NAME
Math::PlanePath::SierpinskiCurveStair -- Sierpinski curve with stair-step diagonals
SYNOPSIS
use Math::PlanePath::SierpinskiCurveStair;
my $path = Math::PlanePath::SierpinskiCurveStair->new (arms => 2);
my ($x, $y) = $path->n_to_xy (123);DESCRIPTION
This is a variation on the SierpinskiCurve with stair-step diagonal parts.
10 | 52-53
| | |
9 | 50-51 54-55
| | |
8 | 49-48 57-56
| | |
7 | 42-43 46-47 58-59 62-63
| | | | | | |
6 | 40-41 44-45 60-61 64-65
| | |
5 | 39-38 35-34 71-70 67-66
| | | | | | |
4 | 12-13 37-36 33-32 73-72 69-68 92-93
| | | | | | |
3 | 10-11 14-15 30-31 74-75 90-91 94-95
| | | | | | |
2 | 9--8 17-16 29-28 77-76 89-88 97-96
| | | | | | |
1 | 2--3 6--7 18-19 22-23 26-27 78-79 82-83 86-87 98-99
| | | | | | | | | | | | |
Y=0 | 0--1 4--5 20-21 24-25 80-81 84-85 ...
|
+-------------------------------------------------------------
^
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19The tiling is the same as the SierpinskiCurve, but each diagonal is a stair step horizontal and vertical. The correspondence is
SierpinskiCurve SierpinskiCurveStair
7-- 12--
/ |
6 10-11
| |
5 9--8
\ |
1--2 4 2--3 6--7
/ \ / | | |
0 3 0--1 4--5So the SierpinskiCurve N=0 to N=1 diagonal corresponds to N=0 to N=2 here, and N=2 to N=3 corresponds to N=3 to N=5. The join section N=3 to N=4 gets an extra point at N=6 here, and later similar N=19, etc.
Diagonal Length
The diagonal_length option can make longer diagonals, still in stair-step style. For example diagonal_length => 4,
10 | 36-37
| | |
9 | 34-35 38-39
| | |
8 | 32-33 40-41
| | |
7 | 30-31 42-43
| | |
6 | 28-29 44-45
| | |
5 | 27-26 47-46
| | |
4 | 8--9 25-24 49-48 ...
| | | | | |
3 | 6--7 10-11 23-22 51-50 62-63
| | | | | |
2 | 4--5 12-13 21-20 53-52 60-61
| | | | | |
1 | 2--3 14-15 18-19 54-55 58-59
| | | | | |
Y=0 | 0--1 16-17 56-57
|
+------------------------------------------------------
^
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17The length is reckoned from N=0 to the end of the first side N=8, which is X=1 to X=5 for length 4 units.
Arms
The optional arms parameter can give up to eight copies of the curve, each advancing successively. For example arms => 8,
98-90 66-58 57-65 89-97 5
| | | | | |
99 82-74 50-42 41-49 73-81 96 4
| | | |
91-83 26-34 33-25 80-88 3
| | | |
67-75 18-10 9-17 72-64 2
| | | |
59-51 27-19 2 1 16-24 48-56 1
| | | | | |
43-35 11--3 . 0--8 32-40 <- Y=0
44-36 12--4 7-15 39-47 -1
| | | | | |
60-52 28-20 5 6 23-31 55-63 -2
| | | |
68-76 21-13 14-22 79-71 -3
| | | |
92-84 29-37 38-30 87-95 -4
| |
85-77 53-45 46-54 78-86 -5
| | | | | |
93 69-61 62-70 94 -6
^
-6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6The multiplies of 8 (or however many arms) N=0,8,16,etc is the original curve, and the further mod 8 parts are the copies.
The middle "." shown is the origin X=0,Y=0. It would be more symmetrical to have the origin the middle of the eight arms, which would be X=-0.5,Y=-0.5 in the above, but that would give fractional X,Y values. Apply an offset X+0.5,Y+0.5 to centre if desired.
Level Ranges
For diagonal_length = L and reckoning the first diagonal side N=0 to N=2L as level 0, a level extends out to a triangle
Nlevel = ((6L+4)*4^level - 4) / 3
Xlevel = (L+2)*2^level - 1For example level 2 in the default L=1 goes to N=((6*1+4)*4^2-4)/3=52 and Xlevel=(1+2)*2^2-1=11. Or in the L=4 sample above level 1 is N=((6*4+4)*4^1-4)/3=36 and Xlevel=(4+2)*2^1-1=11.
The power-of-4 in Nlevel is per the plain SierpinskiCurve, with factor 2L+1 for the points making the diagonal stair. The "/3" arises from the extra points between replications. They become a power-of-4 series
Nextras = 1+4+4^2+...+4^(level-1) = (4^level-1)/3For example level 1 is Nextras=(4^1-1)/3=1, being point N=6 in the default L=1. Or for level 2 Nextras=(4^2-1)/3=5 at N=6 and N=19,26,33,46.
The curve doesn't visit all the points in the eighth of the plane below the X=Y diagonal. In general Nlevel+1 many points of the triangular area Xlevel*(Xlevel-1)/2 are visited, for a filled fraction which approaches a constant
FillFrac = Nlevel / (Xlevel*(Xlevel-1)/2)
-> 4/3 * (3L+2)/(L+2)^2For example the default L=1 has FillFrac=20/27=0.74. Or L=2 FillFrac=2/3=0.66. As the diagonal length increases the fraction decreases due to the growing holes in the pattern.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.
$path = Math::PlanePath::SierpinskiCurveStair->new ()$path = Math::PlanePath::SierpinskiCurveStair->new (diagonal_length => $L, arms => $A)-
Create and return a new path object.
($x,$y) = $path->n_to_xy ($n)-
Return the X,Y coordinates of point number
$non the path. Points begin at 0 and if$n < 0then the return is an empty list.Fractional positions give an X,Y position along a straight line between the integer positions.
$n = $path->n_start()-
Return 0, the first N in the path.
SEE ALSO
Math::PlanePath, Math::PlanePath::SierpinskiCurve
HOME PAGE
http://user42.tuxfamily.org/math-planepath/index.html
LICENSE
Copyright 2011, 2012 Kevin Ryde
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/>.