oracular (3) Math::PlanePath::AR2W2Curve.3pm.gz
NAME
Math::PlanePath::AR2W2Curve -- 2x2 self-similar curve of four patterns
SYNOPSIS
use Math::PlanePath::AR2W2Curve;
my $path = Math::PlanePath::AR2W2Curve->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This is an integer version of the AR2W2 curve per
Asano, Ranjan, Roos, Welzl and Widmayer "Space-Filling Curves and Their Use in the Design of
Geometric Data Structures", Theoretical Computer Science, volume 181, issue 1, pages 3-15, July 1997.
And in LATIN'95 Theoretical Informatics which is at Google Books
<http://books.google.com.au/books?id=_aKhJUJunYwC&pg=PA36>
It traverses the first quadrant in self-similar 2x2 blocks which are a mixture of "U" and "Z" shapes.
The mixture is designed to improve some locality measures (how big the N range for a given region).
|
7 42--43--44 47--48--49 62--63
\ | | | |
6 40--41 45--46 51--50 61--60
| | |
5 39 36--35--34 52 55--56 59
| | / | | | |
4 38--37 33--32 53--54 57--58
\
3 6-- 7-- 8 10 31 28--27--26
| |/ | | | |
2 5-- 4 9 11 30--29 24--25
| | |
1 2-- 3 13--12 17--18 23--22
\ | | | |
Y=0 -> 0-- 1 14--15--16 19--20--21
X=0 1 2 3 4 5 6 7
Shape Parts
There's four base patterns A to D. A2 is a mirror image of A1, B2 a mirror of B1, etc. The start is A1,
and above that D2, then A1 again, alternately.
^----> ^
2---3 C1 | B2 1 3 C2 D1 |
A1 \ | A2 | \ | ----> |
0---1 ^ 0 2 ^ ---->
D2 | B1 |B1 B2
---->| |
1---2 C2 B1 1---2 B2 C1
B1 | | ---->----> B2 | | ---->---->
0 3 ^ | 0 3 ^ |
|D1 B2| |B1 D2|
| v | v
^ \ ^ |
1---2 B1| \A1 1---2 A2/ | B2
C1 | | | v C2 | | / v
0 3 ^ | 0 3 ^ \
/A2 B2| |B1 \A1
/ v | v
^ | ^ \
1---2 A2/ | C2 1---2 C1| \A1
D1 | | / v D2 | | | v
0 3 ^ \ 0 3 ^ |
|D1 \A2 /A1 D2|
| v / v
For parts which fill on the right such as the B1 and B2 sub-parts of A1, the numbering must be reversed.
This doesn't affect the shape of the curve as such, but it matters for enumerating it as done here.
Start Shape
The default starting shape is the A1 "Z" part, and above it D2. Notice the starting sub-part of D2 is A1
and in turn the starting sub-part of A1 is D2, so those two alternate at successive higher levels. Their
sub-parts reach all other parts (in all directions, and forward or reverse).
The "start_shape => $str" option can select a different starting shape. The choices are
"A1" \ pair
"D2" /
"B2" \ pair
"B1rev" /
"D1rev" \ pair
"A2rev" /
B2 begins with a reversed B1 and in turn a B1 reverse begins with B2 (no reverse), so those two
alternate. Similarly D1 reverse starts with A2 reverse, and A2 reverse starts with D1 reverse.
The curve is conceived by the authors as descending into ever-smaller sub-parts and for that any of the
patterns can be a top-level start. But to expand outwards as done here the starting part must be the
start of the pattern above it, and that's so only for the 6 listed. The descent graph is
D2rev -----> D2 <--> A1
B2rev ----->
C2rev --> A1rev -----> B2 <--> B1rev <----- C2
C1rev -----> <----- A2 <-- C1
B1 -----> D1rev <--> A2rev
D1 ----->
So for example B1 is not at the start of anything. Or A1rev is at the start of C2rev, but then nothing
starts with C2rev. Of the 16 total only the three pairs shown "<-->" are cycles and can thus extend
upwards indefinitely.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::AR2W2Curve->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_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)"
The returned range is exact, meaning $n_lo and $n_hi are the smallest and largest in the rectangle.
Level Methods
"($n_lo, $n_hi) = $path->level_to_n_range($level)"
Return "(0, 4**$level - 1)".
SEE ALSO
Math::PlanePath, Math::PlanePath::HilbertCurve, Math::PlanePath::PeanoCurve
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2011, 2012, 2013, 2014, 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/>.