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# NAME

Math::PlanePath::GrayCode -- Gray code coordinates

# SYNOPSIS

`````` use Math::PlanePath::GrayCode;

my \$path = Math::PlanePath::GrayCode->new;
my (\$x, \$y) = \$path->n_to_xy (123);``````

# DESCRIPTION

This path is a mapping of N to X,Y using Gray codes.

``````      7  |  63-62 57-56 39-38 33-32
|      |  |        |  |
6  |  60-61 58-59 36-37 34-35
|
5  |  51-50 53-52 43-42 45-44
|      |  |        |  |
4  |  48-49 54-55 40-41 46-47
|
3  |  15-14  9--8 23-22 17-16
|      |  |        |  |
2  |  12-13 10-11 20-21 18-19
|
1  |   3--2  5--4 27-26 29-28
|      |  |        |  |
Y=0 |   0--1  6--7 24-25 30-31
|
+-------------------------
X=0  1  2  3  4  5  6  7``````

The default is the form by Faloutsos which is an X,Y split in binary reflected Gray code.

Christos Faloutsos, "Gray Codes for Partial Match and Range Queries", IEEE Transactions on Software Engineering (TSE), volume 14, number 10, October 1988, pages 1381-1393. DOI 10.1109/32.6184

N is converted to a Gray code, then split by bits to X,Y, and those X,Y converted back from Gray to integer indices. Stepping from N to N+1 changes just one bit of the Gray code and therefore changes just one of X or Y each time.

Y axis N=0,3,12,15,48,etc are values with only digits 0,3 in base 4. X axis N=0,1,6,7,24,25,etc are values 2k and 2k+1 where k uses only digits 0,3 in base 4.

The default is binary. Option `radix => \$r` can select another radix. This radix is used for both the Gray code and the digit splitting. For example `radix => 4`,

``````    radix => 4

|
127-126-125-124  99--98--97--96--95--94--93--92  67--66--65--64
|   |                           |   |
120-121-122-123 100-101-102-103  88--89--90--91  68--69--70--71
|                           |   |                           |
119-118-117-116 107-106-105-104  87--86--85--84  75--74--73--72
|   |                           |   |
112-113-114-115 108-109-110-111  80--81--82--83  76--77--78--79

15--14--13--12  19--18--17--16  47--46--45--44  51--50--49--48
|   |                           |   |
8-- 9--10--11  20--21--22--23  40--41--42--43  52--53--54--55
|                           |   |                           |
7-- 6-- 5-- 4  27--26--25--24  39--38--37--36  59--58--57--56
|   |                           |   |
0-- 1-- 2-- 3  28--29--30--31--32--33--34--35  60--61--62--63``````

## Apply Type

Option `apply_type => \$str` controls how Gray codes are applied to N and X,Y. It can be one of

``````    "TsF"    to Gray, split, from Gray  (default)
"Ts"     to Gray, split
"Fs"     from Gray, split
"FsT"    from Gray, split, to Gray
"sT"    split, to Gray
"sF"    split, from Gray``````

"T" means integer-to-Gray, "F" means integer-from-Gray, and omitted means no transformation. For example the following is "Ts" which means N to Gray then split, leaving Gray coded values for X,Y.

``````    apply_type => "Ts"

7  |  51--50  52--53  44--45  43--42
|       |       |       |       |
6  |  48--49  55--54  47--46  40--41
|
5  |  60--61  59--58  35--34  36--37  ...-66
|       |       |       |       |       |
4  |  63--62  56--57  32--33  39--38  64--65
|
3  |  12--13  11--10  19--18  20--21
|       |       |       |       |
2  |  15--14   8-- 9  16--17  23--22
|
1  |   3-- 2   4-- 5  28--29  27--26
|       |       |       |       |
Y=0 |   0-- 1   7-- 6  31--30  24--25
|
+---------------------------------
X=0   1   2   3   4   5   6   7``````

This "Ts" is quite attractive because a step from N to N+1 changes just one bit in X or Y alternately, giving 2-D single-bit changes. For example N=19 at X=4 followed by N=20 at X=6 is a single bit change in X.

N=0,2,8,10,etc on the leading diagonal X=Y is numbers using only digits 0,2 in base 4. N=0,3,15,12,etc on the Y axis is numbers using only digits 0,3 in base 4, but in a Gray code order.

The "Fs", "FsT" and "sF" forms effectively treat the input N as a Gray code and convert from it to integers, either before or after split. For "Fs" the effect is little Z parts in various orientations.

