Provided by: libmath-planepath-perl_129-1_all 
NAME
Math::PlanePath::HilbertSides -- sides of Hilbert curve squares
SYNOPSIS
use Math::PlanePath::HilbertSides;
my $path = Math::PlanePath::HilbertSides->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This path is segments along the sides of the Hilbert curve squares as per
F. M. Dekking, "Recurrent Sets", Advances in Mathematics, volume 44, 1982, pages
79-104, section 4.8 "Hilbert Curve"
The base pattern is N=0 to N=4. That pattern repeats transposed as points N=0,4,8,12,16,
etc.
9 | ...
| |
8 | 64----63 49----48 44----43
| | | | | |
7 | 62 50 47----46----45 42
| | | |
6 | 60----61 56 51----52 40---39,41
| | | | |
5 | 59----58---57,55--54---53,33--34----35 38
| | | |
4 | 32 36,28--37,27
| | | |
3 | 5-----6----7,9---10---11,31--30----29 26
| | | | |
2 | 4-----3 8 13----12 24---23,25
| | | |
1 | 2 14 17----18----19 22
| | | | | |
Y=0 | 0-----1 15----16 20----21
+-------------------------------------------------
X=0 1 2 3 4 5 6 7
If each point of the "HilbertCurve" path is taken to be a unit square the effect here is
to go along the sides of those squares. The square for a segment is on the left or right,
-------3. .
v |
|>
|
2 .
|
|>
^ |
0-------1 .
Some points are visited twice. The first is at X=2,Y=3 which is N=7 and N=9 where the two
segments N=7to8 and N=8to9 overlap. These are consecutive segments, and non-consecutive
segments can overlap too, as for example N=27to28 and N=36to37. Double-visited points
occur also as corners touching, for example at X=4,Y=3 the two N=11 N=31 touch without
overlapping segments.
The HilbertCurve squares fall within 2^k x 2^k blocks and so likewise the segments here.
The right side 1 to 2 and 2 to 3 don't touch the 2^k side. This is so of the base figure
N=0 to N=4 which doesn't touch X=2 and higher levels are unrotated replications so for
example in the N=0 to N=64 shown above X=8 is not touched. This creates rectangular
columns up from the X axis. Likewise rectangular rows across from the Y axis, and both
columns and rows inside.
The sides which are N=0 to N=1 and N=3 to N=4 of the base pattern variously touch in
higher levels giving interesting patterns of squares, shapes, notches, etc.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.
"$path = Math::PlanePath::HilbertSides->new ()"
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 = $path->xy_to_n ($x,$y)"
Return the point number for coordinates "$x,$y". If there's nothing at "$x,$y" then
return "undef".
The curve visits an "$x,$y" twice for various points. The smaller of the two N values
is returned.
"@n_list = $path->xy_to_n_list ($x,$y)"
Return a list of N point numbers for coordinates "$x,$y". Points may have up to two
Ns for a given "$x,$y".
FORMULAS
Coordinates
Difference X-Y is the same here as in the "HilbertCurve". The base pattern here has N=3
at 1,2 whereas the HilbertCurve is 0,1 so X-Y is the same. The two then have the same
pattern of rotate 180 and/or transpose in subsequent replications.
3
|
HilbertSides 2 3----2 HilbertCurve
| |
0----1 0----1
Abs dX,dY
abs(dY) is 1 for a vertical segment and 0 for a horizontal segment. For the curve here it
is
abs(dY) = count 1-bits of N, mod 2 = Thue-Morse binary parity
abs(dX) = 1 - abs(dY) = opposite
This is so for the base pattern N=0,1,2, and also at N=3 turning towards N=4. Replication
parts 1 and 2 are transposes where there is a single extra 1-bit in N and dX,dY are
swapped. Replication part 3 is a 180 degree rotation where there are two extra 1-bits in
N and dX,dY are negated so abs(dX),abs(dY) unchanged.
Turn
The path can turn left or right or go straight forward or 180 degree reverse.
Straight,reverse vs left,right is given by
N num trailing 0 bits turn
--------------------- -----------------------
odd straight or 180 reverse (A096268)
even left or right (A035263)
The path goes straight ahead at 2 and reverses 180 at 8 and all subsequent 2*4^k.
Segments on Axes
The number of line segments on the X and Y axes 0 to 2^k, which is N=0 to 4^k, is
Xsegs[k] = 1/3*2^k + 1/2 + 1/6*(-1)^k
= 1, 1, 2, 3, 6, 11, 22, 43, 86 (A005578)
= Ysegs[k] + 1
Ysegs[k] = 1/3*2^k - 1/2 + 1/6*(-1)^k
= 0, 0, 1, 2, 5, 10, 21, 42, 85,... (A000975)
= binary 101010... k-1 many bits alternating
These counts can be calculated from the curve sub-parts
k odd k even
+---+ . . . .
R |>T T T
. . . +---+---+
|>T |> R<|
o---+ . o . +
The block at the origin is X and Y segments of the k-1 level. For k odd, the X axis then
has a transposed block which means the Y segments of k-1. The Y axis has a 180 degree
rotated block R. The curve is symmetric in mirror image across its start to end so the
count of segments it puts on the Y axis is the same as Y of level k-1.
Xsegs[k] = Xsegs[k-1] + Ysegs[k-1] for k odd
Ysegs[k] = 2*Ysegs[k-1]
Then similarly for k even, but the other way around the 2*Y.
Xsegs[k] = 2*Xsegs[k-1] for k even
Ysegs[k] = Xsegs[k-1] + Ysegs[k-1]
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
<http://oeis.org/A059285> (etc)
A059285 X-Y
A010059 abs(dX), 1 - Thue-Morse binary parity
A010060 abs(dY), Thue-Morse binary parity
A096268 turn 1=straight or reverse, 0=left or right
A035263 turn 0=straight or reverse, 1=left or right
A062880 N values on diagonal X=Y (digits 0,2 in base 4)
A005578 count segments on X axis, level k
A000975 count segments on Y axis, level k
SEE ALSO
Math::PlanePath, Math::PlanePath::HilbertCurve, Math::PlanePath::PeanoDiagonals
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2015, 2016, 2017, 2018, 2019, 2020 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/>.