NAME
Math::PlanePath::Staircase  integer points in stairstep 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 3031

6 16 3233
 
5 1718 3435
 
4 7 1920 3637
  
3 8 9 2122 3839
  
2 2 1011 2324 40...
  
1 3 4 1213 2526
  
y=0 > 1 5 6 1415 2728
^
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*k1). 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.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.
$path = Math::PlanePath::Staircase>new ()

Create and return a new staircase path object.
($x,$y) = $path>n_to_xy ($n)

Return the X,Y coordinates of point number
$n
on the path.For
$n < 0.5
the return is an empty list, it being considered the path begins at 1. $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.
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, if Y+1 is in the rectangle. Conversely at the top right if ((X^Y)&1)==0 then it's X,Y1 which is the bigger N, if Y1 is in the rectangle.
SEE ALSO
Math::PlanePath, Math::PlanePath::Diagonals, Math::PlanePath::Corner
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2010, 2011 Kevin Ryde
This file is part of MathPlanePath.
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.