/ 
Math-PlanePath-82
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::Diagonals

NAME

Math::PlanePath::Diagonals -- points in diagonal stripes

SYNOPSIS

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

DESCRIPTION

This path follows successive diagonals going from the Y axis down to the X axis.

  6  |  22
  5  |  16  23
  4  |  11  17  24
  3  |   7  12  18  ...
  2  |   4   8  13  19
  1  |   2   5   9  14  20
Y=0  |   1   3   6  10  15  21
     +-------------------------
       X=0   1   2   3   4   5

The horizontal sequence 1,3,6,10,etc at Y=0 is the triangular numbers s*(s+1)/2. If you plot them on a graph don't confuse that line with the axis or border!

Direction

Option direction => 'up' reverses the order within each diagonal to count upward from the X axis.

  5  |  21 
  4  |  15  20
  3  |  10  14  19 ...
  2  |   6   9  13  18  24 
  1  |   3   5   8  12  17  23 
Y=0  |   1   2   4   7  11  16  22 
     +-----------------------------
      X=0   1   2   3   4   5    6

This is merely a transpose changing X,Y to Y,X, but it's the same as in DiagonalsOctant and can be handy to control the direction when combining Diagonals with some other path or calculation.

FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

$path = Math::PlanePath::Diagonals->new ()
$path = Math::PlanePath::Diagonals->new (direction => $str)

Create and return a new path object. The direction option (a string) can be

direction => "down"       the default
direction => "up"         number upwards from the X axis
($x,$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 each rounded to the nearest integer, 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 entirely 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 N. So in a rectangle the lower left corner is the minimum N and the upper right is the maximum N.

OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

http://oeis.org/A023531  (etc)

direction=down
  A002262    X coordinate, runs 0 to k
  A025581  	 Y coordinate, runs k to 0
  A003056  	 X+Y coordinate sum, k repeated k+1 times
  A114327  	 Y-X coordinate diff
  A049581    abs(X-Y) coordinate diff
  A004247    X*Y coordinate product
  A048147    X^2+Y^2

  A127949    dY, change in Y coordinate

Similar for direction=up but transposing X,Y.

SEE ALSO

Math::PlanePath, Math::PlanePath::Corner, Math::PlanePath::Rows, Math::PlanePath::Columns, Math::PlanePath::DiagonalsAlternating, Math::PlanePath::DiagonalsOctant

HOME PAGE

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

LICENSE

Copyright 2010, 2011, 2012 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/>.