Provided by: libmath-planepath-perl_125-1_all 
NAME
Math::PlanePath::AlternatePaperMidpoint -- alternate paper folding midpoints
SYNOPSIS
use Math::PlanePath::AlternatePaperMidpoint;
my $path = Math::PlanePath::AlternatePaperMidpoint->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This is the midpoints of each alternate paper folding curve
(Math::PlanePath::AlternatePaper).
8 | 64-65-...
| |
7 | 63
| |
6 | 20-21 62
| | | |
5 | 19 22 61-60-59
| | | |
4 | 16-17-18 23 56-57-58
| | | |
3 | 15 26-25-24 55 50-49-48-47
| | | | | |
2 | 4--5 14 27-28-29 54 51 36-37 46
| | | | | | | | | |
1 | 3 6 13-12-11 30 53-52 35 38 45-44-43
| | | | | | | |
Y=0 | 0--1--2 7--8--9-10 31-32-33-34 39-40-41-42
+----------------------------------------------
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
The "AlternatePaper" curve begins as follows and the midpoints are numbered from 0,
|
9
|
--8--
| |
7 |
| |
--2-- --6--
| | |
1 3 5
| | |
*--0-- --4--
These midpoints are on fractions X=0.5,Y=0, X=1,Y=0.5, etc. For this
"AlternatePaperMidpoint" they're turned 45 degrees and mirrored so the 0,1,2 upward
diagonal becomes horizontal along the X axis, and the 2,3,4 downward diagonal becomes a
vertical at X=2, extending to X=2,Y=2 at N=4.
The midpoints are distinct X,Y positions because the alternate paper curve traverses each
edge only once.
The curve is self-similar in 2^level sections due to its unfolding. This can be seen in
the midpoints as for example N=0 to N=16 above is the same shape as N=16 to N=32, but the
latter rotated +90 degrees and numbered in reverse.
Arms
The midpoints fill an eighth of the plane and eight copies can mesh together perfectly
when mirrored and rotated by 90, 180 and 270 degrees. The "arms" parameter can choose 1
to 8 curve arms successively advancing.
For example "arms => 8" begins as follows. N=0,8,16,24,etc is the first arm, the same as
the plain curve above. N=1,9,17,25 is the second, N=2,10,18,26 the third, etc.
90-82 81-89 7
arms => 8 | | | |
... 74 73 ... 6
| |
66 65 5
| |
43-35 42-50-58 57-49-41 4
| | | |
91-.. 51 27 34-26-18 17-25-33 3
| | | | |
83-75-67-59 19-11--3 10 9 32-40 2
| | | |
84-76-68-60 20-12--4 2 1 24 48 ..-88 1
| | | | | |
92-.. 52 28 5 6 0--8-16 56-64-72-80 <- Y=0
| | | |
44-36 13 14 7-15-23 63-71-79-87 -1
| | | | |
37-29-21 22-30-38 31 55 ..-95 -2
| | | |
45-53-61 62-54-46 39-47 -3
| |
69 70 -4
| |
... 77 78 ... -5
| | | |
93-85 86-94 -6
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
-7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6
With eight arms like this every X,Y point is visited exactly once, because the 8-arm
"AlternatePaper" traverses every edge exactly once ("Arms" in
Math::PlanePath::AlternatePaper).
The arm numbering doesn't correspond to the "AlternatePaper", due to the rotate and
reflect of the first arm. It ends up arms 0 and 1 of the "AlternatePaper" corresponding
to arms 7 and 0 of the midpoints here, those two being a pair going horizontally
corresponding to a pair in the "AlternatePaper" going diagonally into a quadrant.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::AlternatePaperMidpoint->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.
Fractional positions give an X,Y position along a straight line between the integer
positions.
"$n = $path->n_start()"
Return 0, the first N in the path.
Level Methods
"($n_lo, $n_hi) = $path->level_to_n_range($level)"
Return "(0, 2**$level - 1)", or for multiple arms return "(0, $arms * (2**$level -
1)*$arms)". This is the same as the "DragonMidpoint".
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
<http://oeis.org/A016116> (etc)
A016116 X/2 at N=2^k, being X/2=2^floor(k/2)
SEE ALSO
Math::PlanePath, Math::PlanePath::AlternatePaper
Math::PlanePath::DragonMidpoint, Math::PlanePath::R5DragonMidpoint,
Math::PlanePath::TerdragonMidpoint
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2012, 2013, 2014, 2015, 2016, 2017 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/>.
perl v5.26.1 2018-01-0Math::PlanePath::AlternatePaperMidpoint(3pm)