allocate memory for the stack (unless a
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sufficiently large stack has been allocated previously). Push the
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original quadratic or cubic curve onto the stack. Mark that segment as
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having a <code>recLevel</code> of zero.</li>
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<li>If the stack is empty, flattening the segment is complete,
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and the next segment is fetched from the base iterator.</li>
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<li>If the stack is not empty, pop a curve segment from the
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stack.
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<ul><li>If its <code>recLevel</code> exceeds the recursion limit,
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hand the current segment to the consumer.</li>
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<li>Calculate the squared flatness of the segment. If it smaller
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than <code>flatnessSq</code>, hand the current segment to the
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consumer.</li>
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<li>Otherwise, split the segment in two halves. Push the right
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half onto the stack. Then, push the left half onto the stack.
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Continue with step two.</li></ul></li>
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</ol>
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<p>The implementation is slightly complicated by the fact that
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consumers <em>pull</em> the flattened segments from the
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<code>FlatteningPathIterator</code>. This means that we actually
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cannot “hand the curent segment over to the consumer.”
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But the algorithm is easier to understand if one assumes a
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<em>push</em> paradigm.</p>
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<h2>Example</h2>
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<p>The following example shows how a
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<code>FlatteningPathIterator</code> processes a
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<code>SEG_QUADTO</code> segment. It is (arbitrarily) assumed that the
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recursion limit was set to 2.</p>
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<blockquote>
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<table border="1" cellspacing="0" cellpadding="8">
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<tr align="center" valign="baseline">
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<th></th><th>A</th><th>B</th><th>C</th>
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<th>D</th><th>E</th><th>F</th><th>G</th><th>H</th>
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</tr>
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<tr align="center" valign="baseline">
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<th><code>stack[0]</code></th>
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<td>—</td>
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<td>—</td>
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<td><i>S<sub>ll</sub>.x</i></td>
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<td>—</td>
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<td>—</td>
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<td>—</td>
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<td>—</td>
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<td>—</td>
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</tr>
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<tr align="center" valign="baseline">
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<th><code>stack[1]</code></th>
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<td>—</td>
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<td>—</td>
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<td><i>S<sub>ll</sub>.y</i></td>
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<td>—</td>
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<td>—</td>
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<td>—</td>
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<td>—</td>
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<td>—</td>
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</tr>
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<tr align="center" valign="baseline">
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<th><code>stack[2]</code></th>
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<td>—</td>
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<td>—</td>
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<td><i>C<sub>ll</sub>.x</i></td>
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<td>—</td>
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<td>—</td>
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<td>—</td>
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<td>—</td>
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<td>—</td>
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</tr>
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<tr align="center" valign="baseline">
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<th><code>stack[3]</code></th>
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<td>—</td>
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<td>—</td>
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<td><i>C<sub>ll</sub>.y</i></td>
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<td>—</td>
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<td>—</td>
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<td>—</td>
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<td>—</td>
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<td>—</td>
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</tr>
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<tr align="center" valign="baseline">
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<th><code>stack[4]</code></th>
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<td>—</td>
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<td><i>S<sub>l</sub>.x</i></td>
