/ 
Math-PlanePath-126
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::DiagonalsOctant

NAME

Math::PlanePath::DiagonalsOctant -- points in diagonal stripes for an eighth of the plane

SYNOPSIS

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

DESCRIPTION

This path follows successive diagonals downwards from the Y axis down to the X=Y centre line, traversing the eighth of the plane on and above X=Y.

  8 |  21 27 33 40 47 55 63 72 81
    |    \  \  \  \  \  \  \
  7 |  17 22 28 34 41 48 56 64
    |    \  \  \  \  \  \
  6 |  13 18 23 29 35 42 49
    |    \  \  \  \  \
  5 |  10 14 19 24 30 36
    |    \  \  \  \
  4 |   7 11 15 20 25
    |    \  \  \
  3 |   5  8 12 16
    |    \  \
  2 |   3  6  9
    |    \
  1 |   2  4
    |
Y=0 |   1
    + ----------------------------
      X=0  1  2  3  4  5  6  7  8

N=1,4,9,16,etc on the X=Y leading diagonal are the perfect squares. N=2,6,12,20,etc at the ends of the other diagonals are the pronic numbers k*(k+1).

Incidentally "octant" usually refers to an eighth of a 3-dimensional coordinate space. Since PlanePath is only 2 dimensions there's no confusion and at the risk of abusing nomenclature half a quadrant is reckoned as an "octant".

Pyramid Rows

Taking two diagonals running from k^2+1 to (k+1)^2 is the same as a row of the step=2 PyramidRows (see Math::PlanePath::PyramidRows). Each endpoint is the same, but here it's two diagonals instead of one row. For example in the PyramidRows the Y=3 row runs from N=10 to N=16 ending at X=3,Y=3. Here that's in two diagonals N=10 to N=12 and then N=13 to N=16, and that N=16 endpoint is the same X=3,Y=3.

Direction

Option direction => 'up' reverses the order within each diagonal and counts upward from the centre to the Y axis.

  8 |  25 29 34 39 45 51 58 65 73 
    |    \  \  \  \  \  \  \
  7 |  20 24 28 33 38 44 50 57
    |    \  \  \  \  \  \
  6 |  16 19 23 27 32 37 43
    |    \  \  \  \  \
  5 |  12 15 18 22 26 31
    |    \  \  \  \
  4 |   9 11 14 17 21             direction => "up"
    |    \  \  \
  3 |   6  8 10 13
    |    \  \
  2 |   4  5  7
    |    \
  1 |   2  3
    |
Y=0 |   1
    +---------------------------
      X=0  1  2  3  4  5  6  7  8

In this arrangement N=1,2,4,6,9,etc on the Y axis are alternately the squares and the pronic numbers. The squares are on even Y and pronic on odd Y.

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 diagonals sequence. For example to start at 0,

n_start => 0                    n_start=>0
direction => "down"             direction=>"up"

  6  | 12                        | 15
  5  |  9 13                     | 11 14
  4  |  6 10 14                  |  8 10 13
  3  |  4  7 11 15               |  5  7  9 12
  2  |  2  5  8                  |  3  4  6
  1  |  1  3                     |  1  2
Y=0  |  0                        |  0
     +--------------             +--------------
      X=0  1  2  3                X=0  1  2  3

FUNCTIONS

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

$path = Math::PlanePath::DiagonalsOctant->new ()
$path = Math::PlanePath::DiagonalsOctant->new (direction => $str, n_start => $n)

Create and return a new 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 each rounded to the nearest integer, which has the effect of treating each point $n as a square of side 1.

($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

N to X,Y

To break N into X,Y it's convenient to take two diagonals at a time, since the length then changes by 1 each pair making a quadratic. Starting at each X=0,Y=odd just after perfect square N allows just a sqrt.

