NAME¶
Math::PlanePath::Staircase -- integer points in stair-step diagonal stripes
SYNOPSIS¶
use Math::PlanePath::Staircase;
my $path = Math::PlanePath::Staircase->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION¶
This path makes a staircase pattern down from the Y axis to the X,
8 29
|
7 30---31
|
6 16 32---33
| |
5 17---18 34---35
| |
4 7 19---20 36---37
| | |
3 8--- 9 21---22 38---39
| | |
2 2 10---11 23---24 40...
| | |
1 3--- 4 12---13 25---26
| | |
Y=0 -> 1 5--- 6 14---15 27---28
^
X=0 1 2 3 4 5 6
The 1,6,15,28,etc along the X axis at the end of each run are the hexagonal
numbers k*(2*k-1). The diagonal 3,10,21,36,etc up from X=0,Y=1 is the second
hexagonal numbers k*(2*k+1), formed by extending the hexagonal numbers to
negative k. The two together are the triangular numbers k*(k+1)/2.
Legendre's prime generating polynomial 2*k^2+29 bounces around for some low
values then makes a steep diagonal upwards from X=19,Y=1, at a slope 3 up for
1 across, but only 2 of each 3 drawn.
N Start¶
The default is to number points starting N=1 as shown above. An optional
"n_start" can give a different start, in the same pattern. For
example to start at 0,
n_start => 0
28
29 30
15 31 32
16 17 33 34
6 18 19 35 36
7 8 20 21 37 38
1 9 10 22 23 ....
2 3 11 12 24 25
0 4 5 13 14 26 27
FUNCTIONS¶
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
classes.
- "$path = Math::PlanePath::Staircase->new ()"
- "$path = Math::PlanePath::AztecDiamondRings->new (n_start =>
$n)"
- Create and return a new staircase path object.
- "$n = $path->xy_to_n ($x,$y)"
- Return the point number for coordinates "$x,$y". $x and $y are
rounded to the nearest integers, which has the effect of treating each
point $n as a square of side 1, so the quadrant x>=-0.5, y>=-0.5 is
covered.
- "($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.
Rectangle to N Range¶
Within each row increasing X is increasing N, and in each column increasing Y is
increasing pairs of N. Thus for "rect_to_n_range()" the lower left
corner vertical pair is the minimum N and the upper right vertical pair is the
maximum N.
A given X,Y is the larger of a vertical pair when ((X^Y)&1)==1. If that
happens at the lower left corner then it's X,Y+1 which is the smaller N, as
long as Y+1 is in the rectangle. Conversely at the top right if
((X^Y)&1)==0 then it's X,Y-1 which is the bigger N, again as long as Y-1
is in the rectangle.
OEIS¶
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this
path include
n_start=1 (the default)
A084849 N on diagonal X=Y
n_start=0
A014105 N on diagonal X=Y, second hexagonal numbers
n_start=2
A128918 N on X axis, except initial 1,1
A096376 N on diagonal X=Y
SEE ALSO¶
Math::PlanePath, Math::PlanePath::Diagonals, Math::PlanePath::Corner,
Math::PlanePath::ToothpickSpiral
HOME PAGE¶
<
http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE¶
Copyright 2010, 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/>.