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b/libjava/classpath/java/awt/geom/doc-files/FlatteningPathIterator-1.html new file mode 100644 index 000000000..5a52d693e --- /dev/null +++ b/libjava/classpath/java/awt/geom/doc-files/FlatteningPathIterator-1.html @@ -0,0 +1,481 @@ + + + + + The GNU Implementation of java.awt.geom.FlatteningPathIterator + + + + + +

The GNU Implementation of FlatteningPathIterator

+ +

Sascha +Brawer, November 2003

+ +

This document describes the GNU implementation of the class +java.awt.geom.FlatteningPathIterator. It does +not 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.

+ + +

Data Structures

+ +

The algorithm uses a stack. Its allocation is delayed to the time +when the source path iterator actually returns the first curved +segment (either SEG_QUADTO or SEG_CUBICTO). +If the input path does not contain any curved segments, the value of +the stack variable stays null. In this quite +common case, the memory consumption is minimal.

+ +
stack
The variable stack is +a double array that holds the start, control and end +points of individual sub-segments.
+ +
recLevel
The variable recLevel +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.
+ +
stackSize
Finally, the variable +stackSize indicates how many sub-segments are stored on +the stack.
+ +

Algorithm

+ +

The implementation separately processes each segment that the +base iterator returns.

+ +

In the case of SEG_CLOSE, +SEG_MOVETO and SEG_LINETO segments, the +implementation simply hands the segment to the consumer, without actually +doing anything.

+ +

Any SEG_QUADTO and SEG_CUBICTO 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 SEG_QUADTO and +SEG_CUBICTO segments, it is recursively flattened as +follows:

+ +
  1. 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 recLevel of zero.
    2. + +
  2. If the stack is empty, flattening the segment is complete, +and the next segment is fetched from the base iterator.
    3. + +
  3. If the stack is not empty, pop a curve segment from the +stack. + +
    • If its recLevel exceeds the recursion limit, + hand the current segment to the consumer.
      • + +
    • Calculate the squared flatness of the segment. If it smaller + than flatnessSq, hand the current segment to the + consumer.
      • + +
    • 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.
    4. +
+ +

The implementation is slightly complicated by the fact that +consumers pull the flattened segments from the +FlatteningPathIterator. 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 +push paradigm.

+ + +

Example

+ +

The following example shows how a +FlatteningPathIterator processes a +SEG_QUADTO segment. It is (arbitrarily) assumed that the +recursion limit was set to 2.

+ +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
ABCDEFGH
stack[0]——Sll.x—————
stack[1]——Sll.y—————
stack[2]——Cll.x—————
stack[3]——Cll.y—————
stack[4]—Sl.xEll.x + = Slr.xSlr.x—Srl.x——
stack[5]—Sl.yEll.x + = Slr.ySlr.y—Srl.y——
stack[6]—Cl.xClr.xClr.x—Crl.x——
stack[7]—Cl.yClr.yClr.y—Crl.y——
stack[8]S.xEl.x + = Sr.xElr.x + = Sr.xElr.x + = Sr.xSr.xErl.x + = Srr.xSrr.x—
stack[9]S.yEl.y + = Sr.yElr.y + = Sr.yElr.y + = Sr.ySr.yErl.y + = Srr.ySrr.y—
stack[10]C.xCr.xCr.xCr.xCr.xCrr.xCrr.x—
stack[11]C.yCr.yCr.yCr.yCr.yCrr.yCrr.y—
stack[12]E.xEr.xEr.xEr.xEr.xErr.xErr.x—
stack[13]E.yEr.yEr.yEr.yEr.yErr.yErr.x—
stackSize12321210
recLevel[2]——2—————
recLevel[1]—122—2——
recLevel[0]0111122—
+
+ +
    + +
  1. The data structures are initialized as follows. + +
    • The segment’s end point E, control point +C, and start point S are pushed onto the stack.
      • + +
    • Currently, the curve in the stack would be approximated by one + single straight line segment (S – E). + Therefore, stackSize is set to 1.
      • + +
    • This single straight line segment is approximating the original + curve, which can be seen as the result of zero recursive + splits. Therefore, recLevel[0] is set to + zero.
    + +Column A shows the state after the initialization step.
    2. + +
  2. The algorithm proceeds by taking the topmost curve segment +(S – C – E) from the stack. + +
    • The recursion level of this segment (stored in + recLevel[0]) is zero, which is smaller than + the limit 2.
      • + +
    • The method java.awt.geom.QuadCurve2D.getFlatnessSq + is called to calculate the squared flatness.
      • + +
    • For the sake of argument, we assume that the squared flatness is + exceeding the threshold stored in flatnessSq. Thus, the + curve segment S – C – E gets + subdivided into a left and a right half, namely + Sl – Cl – + El and Sr – + Cr – Er. Both halves are + pushed onto the stack, so the left half is now on top. + +
       