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<td><i>E<sub>ll</sub>.x</i>
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= <i>S<sub>lr</sub>.x</i></td>
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<td><i>S<sub>lr</sub>.x</i></td>
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<td>—</td>
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<td><i>S<sub>rl</sub>.x</i></td>
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<td>—</td>
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<td>—</td>
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</tr>
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<tr align="center" valign="baseline">
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<th><code>stack[5]</code></th>
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<td>—</td>
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<td><i>S<sub>l</sub>.y</i></td>
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<td><i>E<sub>ll</sub>.x</i>
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= <i>S<sub>lr</sub>.y</i></td>
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<td><i>S<sub>lr</sub>.y</i></td>
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<td>—</td>
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<td><i>S<sub>rl</sub>.y</i></td>
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<td>—</td>
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<td>—</td>
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</tr>
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<tr align="center" valign="baseline">
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<th><code>stack[6]</code></th>
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<td>—</td>
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<td><i>C<sub>l</sub>.x</i></td>
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<td><i>C<sub>lr</sub>.x</i></td>
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<td><i>C<sub>lr</sub>.x</i></td>
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<td>—</td>
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<td><i>C<sub>rl</sub>.x</i></td>
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<td>—</td>
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<td>—</td>
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</tr>
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<tr align="center" valign="baseline">
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<th><code>stack[7]</code></th>
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<td>—</td>
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<td><i>C<sub>l</sub>.y</i></td>
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<td><i>C<sub>lr</sub>.y</i></td>
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<td><i>C<sub>lr</sub>.y</i></td>
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<td>—</td>
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<td><i>C<sub>rl</sub>.y</i></td>
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<td>—</td>
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<td>—</td>
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</tr>
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<tr align="center" valign="baseline">
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<th><code>stack[8]</code></th>
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<td><i>S.x</i></td>
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<td><i>E<sub>l</sub>.x</i>
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= <i>S<sub>r</sub>.x</i></td>
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<td><i>E<sub>lr</sub>.x</i>
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= <i>S<sub>r</sub>.x</i></td>
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<td><i>E<sub>lr</sub>.x</i>
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= <i>S<sub>r</sub>.x</i></td>
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<td><i>S<sub>r</sub>.x</i></td>
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<td><i>E<sub>rl</sub>.x</i>
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= <i>S<sub>rr</sub>.x</i></td>
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<td><i>S<sub>rr</sub>.x</i></td>
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<td>—</td>
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</tr>
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<tr align="center" valign="baseline">
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<th><code>stack[9]</code></th>
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<td><i>S.y</i></td>
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<td><i>E<sub>l</sub>.y</i>
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= <i>S<sub>r</sub>.y</i></td>
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<td><i>E<sub>lr</sub>.y</i>
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= <i>S<sub>r</sub>.y</i></td>
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<td><i>E<sub>lr</sub>.y</i>
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= <i>S<sub>r</sub>.y</i></td>
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<td><i>S<sub>r</sub>.y</i></td>
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<td><i>E<sub>rl</sub>.y</i>
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= <i>S<sub>rr</sub>.y</i></td>
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<td><i>S<sub>rr</sub>.y</i></td>
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<td>—</td>
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</tr>
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<tr align="center" valign="baseline">
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<th><code>stack[10]</code></th>
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<td><i>C.x</i></td>
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<td><i>C<sub>r</sub>.x</i></td>
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<td><i>C<sub>r</sub>.x</i></td>
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<td><i>C<sub>r</sub>.x</i></td>
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<td><i>C<sub>r</sub>.x</i></td>
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<td><i>C<sub>rr</sub>.x</i></td>
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<td><i>C<sub>rr</sub>.x</i></td>
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<td>—</td>
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</tr>
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<tr align="center" valign="baseline">
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<th><code>stack[11]</code></th>
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<td><i>C.y</i></td>
