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### B.2.4 What makes a "perfect" weave?

So what causes the perfect weave cases to be perfect, and the other cases not to be? In all the perfect cases above, @math{S} and @math{J} are relatively prime (i.e. their greatest common divisor (GCD) is 1). As we mentioned above, @math{S=6} and @math{J=4} have a common factor, which causes the overprinting. Where @math{S} and @math{J} have a GCD of 1, they have no common factor other than 1 and, as a result, no overprinting occurs. If @math{S} and @math{J} are not relatively prime, their common factor will cause overprinting.

We can work out the greatest common divisor of a pair of natural numbers using Euler's algorithm:

- Start with the two numbers: (e.g.) 9, 24
- Swap them if necessary so that the larger one comes first: 24, 9
- Subtract the second number from the first: 15, 9
- Repeat until the first number becomes smaller: 6, 9
- Swap the numbers again, so the larger one comes first: 9, 6
- Subtract again: 3, 6
- Swap: 6, 3
- Subtract: 3, 3
- And again: 0, 3
- When one of the numbers becomes 0, the other number is the GCD of the two numbers you started with.

These repeated subtractions can be done with C's ``%'` operator, so we can write this in C as follows:

    unsigned int
    gcd(unsigned int x, unsigned int y)
    {
        if (y == 0)
            return x;
        while (x != 0) {
            if (y > x)
                swap (&x, &y);
            x %= y;
        }
        return y;
    }

``gcd(S,J)'` will feature quite prominently in our weaving algorithm.

If @math{0 <= j < J}, there should only be a single pair @math{(p, j)} for any given row number. If @math{S} and @math{J} are not relatively prime, this assumption breaks down. (For conciseness, let @math{G=GCD(S,J)}.)

@math{S=8}, @math{J=6}, @math{G=2}:

    0 *-------*-------*-------*-------*-------*
    1       *-------*-------*-------*-------*-------*
    2             *-------*-------*-------*-------*-------*
    3                   *-------*-------*-------*-------*-------*
    4                         ^-------^-------^-------*-------*-------*
    5                               ^-------^-------^-------*-------*-------*

In this case, jets 0, 1 and 2 of pass @math{p+4} collide with jets 3, 4 and 5 of pass @math{p}.

How can we calculate these numbers? Suppose we were to print using fewer jets, say @math{J/G} jets. The greatest common divisor of @math{J/G} and @math{S} is 1, enabling a perfect weave. But to get a perfect weave, we also have to advance the paper by a factor of @math{G} less:

    0 *-------*-------*       -       -       -
    1    *-------*-------*       -       -       -
    2       *-------*-------*       -       -       -
    3          *-------*-------*       -       -       -
    4             *-------*-------*       -       -       -
    5                *-------*-------*       -       -       -

If we left the paper advance alone, we'd get a sparse weave; only one row can be printed every @math{G} rows:

    0 *-------*-------*       -       -       -
    1       *-------*-------*       -       -       -
    2             *-------*-------*       -       -       -
    3                   *-------*-------*       -       -       -
    4                         *-------*-------*       -       -       -
    5                               *-------*-------*       -       -       -
                   ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
                  These rows need filling in.

The rows that would have been printed by the jets we've now omitted (shown as ``-'`) are printed by other jets on later passes.

Let's analyse this. Consider how a pass @math{p} could collide with pass 0. Pass @math{p} starts at offset @math{p*J}. Pass 0 prints at rows which are multiples of @math{S}. If @math{p*J} is exactly divisible by @math{S}, a collision has occurred, unless @math{p*J >= J*S} (which will happen when we finish a pass block).

So, we want to find @math{p} and @math{q} such that @math{p*J=q*S} and @math{p} is minimised. Then @math{p} is the number of rows before a collision, and @math{q} is the number of jets in pass 0 which are not involved in the collision. To do this, we find the lowest common multiple of @math{J} and @math{S}, which is @math{L=J*S/G}. @math{L/J} is the number of rows before a collision, and @math{L/S} is the number of jets in the first pass not involved in the collision.

Thus, we see that the first @math{J/G} rows printed by a given pass are not overprinted by any later pass. However, the rest of the rows printed by pass @math{p} are overprinted by the first @math{J-(J/G)} jets of pass @math{p+(S/G)}. We will use @math{C} to refer to @math{S/G}, the number of rows after which a collision occurs.