      The left half starts at the same point + as the original curve, so Sl has the same + coordinates as S. Similarly, the end point of the right + half and of the original curve are identical + (Er = E). More interestingly, the left + half ends where the right half starts. Because + El = Sr, their coordinates need + to be stored only once, which amounts to saving 16 bytes (two + double values) for each iteration.
    + +Column B shows the state after the first iteration.
    3. + +
  3. Again, the topmost curve segment (Sl +– Cl – El) is +taken from the stack. + +
    • The recursion level of this segment (stored in + recLevel[1]) is 1, which is smaller than + the limit 2.
      • + +
    • The method java.awt.geom.QuadCurve2D.getFlatnessSq + is called to calculate the squared flatness.
      • + +
    • Assuming that the segment is still not considered + flat enough, it gets subdivided into a left + (Sll – Cll – + Ell) and a right (Slr + – Clr – Elr) + half.
    + +Column C shows the state after the second iteration.
    4. + +
  4. The topmost curve segment (Sll – +Cll – Ell) is popped from +the stack. + +
    • The recursion level of this segment (stored in + recLevel[2]) is 2, which is not smaller than + the limit 2. Therefore, a SEG_LINETO (from + Sll to Ell) is passed to the + consumer.
    + + The new state is shown in column D.
    5. + + +
  5. The topmost curve segment (Slr – +Clr – Elr) is popped from +the stack. + +
    • The recursion level of this segment (stored in + recLevel[1]) is 2, which is not smaller than + the limit 2. Therefore, a SEG_LINETO (from + Slr to Elr) is passed to the + consumer.
    + + The new state is shown in column E.
    6. + +
  6. The algorithm proceeds by taking the topmost curve segment +(Sr – Cr – +Er) from the stack. + +
    • The recursion level of this segment (stored in + recLevel[0]) is 1, which is smaller than + the limit 2.
      • + +
    • The method java.awt.geom.QuadCurve2D.getFlatnessSq + is called to calculate the squared flatness.
      • + +
    • For the sake of argument, we again assume that the squared + flatness is exceeding the threshold stored in + flatnessSq. Thus, the curve segment + (Sr – Cr – + Er) is subdivided into a left and a right half, + namely + Srl – Crl – + Erl and Srr – + Crr – Err. Both halves + are pushed onto the stack.
    + + The new state is shown in column F.
    7. + +
  7. The topmost curve segment (Srl – +Crl – Erl) is popped from +the stack. + +
    • The recursion level of this segment (stored in + recLevel[2]) is 2, which is not smaller than + the limit 2. Therefore, a SEG_LINETO (from + Srl to Erl) is passed to the + consumer.
    + + The new state is shown in column G.
    8. + +
  8. The topmost curve segment (Srr – +Crr – Err) is popped from +the stack. + +
    • The recursion level of this segment (stored in + recLevel[2]) is 2, which is not smaller than + the limit 2. Therefore, a SEG_LINETO (from + Srr to Err) is passed to the + consumer.
    + + The new state is shown in column H.
    9. + +
  9. The stack is now empty. The FlatteningPathIterator will fetch the +next segment from the base iterator, and process it.
    10. + +
+ +

In order to split the most recently pushed segment, the +subdivideQuadratic() method passes stack +directly to +QuadCurve2D.subdivide(double[],int,double[],int,double[],int). +Because the stack grows towards the beginning of the array, no data +needs to be copied around: subdivide will directly store +the result into the stack, which will have the contents shown to the +right.

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