.\" 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::DekkingCentres 3pm" .TH Math::PlanePath::DekkingCentres 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::DekkingCentres \-\- 5x5 self\-similar .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::DekkingCentres; \& my $path = Math::PlanePath::DekkingCentres\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This is a variation on a 5x5 self-similar curve from .Sp .RS 4 F. M. Dekking, \*(L"Recurrent Sets\*(R", Advances in Mathematics, volume 44, 1982, pages 79\-104, section 4.9 \*(L"Gosper-Type Curves\*(R" .RE .PP and which is a horizontal mirror image of the E\-curve of McKenna 1978. .PP The form visits the \*(L"centres\*(R" of the 5x5 self-similar unit squares of the pattern. The result is some diagonal steps, but replications wholly within 5x5 areas. .PP .Vb 10 \& ... \& | / \& 9 | 115\-116 122\-123\-124 89\-\-88 86\-\-85\-\-84 \& | | | \e | \e | | \& 8 | 114 117\-118 121\-120 90 92 87 82\-\-83 \& | | \e / |/ \e | \& 7 | 113\-112 106 119 102 91 94\-\-93 81 77 \& | / / | / | / / / | \& 6 | 111 107 105 103 101 95\-\-96 80 78 76 \& | | \e \e | | \e \e | | \& 5 | 110\-109\-108 104 100\-\-99\-\-98\-\-97 79 75 \& | \e \& 4 | 10\-\-11 13\-\-14\-\-15 35\-\-36 38\-\-39\-\-40 74 70 66\-\-65\-\-64 \& | | \e | | | \e | | | |\e \e | \& 3 | 9 7 12 17\-\-16 34 32 37 42\-\-41 73 71 69 67 63 \& | |/ \e | |/ \e | |/ |/ / \& 2 | 8 5\-\- 6 18 22 33 30\-\-31 43 47 72 55 68 62\-\-61 \& | / / / | / / / | / \e | \& 1 | 4\-\- 3 19 21 23 29\-\-28 44 46 48 54\-\-53 56\-\-57 60 \& | \e \e | | \e \e | | \e | | \& Y=0 | 0\-\- 1\-\- 2 20 24\-\-25\-\-26\-\-27 45 49\-\-50\-\-51\-\-52 58\-\-59 \& +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 .Ve .PP The base pattern is the N=0 to N=24 section. It repeats with rotations or reversals which make the ends join. For example N=75 to N=99 is the base pattern in reverse. Or N=50 to N=74 is reverse and also rotate by \-90. .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath the behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::DekkingCentres\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::DekkingCentres\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::DekkingCentres->new ()" 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, 25**$level \- 1)\*(C'\fR. .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::DekkingCurve, Math::PlanePath::CincoCurve, Math::PlanePath::PeanoCurve .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 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 .