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, 181(1):3-15, 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 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/>.