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Kevin Ryde
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Math::PlanePath::Staircase -- integer points in stair-step diagonal stripes


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


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.


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.


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,Y-1 which is the bigger N, if Y-1 is in the rectangle.


Math::PlanePath, Math::PlanePath::Diagonals, Math::PlanePath::Corner




Copyright 2010, 2011 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/>.