.\" Automatically generated by Pod::Man 2.28 (Pod::Simple 3.28) .\" .\" 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 turned on, 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 .\" .\" Accent mark definitions (@(#)ms.acc 1.5 88/02/08 SMI; from UCB 4.2). .\" Fear. Run. Save yourself. No user-serviceable parts. . \" fudge factors for nroff and troff .if n \{\ . ds #H 0 . ds #V .8m . ds #F .3m . ds #[ \f1 . ds #] \fP .\} .if t \{\ . ds #H ((1u-(\\\\n(.fu%2u))*.13m) . ds #V .6m . ds #F 0 . ds #[ \& . ds #] \& .\} . \" simple accents for nroff and troff .if n \{\ . ds ' \& . ds ` \& . ds ^ \& . ds , \& . ds ~ ~ . ds / .\} .if t \{\ . ds ' \\k:\h'-(\\n(.wu*8/10-\*(#H)'\'\h"|\\n:u" . ds ` \\k:\h'-(\\n(.wu*8/10-\*(#H)'\`\h'|\\n:u' . ds ^ \\k:\h'-(\\n(.wu*10/11-\*(#H)'^\h'|\\n:u' . ds , \\k:\h'-(\\n(.wu*8/10)',\h'|\\n:u' . ds ~ \\k:\h'-(\\n(.wu-\*(#H-.1m)'~\h'|\\n:u' . ds / \\k:\h'-(\\n(.wu*8/10-\*(#H)'\z\(sl\h'|\\n:u' .\} . \" troff and (daisy-wheel) nroff accents .ds : \\k:\h'-(\\n(.wu*8/10-\*(#H+.1m+\*(#F)'\v'-\*(#V'\z.\h'.2m+\*(#F'.\h'|\\n:u'\v'\*(#V' .ds 8 \h'\*(#H'\(*b\h'-\*(#H' .ds o \\k:\h'-(\\n(.wu+\w'\(de'u-\*(#H)/2u'\v'-.3n'\*(#[\z\(de\v'.3n'\h'|\\n:u'\*(#] .ds d- \h'\*(#H'\(pd\h'-\w'~'u'\v'-.25m'\f2\(hy\fP\v'.25m'\h'-\*(#H' .ds D- D\\k:\h'-\w'D'u'\v'-.11m'\z\(hy\v'.11m'\h'|\\n:u' .ds th \*(#[\v'.3m'\s+1I\s-1\v'-.3m'\h'-(\w'I'u*2/3)'\s-1o\s+1\*(#] .ds Th \*(#[\s+2I\s-2\h'-\w'I'u*3/5'\v'-.3m'o\v'.3m'\*(#] .ds ae a\h'-(\w'a'u*4/10)'e .ds Ae A\h'-(\w'A'u*4/10)'E . \" corrections for vroff .if v .ds ~ \\k:\h'-(\\n(.wu*9/10-\*(#H)'\s-2\u~\d\s+2\h'|\\n:u' .if v .ds ^ \\k:\h'-(\\n(.wu*10/11-\*(#H)'\v'-.4m'^\v'.4m'\h'|\\n:u' . \" for low resolution devices (crt and lpr) .if \n(.H>23 .if \n(.V>19 \ \{\ . ds : e . ds 8 ss . ds o a . ds d- d\h'-1'\(ga . ds D- D\h'-1'\(hy . ds th \o'bp' . ds Th \o'LP' . ds ae ae . ds Ae AE .\} .rm #[ #] #H #V #F C .\" ======================================================================== .\" .IX Title "Math::PlanePath::CubicBase 3pm" .TH Math::PlanePath::CubicBase 3pm "2014-08-26" "perl v5.20.1" "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::CubicBase \-\- replications in three directions .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::CubicBase; \& my $path = Math::PlanePath::CubicBase\->new (radix => 4); \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This is a pattern of replications in three directions 0, 120 and 240 degrees. .PP .Vb 12 \& 18 19 26 27 5 \& 16 17 24 25 4 \& 22 23 30 31 3 \& 20 21 28 29 2 \& 50 51 58 59 2 3 10 11 1 \& 48 49 56 57 0 1 8 9 <\- Y=0 \& 54 55 62 63 6 7 14 15 \-1 \& 52 53 60 61 4 5 12 13 \-2 \& 34 35 42 43 \-3 \& 32 33 40 41 \-4 \& 38 39 46 47 \-5 \& 36 37 44 45 \-6 \& \& ^ \& \-11\-10 \-9 \-8 \-7 \-6 \-5 \-4 \-3 \-2 \-1 X=0 1 2 3 4 5 6 .Ve .PP The points are on a triangular grid by using every second integer X,Y, as per \*(L"Triangular Lattice\*(R" in Math::PlanePath. All points on that triangular grid are visited. .PP The initial N=0,N=1 is replicated at +120 degrees. Then that trapezoid at +240 degrees .PP .Vb 9 \& +\-\-\-\-\-\-\-\-\-\-\-+ +\-\-\-\-\-\-\-\-\-\-\-+ \& \e 2 3 \e \e 2 3 \e \& +\-\-\-\-\-\-\-\-\-\-\-+ \e \e \& \e 0 1 \e \e 0 1 \e \& +\-\-\-\-\-\-\-\-\-\-\-+ \-\-\-\-\-\-\-\-\- \-\-\-\-\-\-\-\-\-\-\-+ \& \e 6 7 \e \& replicate +120deg \e \e rep +240deg \& \e 4 5 \e \& +\-\-\-\-\-\-\-\-\-\-+ .Ve .PP Then that bow-tie N=0to7 is replicated at 0 degrees again. Each replication is 1/3 