Nstart = d*d+1

where d numbers diagonal pairs, eg. d=3 for X=0,Y=5 going down. This is easily reversed as

d = floor sqrt(N-1)

The code reckons the start of the diagonal as 0.5 further back, so that N=9.5 is at X=-.5,Y=5.5. To do that d is formed as

d = floor sqrt(N-0.5)
  = int( sqrt(int(4*$n)-2)/2 )

Taking /2 out of the sqrt helps with Math::BigInt which circa Perl 5.14 doesn't inter-operate with flonums very well.

In any case N-Nstart is an offset into two diagonals, the first of length d many points and the second d+1. For example d=3 starting Y=5 for points N=10,11,12 followed by Y=6 N=13,14,15,16.

The formulas are simplified by calculating a remainder relative to the second diagonal, so it's negative for the first and positive for the second,

Nrem = N - (d*(d+1)+1)

d*(d+1)+1 is 1 past the pronic numbers when end each first diagonal, as described above. In any case for example d=3 is relative to N=13 making Nrem=-3,-2,-1 or Nrem=0,1,2,3.

To include the preceding 0.5 in the second diagonal simply means reckoning Nrem>=-0.5 as belonging to the second. In that base

if Nrem >= -0.5
  X = Nrem            # direction="down"
  Y = 2*d - Nrem
else
  X = Nrem + d
  Y = d - Nrem - 1

For example N=15 Nrem=1 is the first case, X=1, Y=2*3-1=5. Or N=11 Nrem=-2 the second X=-2+3=1, Y=3-(-2)-1=4.

For "up" direction the Nrem and d are the same, but the coordinate directions reverse.

if Nrem >= -0.5
  X = d - Nrem        # direction="up"
  Y = d + Nrem
else
  X = -Nrem - 1
  Y = 2d + Nrem

Another way is to reckon Nstart from the X=0,Y=even diagonals, which is then two diagonals of the same length and d formed by a sqrt inverting a pronic Nstart=d*(d+1).

Rectangle to N Range

Within each row increasing X is increasing N, and in each column increasing Y is increasing N. This is so in both "down" and "up" arrangements. On that basis in a rectangle the lower left corner is the minimum N and the upper right is the maximum N.

If the rectangle is partly outside the covered octant then the corners must be shifted to put them in range, ie. trim off any rows or columns entirely outside the rectangle. For the lower left this means,

  |  |    /
  |  |   /
  +--------          if x1 < 0 then x1 = 0
x1   | /             increase x1 to within octant
     |/
     +

     |  |/
     |  |            if y1 < x1 then y1 = x1
     | /|            increase y1 to bottom-left within octant
     |/ +----y1
     +  x1

And for the top right,

|    /  x2
| ------+ y2    if x2 > y2 then x2 = y2
|  /    |       decrease x2 so top-right within octant
| /     |         (the end of the y2 row)
|/
+

OEIS

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

direction=down
  A002620    N at end each run X=k,Y=k and X=k,Y=k+1
direction=down, n_start=0
  A055087    X coord, runs 0 to k twice
  A082375    Y-X, runs k to 0 or 1 stepping by 2
  A005563    N on X=Y diagonal, X*(X+2)

direction=up
  A002620    N on Y axis, end of each run, quarter squares
direction=up, n_start=0
  A024206    N on Y axis (starting from n=1 is Y=0, so Y=n-1)
  A014616    N in column X=1 (is Y axis N-1, from N=3)
  A002378    N on X=Y diagonal, pronic X*(X+1)

either direction, n_start=0
  A055086    X+Y, k repeating floor(k/2)+1 times

A004652      N start and end of each even-numbered diagonal

permutations
  A056536     N of PyramidRows in DiagonalsOctant order
  A091995      with DiagonalsOctant direction=up
  A091018      N-1, ie. starting from 0
  A090894      N-1 and DiagonalsOctant direction=up

  A056537     N of DiagonalsOctant at X,Y in PyramidRows order
               inverse of A056536

SEE ALSO

Math::PlanePath, Math::PlanePath::Diagonals, Math::PlanePath::DiagonalsAlternating, Math::PlanePath::PyramidRows

HOME PAGE

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

LICENSE

Copyright 2012, 2013, 2014, 2015, 2016, 2017 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/>.