``````    apply_type => "sF"

7  |  32--33  37--36  52--53  49--48
|    /       \       /       \
6  |  34--35  39--38  54--55  51--50
|
5  |  42--43  47--46  62--63  59--58
|    \       /       \       /
4  |  40--41  45--44  60--61  57--56
|
3  |   8-- 9  13--12  28--29  25--24
|    /       \       /       \
2  |  10--11  15--14  30--31  27--26
|
1  |   2-- 3   7-- 6  22--23  19--18
|    \       /       \       /
Y=0 |   0-- 1   5-- 4  20--21  17--16
|
+---------------------------------
X=0   1   2   3   4   5   6   7``````

## Gray Type

The `gray_type` option selects the type of Gray code. The choices are

``````    "reflected"     increment to radix-1 then decrement (default)
"modular"       increment to radix-1 then cycle back to 0``````

For example in decimal,

``````    integer       Gray         Gray
"reflected"   "modular"
-------    -----------   ---------
0            0            0
1            1            1
2            2            2
...          ...          ...
8            8            8
9            9            9
10           19           19
11           18           10
12           17           11
13           16           12
14           15           13
...          ...          ...
17           12           16
18           11           17
19           10           18``````

Notice on reaching "19" the reflected type runs the least significant digit back down from 9 to 0, which is a reverse or reflection of the preceding 0 to 9 upwards. The modular form instead continues to increment that least significant digit, wrapping around from 9 to 0.

In binary, the modular and reflected forms are the same (see "Equivalent Combinations" below).

There's various other systematic ways to make a Gray code changing a single digit successively. But many ways are implicitly based on a pre-determined fixed number of bits or digits, which doesn't suit an unlimited path like here.

## Equivalent Combinations

Some option combinations are equivalent,

``````    condition                  equivalent
---------                  ----------
and TsF==Fs, Ts==FsT

radix>2 odd, reflected     TsF==FsT, Ts==Fs, sT==sF
because T==F

In radix=2 binary, the "modular" and "reflected" Gray codes are the same because there's only digits 0 and 1 so going forward or backward is the same.

For odd radix and reflected Gray code, the "to Gray" and "from Gray" operations are the same. For example the following table is ternary radix=3. Notice how integer value 012 maps to Gray code 010, and in turn integer 010 maps to Gray code 012. All values are either pairs like that or unchanged like 021.

``````    integer      Gray
"reflected"       (written in ternary)
000        000
001        001
002        002
010        012
011        011
012        010
020        020
021        021
022        022``````

For even radix and reflected Gray code, "TsF" is equivalent to "Fs", and also "Ts" equivalent to "FsT". This arises from the way the reversing behaves when split across digits of two X,Y values. (In higher dimensions such as a split to 3-D X,Y,Z it's not the same.)

The net effect for distinct paths is

``````    condition         distinct combinations
---------         ---------------------
radix=2           four TsF==Fs, Ts==FsT, sT, sF
radix>2 odd       / three reflected TsF==FsT, Ts==Fs, sT==sF
\ six modular TsF, Ts, Fs, FsT, sT, sF
radix>2 even      / four reflected TsF==Fs, Ts==FsT, sT, sF
\ six modular TsF, Ts, Fs, FsT, sT, sF``````

## Peano Curve

In `radix => 3` and other odd radices, the "reflected" Gray type gives the Peano curve (see Math::PlanePath::PeanoCurve). This is since the "reflected" encoding is equivalent to Peano's "xk" and "yk" complementing.