Another example:

@math{S=6}, @math{J=9}, @math{G=3}, @math{C=S/G=2}:

    0 *-----*-----*-----*-----*-----*-----*-----*-----*
    1          *-----*-----*-----*-----*-----*-----*-----*-----*
    2                   ^-----^-----^-----^-----^-----^-----*-----*-----*
    3                            ^-----^-----^-----^-----^-----^-----*-----*-----*
    4                                     ^-----^-----^-----^-----^-----^-----*-----
    5                                              ^-----^-----^-----^-----^-----^--
             ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^
                  These rows need filling in.

In this case, the first @math{J-(J/G) = 9-9/3 = 6} jets of pass @math{p+(6/3)=p+2} collide with the last 6 jets of pass @math{p}. Only one row in every @math{G=2} rows is printed by this weave.

@math{S=9}, @math{J=6}, @math{G=3}, @math{C=3}:

    0 *--------*--------*--------*--------*--------*
    1       *--------*--------*--------*--------*--------*
    2             *--------*--------*--------*--------*--------*
    3                   ^--------^--------^--------^--------*--------*
    4                         ^--------^--------^--------^--------*--------*
    5                               ^--------^--------^--------^--------*--------*

Here, the first @math{J-(J/G) = 6-6/3 = 4} jets of pass @math{p+(9/3)=p+3} collide with the last 4 jets of pass @math{p}.

Note that, in these overprinting cases, only rows divisible by @math{G} are ever printed. The other rows, those not divisible by @math{G}, are not touched by this weave.

We can modify our weave pattern to avoid overprinting any rows and simultaneously fill in the missing rows. Instead of using @math{J} alone to determine the start of each pass from the previous pass, we adjust the starting position of some passes. As mentioned before, we will divide the page into pass blocks, with @math{S} passes in each block. This ensures that the first jet of the first pass in a block prints the row which the @math{J}th jet of the first pass of the previous block would have printed, if the print head had one extra jet.

Looking back at an example of a perfect weave, we can divide it into pass blocks:

@math{S=7}, @math{J=2}, @math{G=1}:

                    imaginary extra jet
                    |
    0 *------*      *      <--start of pass block 0
    1   *------*    |
    2     *------*  |
    3       *------*|
    4         *-----|*
    5           *---|--*
    6             *-|----*
                    |
    7               *------*  <--start of pass block 1
    8                 *------*
    9                   *------*

We can now calculate the start of a given pass by reference to its pass block. The first pass of pass block @math{b} always starts at row @math{(b*S*J)}. The start row of each of the other passes in the block are calculated using offsets from this row.

For the example above, there are 7 passes in each pass block, and their offsets are 0, 2, 4, 6, 8, 10 and 12. The next pass block is offset @math{S*J=14} rows from the start of the current pass block.

The simplest way to modify the "perfect" weave pattern to give a correct weave in cases where @math{G!=1} is to simply change any offsets which would result in a collision, until the collision disappears. Every printed row in the weave, as we have shown it up to now, is separated from each of its neighbouring printed rows by @math{G} blank rows. We will add an extra offset to each colliding pass in such a way that we push the pass onto these otherwise blank rows.

We have seen that, unless @math{G=1}, the plain weave pattern results in each pass colliding with the pass @math{S/G} passes before. We will now subdivide our pass block into subblocks, each consisting of @math{B=S/G} passes. There are therefore @math{G} subblocks in a pass block.

For each subblock, the passes in that subblock have a constant offset added to them. The offset is different for each subblock in a block. There are many ways we can choose the offsets, but the simplest is to make the offset equal to the subblock number (starting from 0).

Thus, the passes in the first subblock in each pass block remain at the offsets we've already calculated from @math{J}. The passes in the second subblock each have 1 added to their offset, the passes in the third subblock have 2 added, and so on. Thus, the offset of pass @math{p} (numbered relative to the start of its pass block) is @math{p*J + floor(p/B)}.

This gives us a weave pattern looking like this:

@math{S=6}, @math{J=9}, @math{G=3}, @math{B=2}:

    0 *-----*-----*-----*-----*-----*-----*-----*-----*
    1 ^        *-----*-----*-----*-----*-----*-----*-----*-----*
    2 |              +-> *-----*-----*-----*-----*-----*-----*-----*-----*
    3 |              |            *-----*-----*-----*-----*-----*-----*-----*-----*
    4 |              |                  +-> *-----*-----*-----*-----*-----*-----*---
    5 |              |                  |            *-----*-----*-----*-----*-----*
    6 |              |                  |               +-> *-----*-----*-----*-----
    7 |              |                  |               |            *-----*-----*--
      |              |                  |             start of pass block 1
      |              |                  |             (offset returns to 0)
      |              |                  start of subblock 2 (offset 2 rows)
      |              start of subblock 1 (following passes offset by 1 row)
      start of passblock 0, subblock 0 (pass start calculated as p*J)

