SYNOPSIS
use Math::PlanePath::DiamondSpiral;
my $path = Math::PlanePath::DiamondSpiral>new;
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
This path makes a diamond shaped spiral.
19 3 / \ 20 9 18 2 / / \ \ 21 10 3 8 17 1 / / / \ \ \ 22 11 4 12 7 16 < Y=0 \ \ \ / / 23 12 56 15 ... 1 \ \ / / 24 1314 27 2 \ / 2526 3 ^ 3 2 1 X=0 1 2 3
This is not simply the "SquareSpiral" rotated, it spirals around faster, with side lengths following a pattern 1,1,1,1, 2,2,2,2, 3,3,3,3, etc, if the flat kink at the bottom (like N=13 to N=14) is treated as part of the lower right diagonal.
Going diagonally on the sides as done here is like cutting the corners of the "SquareSpiral", which is how it gets around in fewer steps than the "SquareSpiral". See "PentSpiralSkewed", "HexSpiralSkewed" and "HeptSpiralSkewed" for similar cutting just 3, 2 or 1 of the corners.
N=1,5,13,25,etc on the Y negative axis is the ``centred square numbers'' 2*k*(k+1)+1.
N Start
The default is to number points starting N=1 as shown above. An optional "n_start" can give a different start, with the same shape etc. For example to start at 0,
n_start => 0 18 / \ 19 8 17 / / \ \ 20 9 2 7 16 / / / \ \ \ 21 10 3 0 1 6 15 \ \ \ / / 22 11 4 5 14 ... \ \ / / 23 1213 26 \ / 2425
N=0,1,6,15,28,etc on the X axis is the hexagonal numbers k*(2k1). N=0,3,10,21,36,etc on the negative X axis is the hexagonal numbers of the ``second kind'' k*(2k1) for k<0. Combining those two is the triangular numbers 0,1,3,6,10,15,21,etc, k*(k+1)/2, on the X axis alternately positive and negative.
N=0,2,8,18,etc on the Y axis is 2*squares, 2*Y^2. N=0,4,12,24,etc on the negative Y axis is 2*pronic, 2*Y*(Y+1).
FUNCTIONS
See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes. "$path = Math::PlanePath::DiamondSpiral>new ()"
 "$path = Math::PlanePath::DiamondSpiral>new (n_start => $n)"
 Create and return a new diamond spiral object.
 "($x,$y) = $path>n_to_xy ($n)"

Return the X,Y coordinates of point number $n on the path.
For "$n < 1" the return is an empty list, it being considered the path starts at 1.
 "$n = $path>xy_to_n ($x,$y)"
 Return the point number for coordinates "$x,$y". $x and $y are each rounded to the nearest integer, which has the effect of treating each point in the path as a square of side 1, so the entire plane is covered.
 "($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 N increases as X moves away from the Y axis, and within each column similarly N increases as Y moves away from the X axis. So in a rectangle the maximum N is at one of the four corners.
 x1,y2 MM x2,y2    O       x1,y1 MM x1,y1 
For any two columns x1 and x2 with x1<x2, the values in column x2 are all bigger if x2>x1. This is so even when x1 and x2 are on the same side of the origin, ie. both positive or both negative.
For any two rows y1 and y2, the values in the part of the row with X>0 are bigger if y2>=y1, and in the part of the row with X<=0 it's y2>y1, or equivalently y2>=y1+1. So the biggest corner is at
max_x = (x2 > x1 ? x2 : x1) max_y = (y2 >= y1+(max_x<=0) ? y2 : y1)
The minimum is similar but a little simpler. In any column the minimum is at Y=0, and in any row the minimum is at X=0. So at 0,0 if that's in the rectangle, or the edge on the side nearest the origin when not.
min_x = / if x2 < 0 then x2  if x1 > 0 then x1 \ else 0 min_y = / if y2 < 0 then y2  if y1 > 0 then y1 \ else 0
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
 <http://oeis.org/A010751> (etc)
n_start=1 A130883 N on X axis, 2*n^2n+1 A058331 N on Y axis, 2*n^2 + 1 A001105 N on column X=1, 2*n^2 A084849 N on X negative axis, 2*n^2+n+1 A001844 N on Y negative axis, centred squares 2*n^2+2n+1 A215471 N with >=5 primes among its 8 neighbours A215468 sum 8 neighbours N A217015 N permutation points order SquareSpiral rotate 90, value DiamondSpiral N at each A217296 inverse permutation n_start=0 A010751 X coordinate, runs 1 inc, 2 dec, 3 inc, etc A053616 abs(Y), runs k to 0 to k A000384 N on X axis, hexagonal numbers A001105 N on Y axis, 2*n^2 (and cf similar A184636) A014105 N on X negative axis, second hexagonals A046092 N on Y negative axis, 2*pronic A003982 delta(abs(X)+abs(Y)), 1 when N on Y negative axis which is where move "outward" to next ring n_start=1 A188551 N positions of turns, from N=1 up A188552 and which are primes
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016 Kevin RydeThis 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/>.