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 is a mapping of N to X,Y using Gray codes. The default is the form by
Christos Faloutsos which is an X,Y split in binary reflected Gray code.
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
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.
Radix¶
The default is binary. The "radix => $r" option can select another
radix. This 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 what type of Gray code is used. The
choices are
"reflected" increment to radix-1 then decrement (default)
"modular" cycle from radix-1 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 downwards 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 as given
here.
Equivalent Combinations¶
Some option combinations are equivalent,
condition equivalent
--------- ----------
radix=2 modular==reflected
and TsF==Fs, Ts==FsT
radix>2 odd reflected TsF==FsT, Ts==Fs, sT==sF
because T==F
radix>2 even reflected TsF==Fs, Ts==FsT
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). 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)".
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", radix=2 (the defaults)
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
The 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="sF", radix=2
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
apply_type="FsT", radix=3, gray_type=modular
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 with values written in binary
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
SEE ALSO¶
Math::PlanePath, Math::PlanePath::ZOrderCurve, Math::PlanePath::PeanoCurve,
Math::PlanePath::CornerReplicate
HOME PAGE¶
<
http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE¶
Copyright 2011, 2012, 2013, 2014 Kevin Ryde
This file is part of Math-PlanePath.
Math-PlanePath is free software; you can redistribute it and/or modify it under
the terms of the GNU General Public License as published by the Free Software
Foundation; either version 3, or (at your option) any later version.
Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with
Math-PlanePath. If not, see <
http://www.gnu.org/licenses/>.