.\" 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::CubicBase 3pm" .TH Math::PlanePath::CubicBase 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::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=\fBcuberoot\fR\|(1) = \-1/2+i*\fBsqrt\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, 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 .