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Always turn off hyphenation; it makes .\" way too many mistakes in technical documents. .if n .ad l .nh .SH "NAME" Math::PlanePath::DekkingCurve \-\- 5x5 self\-similar edge curve .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::DekkingCurve; \& my $path = Math::PlanePath::DekkingCurve\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This is an integer version of a 5x5 self-similar curve by Dekking, .PP .Vb 10 \& 10 | 123\-124\-125\-... 86\-\-85 \& | | | | \& 9 | 115\-116\-117 122\-121 90\-\-89\-\-88\-\-87 84 \& | | | | | | \& 8 | 114\-113 118\-119\-120 91\-\-92\-\-93 82\-\-83 \& | | | | \& 7 | 112 107\-106 103\-102 95\-\-94 81 78\-\-77 \& | | | | | | | | | | \& 6 | 111 108 105\-104 101 96\-\-97 80\-\-79 76 \& | | | | | | \& 5 | 110\-109 14\-\-15 100\-\-99\-\-98 39\-\-40 75 66\-\-65 \& | | | | | | | | \& 4 | 10\-\-11\-\-12\-\-13 16 35\-\-36\-\-37\-\-38 41 74 71\-\-70 67 64 \& | | | | | | | | | | \& 3 | 9\-\-\-8\-\-\-7 18\-\-17 34\-\-33\-\-32 43\-\-42 73\-\-72 69\-\-68 63 \& | | | | | | \& 2 | 5\-\-\-6 19 22\-\-23 30\-\-31 44 47\-\-48 55\-\-56\-\-57 62\-\-61 \& | | | | | | | | | | | | \& 1 | 4\-\-\-3 20\-\-21 24 29\-\-28 45\-\-46 49 54\-\-53 58\-\-59\-\-60 \& | | | | | | \& Y=0 | 0\-\-\-1\-\-\-2 25\-\-26\-\-27 50\-\-51\-\-52 \& +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 .Ve .PP The base pattern is the N=0 to N=25 section. It then repeats with rotations or reversals which make the ends join. For example N=75 to N=100 is the base pattern in reverse, ie. from N=25 down to N=0. Or N=50 to N=75 is reverse and also rotate by \-90. .PP The curve segments are edges of squares in a 5x5 arrangement. .PP .Vb 10 \& +\- \- \-+\- \- \-+\- \- 14\-\-\-\-15 \-\-\-+ \& | | | | v |> | \& ^ ^ <| | \& 10\-\-\-\-11\-\-\-\-12\-\-\-\-13\- \- 16 \-\-+ \& | v |> | \& |> ^ ^ | \& 9\-\-\-\-\-8\-\-\-\-\-7 \-\- 18\-\-\-\-17 \-\-+ \& v | | |> | \& | ^ |> | ^ \& +\- \- 5\-\-\-\-\-6 \- 19 22\-\-\-\-23 \& | <| | <| \& | <| ^ | <| | \& +\- \- 4\-\-\-\-\-3 20\-\-\-\-21 \-\- 24 \& | v <| \& ^ ^ |> | | | \& 0\-\-\-\-\-1\-\-\-\-\-2 \-\- + \-\- \-+\- 25 .Ve .PP The little notch marks show which square each edge represents. This is the side the curve expands into at the next level. For example N=1 to N=2 has its notch on the left so the next level N=25 to N=50 expands on the left. .PP An expansion on the left is a repeat of the base shape, possibly rotated 90, 180 or 270 degrees. An expansion on the right is the base shape in reverse, as for example N=2 to N=3 on the right becomes N=50 to N=75 traversing to the right at the next level. .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for the behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::DekkingCurve\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::DekkingCurve\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::DekkingCurve->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::DekkingCentres, Math::PlanePath::CincoCurve, Math::PlanePath::PeanoCurve .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 2011, 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 .