/ 
Math-PlanePath-129
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

N=1,3,6,10,etc on the X axis is the triangular numbers. N=1,2,4,7,11,etc on the Y axis is the triangular plus 1, the next point visited after the X axis.

Direction

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

direction => "up"

  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.

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"

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

N=0,1,3,6,10,etc on the Y axis of "down" or the X axis of "up" is the triangular numbers Y*(Y+1)/2.

X,Y Start

Options x_start => $x and y_start => $y give a starting position for the diagonals. For example to start at X=1,Y=1

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

The effect is merely to add a fixed offset to all X,Y values taken and returned, but it can be handy to have the path do that to step through non-negatives or similar.

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, n_start => $n, x_start => $x, y_start => $y)

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

X,Y to N

The sum d=X+Y numbers each diagonal from d=0 upwards, corresponding to the Y coordinate where the diagonal starts (or X if direction=up).

d=2
    \
d=1  \
    \ \
d=0  \ \
    \ \ \

N is then given by

d = X+Y
N = d*(d+1)/2 + X + Nstart

The d*(d+1)/2 shows how the triangular numbers fall on the Y axis when X=0 and Nstart=0. For the default Nstart=1 it's 1 more than the triangulars, as noted above.

d can be expanded out to the following quite symmetric form. This almost suggests something parabolic but is still the straight line diagonals.

    X^2 + 3X + 2XY + Y + Y^2
N = ------------------------ + Nstart
               2

    (X+Y)^2 + 3X + Y
  = ---------------- + Nstart       (using one square)
            2

N to X,Y

The above formula N=d*(d+1)/2 can be solved for d as

d = floor( (sqrt(8*N+1) - 1)/2 )
# with n_start=0

For example N=12 is d=floor((sqrt(8*12+1)-1)/2)=4 as that N falls in the fifth diagonal. Then the offset from the Y axis NY=d*(d-1)/2 is the X position,

X = N - d*(d+1)/2
Y = X - d

In the code, fractional N is handled by imagining each diagonal beginning 0.5 back from the Y axis. That's handled by adding 0.5 into the sqrt, which is +4 onto the 8*N.

d = floor( (sqrt(8*N+5) - 1)/2 )
# N>=-0.5

The X and Y formulas are unchanged, since N=d*(d-1)/2 is still the Y axis. But each diagonal d begins up to 0.5 before that and therefore X extends back to -0.5.

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 minimum N and the upper right is maximum N.

|            \     \ N max
|       \ ----------+
|        |     \    |\
|        |\     \   |
|       \| \     \  |
|        +----------
|  N min  \  \     \
+-------------------------

OEIS

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

direction=down (the default)
  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
  A101080    HammingDist(X,Y)

  A127949    dY, change in Y coordinate

  A000124    N on Y axis, triangular numbers + 1
  A001844    N on X=Y diagonal

  A185787    total N in row to X=Y diagonal
  A185788    total N in row to X=Y-1
  A100182    total N in column to Y=X diagonal
  A101165    total N in column to Y=X-1
  A185506    total N in rectangle 0,0 to X,Y

either direction=up,down
  A097806    turn 0=straight, 1=not straight

direction=down, x_start=1, y_start=1
  A057555    X,Y pairs
  A057046    X at N=2^k
  A057047    Y at N=2^k

direction=down, n_start=0
  A057554    X,Y pairs
  A023531    dSum = dX+dY, being 1 at N=triangular+1 (and 0)
  A000096    N on X axis, X*(X+3)/2
  A000217    N on Y axis, the triangular numbers
  A129184    turn 1=left,0=right
  A103451    turn 1=left or right,0=straight, but extra initial 1
  A103452    turn 1=left,0=straight,-1=right, but extra initial 1
direction=up, n_start=0
  A129184    turn 0=left,1=right

direction=up, n_start=-1
  A023531    turn 1=left,0=right
direction=down, n_start=-1
  A023531    turn 0=left,1=right

in direction=up the X,Y coordinate forms are the same but swap X,Y

either direction=up,down
  A038722    permutation N at transpose Y,X
               which is direction=down <-> direction=up

either direction, x_start=1, y_start=1
  A003991    X*Y coordinate product
  A003989    GCD(X,Y) greatest common divisor starting (1,1)
  A003983    min(X,Y)
  A051125    max(X,Y)

either direction, n_start=0
  A049581    abs(X-Y) coordinate diff
  A004197    min(X,Y)
  A003984    max(X,Y)
  A004247    X*Y coordinate product
  A048147    X^2+Y^2
  A109004    GCD(X,Y) greatest common divisor starting (0,0)
  A004198    X bit-and Y
  A003986    X bit-or Y
  A003987    X bit-xor Y
  A156319    turn 0=straight,1=left,2=right

  A061579    permutation N at transpose Y,X
               which is direction=down <-> direction=up

SEE ALSO

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

HOME PAGE

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

LICENSE

Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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/>.