NAME¶
Math::PlanePath::AztecDiamondRings -- rings around an Aztec diamond shape
SYNOPSIS¶
use Math::PlanePath::AztecDiamondRings;
my $path = Math::PlanePath::AztecDiamondRings->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION¶
This path makes rings around an Aztec diamond shape,
46-45 4
/ \
47 29-28 44 3
/ / \ \
48 30 16-15 27 43 ... 2
/ / / \ \ \ \
49 31 17 7--6 14 26 42 62 1
/ / / / \ \ \ \ \
50 32 18 8 2--1 5 13 25 41 61 <- Y=0
| | | | | | | | | |
51 33 19 9 3--4 12 24 40 60 -1
\ \ \ \ / / / /
52 34 20 10-11 23 39 59 -2
\ \ \ / / /
53 35 21-22 38 58 -3
\ \ / /
54 36-37 57 -4
\ /
55-56 -5
^
-5 -4 -3 -2 -1 X=0 1 2 3 4 5
This is similar to the "DiamondSpiral", but has all four corners
flattened to 2 vertical or horizontal, instead of just one in the
"DiamondSpiral". This is only a small change to the alignment of
numbers in the sides, but is more symmetric.
Y axis N=1,6,15,28,45,66,etc are the hexagonal numbers k*(2k-1). The hexagonal
numbers of the "second kind" 3,10,21,36,55,78, etc k*(2k+1), are the
vertical at X=-1 going downwards. Combining those two is the triangular
numbers 3,6,10,15,21,etc, k*(k+1)/2, alternately on one line and the other.
Those are the positions of all the horizontal steps, ie. where dY=0.
X axis N=1,5,13,25,etc is the "centred square numbers". Those numbers
are made by drawing concentric squares with an extra point on each side each
time. The path here grows the same way, adding one extra point to each of the
four sides.
*---*---*---*
| |
| *---*---* | count total "*"s for
| | | | centred square numbers
* | *---* | *
| | | | | |
| * | * | * |
| | | | | |
| | *---* | |
* | | *
| *---*---* |
| |
*---*---*---*
N Start¶
The default is to number points starting N=1 as shown above. An optional
"n_start" can give a different start, in the same pattern. For
example to start at 0,
n_start => 0
45 44
46 28 27 43
47 29 15 14 26 42
48 30 16 6 5 13 25 41
49 31 17 7 1 0 4 12 24 40
50 32 18 8 2 3 11 23 39 59
51 33 19 9 10 22 38 58
52 34 20 21 37 57
53 35 36 56
54 55
FUNCTIONS¶
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
classes.
- "$path = Math::PlanePath::AztecDiamondRings->new ()"
- "$path = Math::PlanePath::AztecDiamondRings->new (n_start =>
$n)"
- Create and return a new Aztec diamond spiral object.
- "($x,$y) = $path->n_to_xy ($n)"
- Return the X,Y coordinates of point number $n on the path.
For "$n < 1" the return is an empty list, it being considered
the path starts at 1.
- "$n = $path->xy_to_n ($x,$y)"
- Return the point number for coordinates "$x,$y". $x and $y are
each rounded to the nearest integer, which has the effect of treating each
point in the path as a square of side 1, so the entire plane is
covered.
- "($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
biggest in the rectangle.
X,Y to N¶
The path makes lines in each quadrant. The quadrant is determined by the signs
of X and Y, then the line in that quadrant is either d=X+Y or d=X-Y. A
quadratic in d gives a starting N for the line and Y (or X if desired) is an
offset from there,
Y>=0 X>=0 d=X+Y N=(2d+2)*d+1 + Y
Y>=0 X<0 d=Y-X N=2d^2 - Y
Y<0 X>=0 d=X-Y N=(2d+2)*d+1 + Y
Y<0 X<0 d=X+Y N=(2d+4)*d+2 - Y
For example
Y=2 X=3 d=2+3=5 N=(2*5+2)*5+1 + 2 = 63
Y=2 X=-1 d=2-(-1)=3 N=2*3*3 - 2 = 16
Y=-1 X=4 d=4-(-1)=5 N=(2*5+2)*5+1 + -1 = 60
Y=-2 X=-3 d=-3+(-2)=-5 N=(2*-5+4)*-5+2 - (-2) = 34
The two X>=0 cases are the same N formula and can be combined with an abs,
X>=0 d=X+abs(Y) N=(2d+2)*d+1 + Y
This works because at Y=0 the last line of one ring joins up to the start of the
next. For example N=11 to N=15,
15 2
\
14 1
\
13 <- Y=0
12 -1
/
11 -2
^
X=0 1 2
Rectangle to N Range¶
Within each row N increases as X increases away from the Y axis, and within each
column similarly N increases as Y increases away from the X axis. So in a
rectangle the maximum N is at one of the four corners of the rectangle.
|
x1,y2 M---|----M x2,y2
| | |
-------O---------
| | |
| | |
x1,y1 M---|----M x1,y1
|
For any two rows y1 and y2, the values in row y2 are all bigger than in y1 if
y2>=-y1. This is so even when y1 and y2 are on the same side of the origin,
ie. both positive or both negative.
For any two columns x1 and x2, the values in the part with Y>=0 are all
bigger if x2>=-x1, or in the part of the columns with Y<0 it's
x2>=-x1-1. So the biggest corner is at
max_y = (y2 >= -y1 ? y2 ? y1)
max_x = (x2 >= -x1 - (max_y<0) ? x2 : x1)
The difference in the X handling for Y positive or negative is due to the
quadrant ordering. When Y>=0, at X and -X the bigger N is the X negative
side, but when Y<0 it's the X positive side.
A similar approach gives the minimum N in a rectangle.
min_y = / y2 if y2 < 0, and set xbase=-1
| y1 if y1 > 0, and set xbase=0
\ 0 otherwise, and set xbase=0
min_x = / x2 if x2 < xbase
| x1 if x1 > xbase
\ xbase otherwise
The minimum row is Y=0, but if that's not in the rectangle then the y2 or y1 top
or bottom edge is the minimum. Then within any row the minimum N is at xbase=0
if Y<0 or xbase=-1 if Y>=0. If that xbase is not in range then the x2 or
x1 left or right edge is the minimum.
OEIS¶
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this
path include
n_start=1 (the default)
A001844 N on X axis, the centred squares 2k(k+1)+1
n_start=0
A046092 N on X axis, 4*triangular
A139277 N on diagonal X=Y
A023532 abs(dY), being 0 if N=k*(k+3)/2
SEE ALSO¶
Math::PlanePath, Math::PlanePath::DiamondSpiral
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/>.