SYNOPSIS
use Math::PlanePath::TriangleSpiralSkewed;
my $path = Math::PlanePath::TriangleSpiralSkewed>new;
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
This path makes an spiral shaped as an equilateral triangle (each side the same length), but skewed to the left to fit on a square grid,
16 4 \ 17 15 3  \ 18 4 14 2  \ \ 19 5 3 13 1   \ \ 20 6 12 12 ... < Y=0   \ \ 21 7891011 30 1  \ 2223242526272829 2 ^ 2 1 X=0 1 2 3 4 5
The properties are the same as the spreadout "TriangleSpiral". The triangle numbers fall on straight lines as the do in the "TriangleSpiral" but the skew means the top corner goes up at an angle to the vertical and the left and right downwards are different angles plotted (but are symmetric by N count).
Skew Right
Option "skew => 'right'" directs the skew towards the right, giving
4 16 skew="right" /  3 17 15 /  2 18 4 14 / /   1 ... 5 3 13 /   Y=0 > 6 12 12 /  1 7891011 ^ 2 1 X=0 1 2
This is a shear ``X > X+Y'' of the default skew=``left'' shown above. The coordinates are related by
Xright = Xleft + Yleft Xleft = Xright  Yright Yright = Yleft Yleft = Yright
Skew Up
2 161514131211 skew="up"  / 1 17 432 10   / / Y=0 > 18 5 1 9   / 1 ... 6 8 / 2 7 ^ 2 1 X=0 1 2
This is a shear ``Y > X+Y'' of the default skew=``left'' shown above. The coordinates are related by
Xup = Xleft Xleft = Xup Yup = Yleft + Xleft Yleft = Yup  Xup
Skew Down
2 ..181716 skew="down"  1 7654 15 \   Y=0 > 8 1 3 14 \ \   1 9 2 13 \  2 10 12 \  11 ^ 2 1 X=0 1 2
This is a rotate by 90 degrees of the skew=``up'' above. The coordinates are related
Xdown = Yup Xup =  Ydown Ydown =  Xup Yup = Xdown
Or related to the default skew=``left'' by
Xdown = Yleft + Xleft Xleft =  Ydown Ydown =  Xleft Yleft = Xdown + Ydown
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,
15 n_start => 0 \ 16 14  \ 17 3 13 ...  \ \ \ 18 4 2 12 31   \ \ \ 19 5 01 11 30   \ \ 20 678910 29  \ 2122232425262728
With this adjustment for example the X axis N=0,1,11,30,etc is (9X7)*X/2, the hendecagonal numbers (11gonals). And SouthEast N=0,8,25,etc is the hendecagonals of the second kind, (9Y7)*Y/2 with Y negative.
FUNCTIONS
See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes. "$path = Math::PlanePath::TriangleSpiralSkewed>new ()"
 "$path = Math::PlanePath::TriangleSpiralSkewed>new (skew => $str, n_start => $n)"

Create and return a new skewed triangle spiral object. The "skew"
parameter can be
"left" (the default) "right" "up" "down"
 "$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 N in the path as centred in a square of side 1, so the entire plane is covered.
FORMULAS
Rectangle to N Range
Within each row there's a minimum N and the N values then increase monotonically away from that minimum point. Likewise in each column. This means in a rectangle the maximum N is at one of the four corners of the rectangle.
 x1,y2 MM x2,y2 maximum N at one of    the four corners O of the rectangle       x1,y1 MM x1,y1 
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
 <http://oeis.org/A117625> (etc)
n_start=1, skew="left" (the defaults) A204439 abs(dX) A204437 abs(dY) A010054 turn 1=left,0=straight, extra initial 1 A117625 N on X axis A064226 N on Y axis, but without initial value=1 A006137 N on X negative A064225 N on Y negative A081589 N on X=Y leading diagonal A038764 N on X=Y negative SouthWest diagonal A081267 N on X=Y negative SouthEast diagonal A060544 N on ESE slope dX=+2,dY=1 A081272 N on SSE slope dX=+1,dY=2 A217010 permutation N values of points in SquareSpiral order A217291 inverse A214230 sum of 8 surrounding N A214231 sum of 4 surrounding N n_start=0 A051682 N on X axis (11gonal numbers) A081268 N on X=1 vertical (next to Y axis) A062708 N on Y axis A062725 N on Y negative axis A081275 N on X=Y+1 NorthEast diagonal A062728 N on SouthEast diagonal (11gonal second kind) A081266 N on X=Y negative SouthWest diagonal A081270 N on X=1Y NorthWest diagonal, starting N=3 A081271 N on dX=1,dY=2 NNW slope up from N=1 at X=1,Y=0 n_start=1 A023531 turn 1=left,0=straight, being 1 at N=k*(k+3)/2 A023532 turn 1=straight,0=left n_start=1, skew="right" A204435 abs(dX) A204437 abs(dY) A217011 permutation N values of points in SquareSpiral order but with 90degree rotation A217292 inverse A214251 sum of 8 surrounding N n_start=1, skew="up" A204439 abs(dX) A204435 abs(dY) A217012 permutation N values of points in SquareSpiral order but with 90degree rotation A217293 inverse A214252 sum of 8 surrounding N n_start=1, skew="down" A204435 abs(dX) A204439 abs(dY)
The square spiral order in A217011,A217012 and their inverses has first step at 90degrees to the first step of the triangle spiral, hence the rotation by 90 degrees when relating to the "SquareSpiral" path. A217010 on the other hand has no such rotation since it reckons the square and triangle spirals starting in the same direction.
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/>.