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Math::PlanePath::Diagonals -- points in diagonal stripes
Contents
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.
NStart
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,YStart
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.
Formulas
X,YtoN
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
NtoX,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.
RectangletoNRange
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 \ \ \
+-------------------------
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.
Home Page
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/>.
perl v5.32.0 2021-01-23 Math::PlanePath::Diagonals(3pm)
Name
Math::PlanePath::Diagonals -- points in diagonal stripes
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 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
Synopsis
use Math::PlanePath::Diagonals;
my $path = Math::PlanePath::Diagonals->new;
my ($x, $y) = $path->n_to_xy (123);
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