/ 
math-image-38
River stage zero No dependents
/ App::MathImage::PlanePath::Staircase

NAME

App::MathImage::PlanePath::Staircase -- integer points in a diamond shape

SYNOPSIS

use App::MathImage::PlanePath::Staircase;
my $path = App::MathImage::PlanePath::Staircase->new;
my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

This path makes a staircase pattern of steps down from the Y axis to the X,

 8      29
         |
 7      30---31
              |
 6      16   32---33
         |         |
 5      17---18   34---...
              |
 4       7   19---20
         |         |
 3       8--- 9   21---22 
              |         |
 2       2   10---11   23---24
         |         |         |
 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 last of each staircase along the X axis are the hexagonal numbers k*(2*k-1). The diagonal 3,10,21,36,etc up to the right from x=0,y=1 is the second hexagonal numbers k*(2*k+1), as obtained 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.

FORMULAS

Within each row increasing X is increasing N, and each column increasing Y is increasing N pairs. On that basis in a rectangle for rect_to_n_range the lower left corner pair is the minimum N and the upper right pair is the maximum N.

A given X,Y is the larger of an N pair when ((X^Y)&1)==1. If that happens at the lower left corner then X,Y+1 is the smallest N in the rectangle, when Y+1 is also in the rectangle. Conversely at the top right if ((X^Y)&1)==0 then it's the smaller of a pair and X,Y-1 is the bigger N, when Y-1 is in the rectangle too.

FUNCTIONS

$path = App::MathImage::PlanePath::Staircase->new ()

Create and return a new Staircase spiral object.

SEE ALSO

Math::PlanePath, Math::PlanePath::SquareSpiral, Math::PlanePath::HexSpiralSkewed, Math::PlanePath::PyramidSides

HOME PAGE

http://user42.tuxfamily.org/math-image/index.html

LICENSE

Math-Image is Copyright 2010 Kevin Ryde

Math-Image 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-Image 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-Image. If not, see <http://www.gnu.org/licenses/>.