``````    radix => 3, gray_type => "reflected"

|
53--52--51  38--37--36--35--34--33
|   |                   |
48--49--50  39--40--41  30--31--32
|                   |   |
47--46--45--44--43--42  29--28--27
|
6-- 7-- 8-- 9--10--11  24--25--26
|                   |   |
5-- 4-- 3  14--13--12  23--22--21
|   |                   |
0-- 1-- 2  15--16--17--18--19--20``````

# FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.

`\$path = Math::PlanePath::GrayCode->new ()`
`\$path = Math::PlanePath::GrayCode->new (radix => \$r, apply_type => \$str, gray_type => \$str)`

Create and return a new path object.

`(\$x,\$y) = \$path->n_to_xy (\$n)`

Return the X,Y coordinates of point number `\$n` on the path. Points begin at 0 and if `\$n < 0` then the return is an empty list.

`\$n = \$path->n_start ()`

Return the first N on the path, which is 0.

## Level Methods

`(\$n_lo, \$n_hi) = \$path->level_to_n_range(\$level)`

Return `(0, \$radix**(2*\$level) - 1)`.

# FORMULAS

## Turn

The turns in the default binary TsF curve are either to the left +90 or a reverse 180. For example at N=2 the curve turns left, then at N=3 it reverses back 180 to go to N=4. The turn is given by the low zero bits of (N+1)/2,

``````    count_low_0_bits(floor((N+1)/2))
if even then turn 90 left
if odd  then turn 180 reverse``````

Or equivalently

``````    floor((N+1)/2) lowest non-zero digit in base 4,
1 or 3 = turn 90 left
2      = turn 180 reverse``````

The 180 degree reversals are all horizontal. They occur because at those N the three N-1,N,N+1 converted to Gray code have the same bits at odd positions and therefore the same Y coordinate.

See "N to Turn" in Math::PlanePath::KochCurve for similar turns based on low zero bits (but by +60 and -120 degrees).

# OEIS

This path is in Sloane's Online Encyclopedia of Integer Sequences in a few forms,

``````    apply_type="TsF" (="Fs"), radix=2  (the defaults)
A059905    X xor Y
A039963    turn sequence, 1=+90 left, 0=180 reverse
A035263    turn undoubled, at N=2n and N=2n+1
A065882    base4 lowest non-zero,
turn undoubled 1,3=left 2=180rev at N=2n,2n+1
A003159    (N+1)/2 of positions of Left turns,
being n with even number of low 0 bits
A036554    (N+1)/2 of positions of Right turns
being n with odd number of low 0 bits``````

TsF turn sequence goes in pairs, so N=1 and N=2 left then N=3 and N=4 reverse. A039963 includes that repetition, A035263 is just one copy of each and so is the turn at each pair N=2k and N=2k+1. There's many sequences like A065882 which when taken mod2 equal the "count low 0-bits odd/even" which is the same undoubled turn sequence.

``````    apply_type="Ts", radix=2
A309952    X coordinate (XOR bit pairs)

A163233    N values by diagonals, same axis start
A163234     inverse permutation
A163235    N values by diagonals, opp axis start
A163236     inverse permutation
A163242    N sums along diagonals
A163478     those sums divided by 3

A163237    N values by diagonals, same axis, flip digits 2,3
A163238     inverse permutation
A163239    N values by diagonals, opp axis, flip digits 2,3
A163240     inverse permutation

A099896    N values by PeanoCurve radix=2 order
A100280     inverse permutation

A208665    N values on X=Y diagonal, base 9 digits 0,3,6``````

Gray code conversions themselves (not directly offered by the PlanePath code here) are variously

``````    A003188  binary
A014550  binary written in binary
A055975    increments
A006068    inverse, Gray->integer
A128173  ternary reflected (its own inverse)
A105530  ternary modular
A105529    inverse, Gray->integer
A003100  decimal reflected
A174025    inverse, Gray->integer
A098488  decimal modular
A226134    inverse, Gray->integer``````

http://user42.tuxfamily.org/math-planepath/index.html