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<td><i>C<sub>r</sub>.y</i></td>
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<td><i>C<sub>r</sub>.y</i></td>
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<td><i>C<sub>r</sub>.y</i></td>
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<td><i>C<sub>r</sub>.y</i></td>
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<td><i>C<sub>rr</sub>.y</i></td>
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<td><i>C<sub>rr</sub>.y</i></td>
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<td>—</td>
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</tr>
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<tr align="center" valign="baseline">
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<th><code>stack[12]</code></th>
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<td><i>E.x</i></td>
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<td><i>E<sub>r</sub>.x</i></td>
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|
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<td><i>E<sub>r</sub>.x</i></td>
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|
|
<td><i>E<sub>r</sub>.x</i></td>
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<td><i>E<sub>r</sub>.x</i></td>
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<td><i>E<sub>rr</sub>.x</i></td>
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<td><i>E<sub>rr</sub>.x</i></td>
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| 264 |
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<td>—</td>
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</tr>
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<tr align="center" valign="baseline">
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<th><code>stack[13]</code></th>
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<td><i>E.y</i></td>
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<td><i>E<sub>r</sub>.y</i></td>
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<td><i>E<sub>r</sub>.y</i></td>
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<td><i>E<sub>r</sub>.y</i></td>
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<td><i>E<sub>r</sub>.y</i></td>
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<td><i>E<sub>rr</sub>.y</i></td>
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<td><i>E<sub>rr</sub>.x</i></td>
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<td>—</td>
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</tr>
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<tr align="center" valign="baseline">
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<th><code>stackSize</code></th>
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<td>1</td>
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<td>2</td>
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<td>3</td>
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<td>2</td>
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<td>1</td>
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<td>2</td>
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<td>1</td>
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<td>0</td>
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</tr>
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<tr align="center" valign="baseline">
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<th><code>recLevel[2]</code></th>
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<td>—</td>
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<td>—</td>
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<td>2</td>
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<td>—</td>
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|
<td>—</td>
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|
<td>—</td>
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|
<td>—</td>
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|
<td>—</td>
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</tr>
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<tr align="center" valign="baseline">
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|
<th><code>recLevel[1]</code></th>
|
| 301 |
|
|
<td>—</td>
|
| 302 |
|
|
<td>1</td>
|
| 303 |
|
|
<td>2</td>
|
| 304 |
|
|
<td>2</td>
|
| 305 |
|
|
<td>—</td>
|
| 306 |
|
|
<td>2</td>
|
| 307 |
|
|
<td>—</td>
|
| 308 |
|
|
<td>—</td>
|
| 309 |
|
|
</tr>
|
| 310 |
|
|
<tr align="center" valign="baseline">
|
| 311 |
|
|
<th><code>recLevel[0]</code></th>
|
| 312 |
|
|
<td>0</td>
|
| 313 |
|
|
<td>1</td>
|
| 314 |
|
|
<td>1</td>
|
| 315 |
|
|
<td>1</td>
|
| 316 |
|
|
<td>1</td>
|
| 317 |
|
|
<td>2</td>
|
| 318 |
|
|
<td>2</td>
|
| 319 |
|
|
<td>—</td>
|
| 320 |
|
|
</tr>
|
| 321 |
|
|
</table>
|
| 322 |
|
|
</blockquote>
|
| 323 |
|
|
|
| 324 |
|
|
<ol>
|
| 325 |
|
|
|
| 326 |
|
|
<li>The data structures are initialized as follows.
|
| 327 |
|
|
|
| 328 |
|
|
<ul><li>The segment’s end point <i>E</i>, control point
|
| 329 |
|
|
<i>C</i>, and start point <i>S</i> are pushed onto the stack.</li>
|
| 330 |
|
|
|
| 331 |
|
|
<li>Currently, the curve in the stack would be approximated by one
|
| 332 |
|
|
single straight line segment (<i>S</i> – <i>E</i>).
|
| 333 |
|
|
Therefore, <code>stackSize</code> is set to 1.</li>
|
| 334 |
|
|
|
| 335 |
|
|
<li>This single straight line segment is approximating the original
|
| 336 |
|
|
curve, which can be seen as the result of zero recursive
|
| 337 |
|
|
splits. Therefore, <code>recLevel[0]</code> is set to
|
| 338 |
|
|
zero.</li></ul>
|
| 339 |
|
|
|
| 340 |
|
|
Column A shows the state after the initialization step.</li>
|
| 341 |
|
|
|
| 342 |
|
|
<li>The algorithm proceeds by taking the topmost curve segment
|
| 343 |
|
|
(<i>S</i> – <i>C</i> – <i>E</i>) from the stack.
|
| 344 |
|
|
|
| 345 |
|
|
<ul><li>The recursion level of this segment (stored in
|
| 346 |
|
|
<code>recLevel[0]</code>) is zero, which is smaller than
|
| 347 |
|
|
the limit 2.</li>
|
| 348 |
|
|
|
| 349 |
|
|
<li>The method <code>java.awt.geom.QuadCurve2D.getFlatnessSq</code>
|
| 350 |
|
|
is called to calculate the squared flatness.</li>
|
| 351 |
|
|
|
| 352 |
|
|
<li>For the sake of argument, we assume that the squared flatness is
|
| 353 |
|
|
exceeding the threshold stored in <code>flatnessSq</code>. Thus, the
|
| 354 |
|
|
curve segment <i>S</i> – <i>C</i> – <i>E</i> gets
|
| 355 |
|
|
subdivided into a left and a right half, namely
|
| 356 |
|
|
<i>S<sub>l</sub></i> – <i>C<sub>l</sub></i> –
|
| 357 |
|
|
<i>E<sub>l</sub></i> and <i>S<sub>r</sub></i> –
|
| 358 |
|
|
<i>C<sub>r</sub></i> – <i>E<sub>r</sub></i>. Both halves are
|
| 359 |
|
|
pushed onto the stack, so the left half is now on top.
