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.
FORMULAS
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
- <http://oeis.org/A084849> (etc)
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
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016 Kevin RydeThis 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/>.