@math{S=9}, @math{J=6}, @math{G=3}, @math{B=3}:

    0 *--------*--------*--------*--------*--------*
    1       *--------*--------*--------*--------*--------*
    2             *--------*--------*--------*--------*--------*
    3                    *--------*--------*--------*--------*--------*
    4                          *--------*--------*--------*--------*--------*
    5                                *--------*--------*--------*--------*--------*
    6                                       *--------*--------*--------*--------*---
    7                                             *--------*--------*--------*------
    8                                                   *--------*--------*--------*
    9                                                       *--------*--------*-----
    10                                                  \---/     *--------*--------
    11                                               small offset       *--------*--
    12                                                                         *----

This method of choosing offsets for subblocks can result in an occasional small offset (as shown above) between one pass and the next, particularly when @math{G} is large compared to @math{J}. For example:

@math{S=8}, @math{J=4}, @math{G=4}, @math{B=2}:

    0 *-------*-------*-------*
    1     *-------*-------*-------*
    2          *-------*-------*-------*
    3              *-------*-------*-------*
    4                   *-------*-------*-------*
    5                       *-------*-------*-------*
    6                            *-------*-------*-------*
    7                                *-------*-------*-------*
    8                                 *-------*-------*-------*
    9                                \/   *-------*-------*-------*
                                  very small offset!

We can plot the offset against the subblock number as follows:

    subblock number
    | offset
    | |
    | 0123
    0 *
    1  *
    2   *
    3    *
    0 *
    1  *
    2   *
    3    *

The discontinuity in this plot results in the small offset between passes.

As we said at the beginning, we want the offsets from each pass to the next to be as similar as possible. We can fix this by calculating the offset for a given subblock b as follows:

      offset(b) = 2*b             , if b < ceiling(G/2)
                = 2*(G-b)-1       , otherwise

We can visualise this as follows, for @math{G=10}:

      0123456789
    0 *
    1   *
    2     *
    3       *
    4         *
    5          *
    6        *
    7      *
    8    *
    9  *
    0 *
    1   *
    2     *
    3       *
    4         *
    5          *
    6        *
    7      *
    8    *
    9  *

and for @math{G=11}:

                 1
       01234567890
     0 *
     1   *
     2     *
     3       *
     4         *
     5           *
     6          *
     7        *
     8      *
     9    *
    10  *
     0 *
     1   *
     2     *
     3       *
     4         *
     5           *
     6          *
     7        *
     8      *
     9    *
    10  *

This gives a weave looking like this:

@math{S=12}, @math{J=6}, @math{G=6}, @math{B=2}:

    0 *-----------*-----------*-----------*-----------*-----------*
    1       *-----------*-----------*-----------*-----------*-----------*
    2               *-----------*-----------*-----------*-----------*-----------*
    3                     *-----------*-----------*-----------*-----------*---------
    4                             *-----------*-----------*-----------*-----------*-
    5                                   *-----------*-----------*-----------*-------
    6                                          *-----------*-----------*-----------*
    7                                                *-----------*-----------*------
    8                                                    *-----------*-----------*--
    9                                                          *-----------*--------
    10                                                             *-----------*----
    11                                                                   *----------
    12                                                                        *-----

This method ensures that the offset between passes is always in the range @math{[J-2,J+2]}.

(This might seem odd, but it occurs to me that a good weave pattern might also make a good score for bell ringers. When church bells are rung, a list of "changes" are used. For example, if 8 bells are being used, they will, at first, be rung in order: 12345678. If the first change is for bells 5 and 6, the bells will then be rung in the order 12346578. If the second change is 1 and 2, the next notes are 21346578. After a long list of changes, the order the bells are rung in can become quite complex.

For a group of bell-ringers to change the order of the notes, they must each either delay their bell's next ring, hasten it, or keep it the same as the time it takes to ring all the bells once. The length of time between each ring of a given bell can only be changed a little each time, though; with an ink-jet weave pattern, we want the same to apply to the distance between passes.)