|
| 360 |
|
|
|
| 361 |
|
|
<br /> <br />The left half starts at the same point
|
| 362 |
|
|
as the original curve, so <i>S<sub>l</sub></i> has the same
|
| 363 |
|
|
coordinates as <i>S</i>. Similarly, the end point of the right
|
| 364 |
|
|
half and of the original curve are identical
|
| 365 |
|
|
(<i>E<sub>r</sub></i> = <i>E</i>). More interestingly, the left
|
| 366 |
|
|
half ends where the right half starts. Because
|
| 367 |
|
|
<i>E<sub>l</sub></i> = <i>S<sub>r</sub></i>, their coordinates need
|
| 368 |
|
|
to be stored only once, which amounts to saving 16 bytes (two
|
| 369 |
|
|
<code>double</code> values) for each iteration.</li></ul>
|
| 370 |
|
|
|
| 371 |
|
|
Column B shows the state after the first iteration.</li>
|
| 372 |
|
|
|
| 373 |
|
|
<li>Again, the topmost curve segment (<i>S<sub>l</sub></i>
|
| 374 |
|
|
– <i>C<sub>l</sub></i> – <i>E<sub>l</sub></i>) is
|
| 375 |
|
|
taken from the stack.
|
| 376 |
|
|
|
| 377 |
|
|
<ul><li>The recursion level of this segment (stored in
|
| 378 |
|
|
<code>recLevel[1]</code>) is 1, which is smaller than
|
| 379 |
|
|
the limit 2.</li>
|
| 380 |
|
|
|
| 381 |
|
|
<li>The method <code>java.awt.geom.QuadCurve2D.getFlatnessSq</code>
|
| 382 |
|
|
is called to calculate the squared flatness.</li>
|
| 383 |
|
|
|
| 384 |
|
|
<li>Assuming that the segment is still not considered
|
| 385 |
|
|
flat enough, it gets subdivided into a left
|
| 386 |
|
|
(<i>S<sub>ll</sub></i> – <i>C<sub>ll</sub></i> –
|
| 387 |
|
|
<i>E<sub>ll</sub></i>) and a right (<i>S<sub>lr</sub></i>
|
| 388 |
|
|
– <i>C<sub>lr</sub></i> – <i>E<sub>lr</sub></i>)
|
| 389 |
|
|
half.</li></ul>
|
| 390 |
|
|
|
| 391 |
|
|
Column C shows the state after the second iteration.</li>
|
| 392 |
|
|
|
| 393 |
|
|
<li>The topmost curve segment (<i>S<sub>ll</sub></i> –
|
| 394 |
|
|
<i>C<sub>ll</sub></i> – <i>E<sub>ll</sub></i>) is popped from
|
| 395 |
|
|
the stack.
|
| 396 |
|
|
|
| 397 |
|
|
<ul><li>The recursion level of this segment (stored in
|
| 398 |
|
|
<code>recLevel[2]</code>) is 2, which is <em>not</em> smaller than
|
| 399 |
|
|
the limit 2. Therefore, a <code>SEG_LINETO</code> (from
|
| 400 |
|
|
<i>S<sub>ll</sub></i> to <i>E<sub>ll</sub></i>) is passed to the
|
| 401 |
|
|
consumer.</li></ul>
|
| 402 |
|
|
|
| 403 |
|
|
The new state is shown in column D.</li>
|
| 404 |
|
|
|
| 405 |
|
|
|
| 406 |
|
|
<li>The topmost curve segment (<i>S<sub>lr</sub></i> –
|
| 407 |
|
|
<i>C<sub>lr</sub></i> – <i>E<sub>lr</sub></i>) is popped from
|
| 408 |
|
|
the stack.
|
| 409 |
|
|
|
| 410 |
|
|
<ul><li>The recursion level of this segment (stored in
|
| 411 |
|
|
<code>recLevel[1]</code>) is 2, which is <em>not</em> smaller than
|
| 412 |
|
|
the limit 2. Therefore, a <code>SEG_LINETO</code> (from
|
| 413 |
|
|
<i>S<sub>lr</sub></i> to <i>E<sub>lr</sub></i>) is passed to the
|
| 414 |
|
|
consumer.</li></ul>
|
| 415 |
|
|
|
| 416 |
|
|
The new state is shown in column E.</li>
|
| 417 |
|
|
|
| 418 |
|
|
<li>The algorithm proceeds by taking the topmost curve segment
|
| 419 |
|
|
(<i>S<sub>r</sub></i> – <i>C<sub>r</sub></i> –
|
| 420 |
|
|
<i>E<sub>r</sub></i>) from the stack.
