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 nonnegatives 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 bedirection => "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*(d1)/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*(d1)/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
http://oeis.org/A002262 (etc)
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 YX 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=Y1
A100182 total N in column to Y=X diagonal
A101165 total N in column to Y=X1
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(XY) 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 bitand Y
A003986 X bitor Y
A003987 X bitxor 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/mathplanepath/index.html
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
This file is part of MathPlanePath.
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.