.\" Automatically generated by Pod::Man 4.09 (Pod::Simple 3.35) .\" .\" 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 .. .if !\nF .nr F 0 .if \nF>0 \{\ . de IX . tm Index:\\$1\t\\n%\t"\\$2" .. . if !\nF==2 \{\ . nr % 0 . nr F 2 . \} .\} .\" ======================================================================== .\" .IX Title "Math::PlanePath::DekkingCentres 3pm" .TH Math::PlanePath::DekkingCentres 3pm "2018-03-18" "perl v5.26.1" "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::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 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 .