.\" Automatically generated by Pod::Man 4.14 (Pod::Simple 3.40) .\" .\" Standard preamble: .\" ======================================================================== .de Sp \" Vertical space (when we can't use .PP) .if t .sp .5v .if n .sp .. .de Vb \" Begin verbatim text .ft CW .nf .ne \\$1 .. .de Ve \" End verbatim text .ft R .fi .. .\" Set up some character translations and predefined strings. \*(-- will .\" give an unbreakable dash, \*(PI will give pi, \*(L" will give a left .\" double quote, and \*(R" will give a right double quote. \*(C+ will .\" give a nicer C++. Capital omega is used to do unbreakable dashes and .\" therefore won't be available. \*(C` and \*(C' expand to `' in nroff, .\" nothing in troff, for use with C<>. .tr \(*W- .ds C+ C\v'-.1v'\h'-1p'\s-2+\h'-1p'+\s0\v'.1v'\h'-1p' .ie n \{\ . ds -- \(*W- . ds PI pi . if (\n(.H=4u)&(1m=24u) .ds -- \(*W\h'-12u'\(*W\h'-12u'-\" diablo 10 pitch . if (\n(.H=4u)&(1m=20u) .ds -- \(*W\h'-12u'\(*W\h'-8u'-\" diablo 12 pitch . ds L" "" . ds R" "" . ds C` "" . ds C' "" 'br\} .el\{\ . ds -- \|\(em\| . ds PI \(*p . ds L" `` . ds R" '' . ds C` . ds C' 'br\} .\" .\" Escape single quotes in literal strings from groff's Unicode transform. .ie \n(.g .ds Aq \(aq .el .ds Aq ' .\" .\" If the F register is >0, we'll generate index entries on stderr for .\" titles (.TH), headers (.SH), subsections (.SS), items (.Ip), and index .\" entries marked with X<> in POD. Of course, you'll have to process the .\" output yourself in some meaningful fashion. .\" .\" Avoid warning from groff about undefined register 'F'. .de IX .. .nr rF 0 .if \n(.g .if rF .nr rF 1 .if (\n(rF:(\n(.g==0)) \{\ . if \nF \{\ . de IX . tm Index:\\$1\t\\n%\t"\\$2" .. . if !\nF==2 \{\ . nr % 0 . nr F 2 . \} . \} .\} .rr rF .\" ======================================================================== .\" .IX Title "Math::PlanePath::HexSpiral 3pm" .TH Math::PlanePath::HexSpiral 3pm "2021-01-23" "perl v5.32.0" "User Contributed Perl Documentation" .\" For nroff, turn off justification. Always turn off hyphenation; it makes .\" way too many mistakes in technical documents. .if n .ad l .nh .SH "NAME" Math::PlanePath::HexSpiral \-\- integer points around a hexagonal spiral .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::HexSpiral; \& my $path = Math::PlanePath::HexSpiral\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This path makes a hexagonal spiral, with points spread out horizontally to fit on a square grid. .PP .Vb 10 \& 28 \-\- 27 \-\- 26 \-\- 25 3 \& / \e \& 29 13 \-\- 12 \-\- 11 24 2 \& / / \e \e \& 30 14 4 \-\-\- 3 10 23 1 \& / / / \e \e \e \& 31 15 5 1 \-\-\- 2 9 22 <\- Y=0 \& \e \e \e / / \& 32 16 6 \-\-\- 7 \-\-\- 8 21 \-1 \& \e \e / \& 33 17 \-\- 18 \-\- 19 \-\- 20 \-2 \& \e \& 34 \-\- 35 ... \-3 \& \& ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ \& \-6 \-5 \-4 \-3 \-2 \-1 X=0 1 2 3 4 5 6 .Ve .PP Each horizontal gap is 2, so for instance n=1 is at X=0,Y=0 then n=2 is at X=2,Y=0. The diagonals are just 1 across, so n=3 is at X=1,Y=1. Each alternate row is offset from the one above or below. The result is a triangular lattice per \*(L"Triangular Lattice\*(R" in Math::PlanePath. .PP The octagonal numbers 8,21,40,65, etc 3*k^2\-2*k fall on a horizontal straight line at Y=\-1. In general straight lines are 3*k^2 + b*k + c. A plain 3*k^2 goes diagonally up to the left, then b is a 1/6 turn anti-clockwise, or clockwise if negative. So b=1 goes horizontally to the left, b=2 diagonally down to the left, b=3 diagonally down to the right, etc. .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 \& $path = Math::PlanePath::HexSpiral\->new (wider => 2); .Ve .PP gives .PP .Vb 11 \& ... 