Finally, knowing the number of jets @math{J} and their separation @math{S}, we can calculate the starting row of any given pass @math{p} as follows:

    passesperblock = S
    passblock = floor(p / passesperblock)
    offsetinpassblock = p - passblock * passesperblock
    subblocksperblock = gcd(S, J)
    passespersubblock = S / subblocksperblock
    subpassblock = floor(offsetinpassblock / passespersubblock)
    if subpassblock < ceiling(subblocksperblock/2)
        subblockoffset = 2*subpassblock
    else
        subblockoffset = 2*(subblocksperblock-subpassblock)-1
    startingrow = passblock * S * J + offsetinpassblock * J + subblockoffset

We can simplify this down to the following:

    subblocksperblock = gcd(S, J)
    subpassblock = floor((p % S) * subblocksperblock / S)
    if subpassblock * 2 < subblocksperblock
        subblockoffset = 2*subpassblock
    else
        subblockoffset = 2*(subblocksperblock-subpassblock)-1
    startingrow = p * J + subblockoffset

So the row number of jet @math{j} of pass @math{p} is

    subblocksperblock = gcd(S, J)

    subblockoffset(p)
        = 2*subpassblock       , if subpassblock * 2 < subblocksperblock
        = 2*(subblocksperblock-subpassblock)-1      , otherwise
          where
          subpassblock = floor((p % S) * subblocksperblock / S)

    row(j, p) = p * J + subblockoffset(p) + j * S

Together with the inequality @math{0 <= j < J}, we can use this definition in reverse to calculate the pass number containing a given row, @math{r}. Working out the inverse definition involves a little guesswork, but one possible result is as follows. Given a row, @math{r}, which is known to be the first row of a pass, we can calculate the pass number as follows:

    subblocksperblock = gcd(S, J)
    subblockoffset = r % subblocksperblock
    pass = (r - subblockoffset) / J

If @math{G==1}, we can determine the pass number with this algorithm:

    offset = r % J
    pass = (r - offset) / J
    while (offset % S != 0)
    {
      pass--
      offset += J
    }
    jet = offset / S

Generalising, we come up with this algorithm. Given @math{r}, @math{S} and @math{J}:

    G = gcd(S, J)
    passespersubblock = S/G
    subblockoffset = r % G
    subpassblock = subblockoffset / 2  , if subblockoffset % 2 == 0
                 = G - (subblockoffset+1)/2    , otherwise
    baserow = r - subblockoffset - (subpassblock * passespersubblock * J)
    offset = baserow % J
    pass = (baserow - offset) / J
    while (offset % S != 0)
    {
      offset += J
      pass -= 1
    }
    subblockretreat = floor(pass / passespersubblock) % G
    pass -= subblockretreat * passespersubblock
    pass += subpassblock * passespersubblock
    jet = (r - subblockoffset - pass * J) / S

Let's look at some examples of imperfect but correct weave patterns:

@math{S=6}, @math{J=4}, @math{GCD=2},
passesperblock=@math{S}=6,
passespersubblock=@math{S/G}=6/2=3:

    0 *-----*-----*-----*
    1     *-----*-----*-----*
    2         *-----*-----*-----*
    3              *-----*-----*-----*
    4                  *-----*-----*-----*
    5                      *-----*-----*-----*
    6                         *-----*-----*-----*
    7                             *-----*-----*-----*
    8                                 *-----*-----*-----*
    9                                      *-----*-----*-----*
    10                                         *-----*-----*-----*
    11                                             *-----*-----*-----*
    12                                                *-----*-----*-----*
    13                                                    *-----*-----*-----*
    14                                                        *-----*-----*-----*
    15                                                             *-----*-----*----
    16                                                                 *-----*-----*
    17                                                                     *-----*--
    18                                                                        *-----
    19                                                                            *-

@math{S=8}, @math{J=6}, @math{G=2},
passesperblock=@math{S}=8,
passespersubblock=@math{S/G}=8/2=4:

    0 *-------*-------*-------*-------*-------*
    1       *-------*-------*-------*-------*-------*
    2             *-------*-------*-------*-------*-------*
    3                   *-------*-------*-------*-------*-------*
    4                          *-------*-------*-------*-------*-------*
    5                                *-------*-------*-------*-------*-------*
    6                                      *-------*-------*-------*-------*-------*
    7                                            *-------*-------*-------*-------*--
    8                                                 *-------*-------*-------*-----
    9                                                       *-------*-------*-------
    10                                                            *-------*-------*-
    11                                                                  *-------*---
    12                                                                         *----

@math{S=6}, @math{J=12}, @math{G=6},
passesperblock=@math{S}=6,
passespersubblock=@math{S/G}=6/6=1:

    0 *-----*-----*-----*-----*-----*-----*-----*-----*-----*-----*-----*
    1               *-----*-----*-----*-----*-----*-----*-----*-----*-----*-----*---
    2                             *-----*-----*-----*-----*-----*-----*-----*-----*-
    3                                          *-----*-----*-----*-----*-----*-----*
    4                                                    *-----*-----*-----*-----*--
    5                                                              *-----*-----*----
    6                                                                         *-----

We have now solved the basic weaving problem. There are two further refinements we need to consider: oversampling, and filling in the missing rows at the start of the weave.

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