|
| 421 |
|
|
|
| 422 |
|
|
<ul><li>The recursion level of this segment (stored in
|
| 423 |
|
|
<code>recLevel[0]</code>) is 1, which is smaller than
|
| 424 |
|
|
the limit 2.</li>
|
| 425 |
|
|
|
| 426 |
|
|
<li>The method <code>java.awt.geom.QuadCurve2D.getFlatnessSq</code>
|
| 427 |
|
|
is called to calculate the squared flatness.</li>
|
| 428 |
|
|
|
| 429 |
|
|
<li>For the sake of argument, we again assume that the squared
|
| 430 |
|
|
flatness is exceeding the threshold stored in
|
| 431 |
|
|
<code>flatnessSq</code>. Thus, the curve segment
|
| 432 |
|
|
(<i>S<sub>r</sub></i> – <i>C<sub>r</sub></i> –
|
| 433 |
|
|
<i>E<sub>r</sub></i>) is subdivided into a left and a right half,
|
| 434 |
|
|
namely
|
| 435 |
|
|
<i>S<sub>rl</sub></i> – <i>C<sub>rl</sub></i> –
|
| 436 |
|
|
<i>E<sub>rl</sub></i> and <i>S<sub>rr</sub></i> –
|
| 437 |
|
|
<i>C<sub>rr</sub></i> – <i>E<sub>rr</sub></i>. Both halves
|
| 438 |
|
|
are pushed onto the stack.</li></ul>
|
| 439 |
|
|
|
| 440 |
|
|
The new state is shown in column F.</li>
|
| 441 |
|
|
|
| 442 |
|
|
<li>The topmost curve segment (<i>S<sub>rl</sub></i> –
|
| 443 |
|
|
<i>C<sub>rl</sub></i> – <i>E<sub>rl</sub></i>) is popped from
|
| 444 |
|
|
the stack.
|
| 445 |
|
|
|
| 446 |
|
|
<ul><li>The recursion level of this segment (stored in
|
| 447 |
|
|
<code>recLevel[2]</code>) is 2, which is <em>not</em> smaller than
|
| 448 |
|
|
the limit 2. Therefore, a <code>SEG_LINETO</code> (from
|
| 449 |
|
|
<i>S<sub>rl</sub></i> to <i>E<sub>rl</sub></i>) is passed to the
|
| 450 |
|
|
consumer.</li></ul>
|
| 451 |
|
|
|
| 452 |
|
|
The new state is shown in column G.</li>
|
| 453 |
|
|
|
| 454 |
|
|
<li>The topmost curve segment (<i>S<sub>rr</sub></i> –
|
| 455 |
|
|
<i>C<sub>rr</sub></i> – <i>E<sub>rr</sub></i>) is popped from
|
| 456 |
|
|
the stack.
|
| 457 |
|
|
|
| 458 |
|
|
<ul><li>The recursion level of this segment (stored in
|
| 459 |
|
|
<code>recLevel[2]</code>) is 2, which is <em>not</em> smaller than
|
| 460 |
|
|
the limit 2. Therefore, a <code>SEG_LINETO</code> (from
|
| 461 |
|
|
<i>S<sub>rr</sub></i> to <i>E<sub>rr</sub></i>) is passed to the
|
| 462 |
|
|
consumer.</li></ul>
|
| 463 |
|
|
|
| 464 |
|
|
The new state is shown in column H.</li>
|
| 465 |
|
|
|
| 466 |
|
|
<li>The stack is now empty. The FlatteningPathIterator will fetch the
|
| 467 |
|
|
next segment from the base iterator, and process it.</li>
|
| 468 |
|
|
|
| 469 |
|
|
</ol>
|
| 470 |
|
|
|
| 471 |
|
|
<p>In order to split the most recently pushed segment, the
|
| 472 |
|
|
<code>subdivideQuadratic()</code> method passes <code>stack</code>
|
| 473 |
|
|
directly to
|
| 474 |
|
|
<code>QuadCurve2D.subdivide(double[],int,double[],int,double[],int)</code>.
|
| 475 |
|
|
Because the stack grows towards the beginning of the array, no data
|
| 476 |
|
|
needs to be copied around: <code>subdivide</code> will directly store
|
| 477 |
|
|
the result into the stack, which will have the contents shown to the
|
| 478 |
|
|
right.</p>
|
| 479 |
|
|
|
| 480 |
|
|
</body>
|
| 481 |
|
|
</html>
|