36\-\-\-\-35 3 \& \e \& 21\-\-\-\-20\-\-\-\-19\-\-\-\-18\-\-\-\-17 34 2 \& / \e \e \& 22 8\-\-\-\- 7\-\-\-\- 6\-\-\-\- 5 16 33 1 \& / / \e \e \e \& 23 9 1\-\-\-\- 2\-\-\-\- 3\-\-\-\- 4 15 32 <\- Y=0 \& \e \e / / \& 24 10\-\-\-\-11\-\-\-\-12\-\-\-\-13\-\-\-\-14 31 \-1 \& \e / \& 25\-\-\-\-26\-\-\-\-27\-\-\-\-28\-\-\-29\-\-\-\-30 \-2 \& \& ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ \& \-7 \-6 \-5 \-4 \-3 \-2 \-1 X=0 1 2 3 4 5 6 7 .Ve .PP The centre horizontal from N=1 is extended by \f(CW\*(C`wider\*(C'\fR many extra places, then the path loops around that shape. The starting point N=1 is shifted to the left by wider many places to keep the spiral centred on the origin X=0,Y=0. Each horizontal gap is still 2. .PP Each loop is still 6 longer than the previous, since the widening is basically a constant amount added into each loop. .SS "N Start" .IX Subsection "N Start" The default is to number points starting N=1 as shown above. An optional \&\f(CW\*(C`n_start\*(C'\fR can give a different start with the same shape etc. For example to start at 0, .PP .Vb 1 \& n_start => 0 \& \& 27 26 25 24 3 \& 28 12 11 10 23 2 \& 29 13 3 2 9 22 1 \& 30 14 4 0 1 8 21 <\- Y=0 \& 31 15 5 6 7 20 ... \-1 \& 32 16 17 18 19 38 \-2 \& 33 34 35 36 37 \-3 \& ^ \& \-6 \-5 \-4 \-3 \-2 \-1 X=0 1 2 3 4 5 6 .Ve .PP In this numbering the X axis N=0,1,8,21,etc is the octagonal numbers 3*X*(X+1). .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::HexSpiral\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::HexSpiral\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::HexSpiral->new ()" .PD 0 .ie n .IP """$path = Math::PlanePath::HexSpiral\->new (wider => $w)""" 4 .el .IP "\f(CW$path = Math::PlanePath::HexSpiral\->new (wider => $w)\fR" 4 .IX Item "$path = Math::PlanePath::HexSpiral->new (wider => $w)" .PD Create and return a new hex spiral object. An optional \f(CW\*(C`wider\*(C'\fR parameter widens the path, it defaults to 0 which is no widening. .ie n .IP """($x,$y) = $path\->n_to_xy ($n)""" 4 .el .IP "\f(CW($x,$y) = $path\->n_to_xy ($n)\fR" 4 .IX Item "($x,$y) = $path->n_to_xy ($n)" Return the X,Y coordinates of point number \f(CW$n\fR on the path. .Sp For \f(CW\*(C`$n < 1\*(C'\fR the return is an empty list, it being considered the path starts at 1. .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$y\fR are each rounded to the nearest integer, which has the effect of treating each \&\f(CW$n\fR in the path as a square of side 1. .Sp Only every second square in the plane has an N, being those where X,Y both odd or both even. If \f(CW\*(C`$x,$y\*(C'\fR is a position without an N, ie. one of X,Y odd the other even, then the return is \f(CW\*(C`undef\*(C'\fR. .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 4 \& A328818 X coordinate \& A307012 Y coordinate \& A307011 (X\-Y)/2 \& A307013 (X+Y)/2 \& \& A056105 N on X axis \& A056106 N on X=Y diagonal \& A056107 N on North\-West diagonal \& A056108 N on negative X axis \& A056109 N on South\-West diagonal \& A003215 N on South\-East diagonal \& \& A063178 total sum N previous row or diagonal \& A135711 boundary length of N hexagons \& A135708 grid sticks of N hexagons \& \& n_start=0 \& A001399 N positions of turns (extra initial 1) \& A000567 N on X axis, octagonal numbers \& A049451 N on X negative axis \& A049450 N on X=Y diagonal north\-east \& A033428 N on north\-west diagonal, 3*k^2 \& A045944 N on south\-west diagonal, octagonal numbers second kind \& A063436 N on WSW slope dX=\-3,dY=\-1 \& A028896 N on south\-east diagonal .Ve .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::HexSpiralSkewed, Math::PlanePath::HexArms, Math::PlanePath::TriangleSpiral, Math::PlanePath::TriangularHypot .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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 .