of the circle around, 0, 120, 240 degrees repeating. The relative layout within a replication is unchanged. .PP .Vb 10 \& \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& \e 18 19 26 27 \e \& \e \e \& \e 16 17 24 25 \e \& \-\-\-\-\-\-\-\-\-\- \-\-\-\-\-\-\-\-\-\- \& \e 22 23 30 31 \e \& \e \e \& \e 20 21 28 29 \e \& \-\-\-\-\-\-\-\-\- \-\-\-\-\-\-\-\-\-\-\-\- +\-\-\-\-\-\-\-\-\-\-\- \-\-\-\-\-\-\-\-\-\-\- \& \e 50 51 58 59 \e 2 3 \e 10 11 \e \& \e +\-\-\-\-\-\-\-\-\-\-\-+ \e \& \e 48 49 56 57 \e 0 1 \e 8 9 \e \& \-\-\-\-\-\-\-\-\-\- \-\-\-\-\-\-\-\-\- +\-\-\-\-\-\-\-\-\-\-\- \-\-\-\-\-\-\-\-\-+ \& \e 54 55 62 63 \e 6 7 \e 14 15 \e \& \e \e \e \e \& \e 52 53 60 61 \e 4 5 \e 12 13 \e \& \-\-\-\-\-\-\-\-\-\-\-\-\-\- +\-\-\-\-\-\-\-\-\-\-+\-\-\-\-\-\-\-\-\-\-\-\- \& \e 34 35 42 43 \e \& \e \e \& \e 32 33 40 41 \e \& \-\-\-\-\-\-\-\-\-+ \-\-\-\-\-\-\-\-\-\-\- \& \e 38 39 46 47 \e \& \e \e \& \e 36 37 44 45 \e \& \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- .Ve .PP The radial distance doubles on every second replication, so N=1 and N=2 are at 1 unit from the origin, then N=4 and N=8 at 2 units, then N=16 and N=32 at 4 units. N=64 is not shown but is then at 8 units away (X=8,Y=0). .PP This is similar to the \f(CW\*(C`ImaginaryBase\*(C'\fR, but where that path repeats in 4 directions based on i=squareroot(\-1), here it's 3 directions based on w=\fIcuberoot\fR\|(1) = \-1/2+i*\fIsqrt\fR\|(3)/2. .SS "Radix" .IX Subsection "Radix" The \f(CW\*(C`radix\*(C'\fR parameter controls the \*(L"r\*(R" used to break N into X,Y. For example radix 4 gives 4x4 blocks, with r\-1 replications of the preceding level at each stage. .PP .Vb 10 \& 3 radix => 4 12 13 14 15 \& 2 8 9 10 11 \& 1 4 5 6 7 \& Y=0 \-> 0 1 2 3 \& \-1 28 29 30 31 \& \-2 24 25 26 27 \& \-3 20 21 22 23 \& \-4 16 17 18 19 \& \-5 44 45 46 47 \& ... 40 41 42 43 \& 36 37 38 39 \& 32 33 34 35 \& 60 61 62 63 \& 56 57 58 59 \& 52 53 54 55 \& 48 49 50 51 \& \& ^ \& \-15\-14\-13\-12\-11\-10 \-9 \-8 \-7 \-6 \-5 \-4 \-3 \-2 \-1 X=0 1 2 3 4 5 6 .Ve .PP Notice the parts always replicate away from the origin, so the block N=16 to N=31 is near the origin at X=\-4, then N=32,48,64 are further away. .PP In this layout the replications still mesh together perfectly and all points on the triangular grid are visited. .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::CubicBase\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::CubicBase\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::CubicBase->new ()" .PD 0 .ie n .IP """$path = Math::PlanePath::CubicBase\->new (radix => $r)""" 4 .el .IP "\f(CW$path = Math::PlanePath::CubicBase\->new (radix => $r)\fR" 4 .IX Item "$path = Math::PlanePath::CubicBase->new (radix => $r)" .PD Create and return a new path object. .ie n .IP """($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$n\fR on the path. Points begin at 0 and if \f(CW\*(C`$n < 0\*(C'\fR then the return is an empty list. .SS "Level Methods" .IX Subsection "Level Methods" .ie n .IP """($n_lo, $n_hi) = $path\->level_to_n_range($level)""" 4 .el .IP "\f(CW($n_lo, $n_hi) = $path\->level_to_n_range($level)\fR" 4 .IX Item "($n_lo, $n_hi) = $path->level_to_n_range($level)" Return \f(CW\*(C`(0, $radix**$level \- 1)\*(C'\fR. .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::ImaginaryBase, Math::PlanePath::ImaginaryHalf .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 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 .