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Always turn off hyphenation; it makes .\" way too many mistakes in technical documents. .if n .ad l .nh .SH "NAME" Math::PlanePath::HexSpiralSkewed \-\- integer points around a skewed hexagonal spiral .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::HexSpiralSkewed; \& my $path = Math::PlanePath::HexSpiralSkewed\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This path makes a hexagonal spiral with points skewed so as to fit a square grid and fully cover the plane. .PP .Vb 9 \& 13\-\-12\-\-11 ... 2 \& | \e \e \& 14 4\-\-\-3 10 23 1 \& | | \e \e \e \& 15 5 1\-\-\-2 9 22 <\- Y=0 \& \e \e | | \& 16 6\-\-\-7\-\-\-8 21 \-1 \& \e | \& 17\-\-18\-\-19\-\-20 \-2 \& \& ^ ^ ^ ^ ^ ^ \& \-2 \-1 X=0 1 2 3 ... .Ve .PP The kinds of N=3*k^2 numbers which fall on straight lines in the plain \&\f(CW\*(C`HexSpiral\*(C'\fR also fall on straight lines when skewed. See Math::PlanePath::HexSpiral for notes on this. .SS "Skew" .IX Subsection "Skew" The skewed path is the same shape as the plain \f(CW\*(C`HexSpiral\*(C'\fR, but fits more points on a square grid. The skew pushes the top horizontal to the left, as shown by the following parts, and the bottom horizontal is similarly skewed but to the right. .PP .Vb 1 \& HexSpiralSkewed HexSpiral \& \& 13\-\-12\-\-11 13\-\-12\-\-11 \& | \e / \e \& 14 10 14 10 \& | \e / \e \& 15 9 15 9 \& \& \-2 \-1 X=0 1 2 \-4 \-3 \-2 X=0 2 3 4 .Ve .PP In general the coordinates can be converted each way by .PP .Vb 1 \& plain X,Y \-> skewed (X\-Y)/2, Y \& \& skewed X,Y \-> plain 2*X+Y, Y .Ve .SH "Corners" .IX Header "Corners" \&\f(CW\*(C`HexSpiralSkewed\*(C'\fR is similar to the \f(CW\*(C`SquareSpiral\*(C'\fR but cuts off the top-right and bottom-left corners so that each loop is 6 steps longer than the previous, whereas for the \f(CW\*(C`SquareSpiral\*(C'\fR it's 8. See \&\*(L"Corners\*(R" in Math::PlanePath::SquareSpiral for other corner cutting. .SS "Wider" .IX Subsection "Wider" An optional \f(CW\*(C`wider\*(C'\fR parameter makes the path wider, stretched along the top and bottom horizontals. For example .PP .Vb 1 \& $path = Math::PlanePath::HexSpiralSkewed\->new (wider => 2); .Ve .PP gives .PP .Vb 9 \& 21\-\-20\-\-19\-\-18\-\-17 2 \& | \e \& 22 8\-\-\-7\-\-\-6\-\-\-5 16 1 \& | | \e \e \& 23 9 1\-\-\-2\-\-\-3\-\-\-4 15 <\- Y=0 \& \e \e | \& 24 10\-\-11\-\-12\-\-13\-\-14 ... \-1 \& \e | \& 25\-\-26\-\-27\-\-28\-\-29\-\-30 \-2 \& \& ^ ^ ^ ^ ^ ^ ^ ^ \& \-4 \-3 \-2 \-1 X=0 1 2 3 ... .Ve .PP The centre horizontal from N=1 is extended by \f(CW\*(C`wider\*(C'\fR many further places, then the path loops around that shape. The starting point 1 is shifted to the left by wider/2 places (rounded up to an integer) to keep the spiral centred on the origin X=0,Y=0. .PP Each loop is still 6 longer than the previous, since the widening is basically a constant amount added into each loop. The result is the same as the plain \f(CW\*(C`HexSpiral\*(C'\fR of the same widening too. The effect looks better in the plain \f(CW\*(C`HexSpiral\*(C'\fR. .SS "N Start" .IX Subsection "N Start" The default is to number points starting N=1 as shown above. An optional \&\f(CW\*(C`n_start\*(C'\fR can give a different start with the same shape etc. For example to start at 0, .PP .Vb 1 \& n_start => 0 \& \& 27 26 25 24 3 \& 28 12 11 10 23 2 \& 29 13 3 2 9 22 1 \& 30 14 4 0 1 8 21 ... <\- Y=0 \& 31 15 5 6 7 20 39 \-1 \& 32 16 17 18 19 38 \-2 \& 33 34 35 36 37 \-3 \& \& \-3 \-2 \-1 X=0 1 2 3 4 .Ve .PP In this numbering the X axis N=0,1,8,21,etc is the octagonal numbers 3*X*(X+1). .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::HexSpiralSkewed\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::HexSpiralSkewed\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::HexSpiralSkewed->new ()" .PD 0 .ie n .IP """$path = Math::PlanePath::HexSpiralSkewed\->new (wider => $w)""" 4 .el .IP "\f(CW$path = Math::PlanePath::HexSpiralSkewed\->new (wider => $w)\fR" 4 .IX Item "$path = Math::PlanePath::HexSpiralSkewed->new (wider => $w)" .PD Create and return a new hexagon spiral object. An optional \f(CW\*(C`wider\*(C'\fR parameter widens the spiral path, it defaults to 0 which is no widening. .ie n .IP """$n = $path\->xy_to_n ($x,$y)""" 4 .el .IP "\f(CW$n = $path\->xy_to_n ($x,$y)\fR" 4 .IX Item "$n = $path->xy_to_n ($x,$y)" Return the point number for coordinates \f(CW\*(C`$x,$y\*(C'\fR. \f(CW$x\fR and \f(CW$y\fR are each rounded to the nearest integer, which has the effect of treating each point in the path as a square of side 1. .SH "OEIS" .IX Header "OEIS" Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include .Sp .RS 4 (etc) .RE .PP .Vb 6 \& A056105 N on X axis, 3n^2\-2n+1 \& A056106 N on Y axis, 3n^2\-n+1 \& A056107 N on North\-West diagonal, 3n^2+1 \& A056108 N on X negative axis, 3n^2+n+1 \& A056109 N on Y negative axis, 3n^2+2n+1 \& A003215 N on South\-East diagonal, centred hexagonals \& \& n_start=0 \& A000567 N on X axis, octagonal numbers \& A049450 N on Y axis \& A049451 N on X negative axis \& A045944 N on Y negative axis, octagonal numbers second kind \& A062783 N on X=Y diagonal north\-east \& A033428 N on north\-west diagonal, 3*k^2 \& A063436 N on south\-west diagonal \& A028896 N on south\-east diagonal .Ve .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::HexSpiral, Math::PlanePath::HeptSpiralSkewed, Math::PlanePath::PentSpiralSkewed, Math::PlanePath::DiamondSpiral .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 2010, 2011, 2012, 2013, 2014 Kevin Ryde .PP This file is part of Math-PlanePath. .PP Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the \s-1GNU\s0 General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version. .PP Math-PlanePath is distributed in the hope that it will be useful, but \&\s-1WITHOUT ANY WARRANTY\s0; without even the implied warranty of \s-1MERCHANTABILITY\s0 or \s-1FITNESS FOR A PARTICULAR PURPOSE. \s0 See the \s-1GNU\s0 General Public License for more details. .PP You should have received a copy of the \s-1GNU\s0 General Public License along with Math-PlanePath. If not, see .