.\" 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::HilbertSpiral 3pm" .TH Math::PlanePath::HilbertSpiral 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::HilbertSpiral \-\- 2x2 self\-similar spiral .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::HilbertSpiral; \& my $path = Math::PlanePath::HilbertSpiral\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This is a Hilbert curve variation which fills the plane by spiralling around into negative X,Y on every second replication level. .PP .Vb 10 \& ..\-\-63\-\-62 49\-\-48\-\-47 44\-\-43\-\-42 5 \& | | | | | \& 60\-\-61 50\-\-51 46\-\-45 40\-\-41 4 \& | | | \& 59 56\-\-55 52 33\-\-34 39\-\-38 3 \& | | | | | | | \& 58\-\-57 54\-\-53 32 35\-\-36\-\-37 2 \& | \& 5\-\- 4\-\- 3\-\- 2 31 28\-\-27\-\-26 1 \& | | | | | \& 6\-\- 7 0\-\- 1 30\-\-29 24\-\-25 <\- Y=0 \& | | \& 9\-\- 8 13\-\-14 17\-\-18 23\-\-22 \-1 \& | | | | | | \& 10\-\-11\-\-12 15\-\-16 19\-\-20\-\-21 \-2 \& \& \-2 \-1 X=0 1 2 3 4 5 .Ve .PP The curve starts with the same N=0 to N=3 as the \f(CW\*(C`HilbertCurve\*(C'\fR, then the following 2x2 blocks N=4 to N=15 go around in negative X,Y. The top-left corner for this negative direction is at Ntopleft=4^level\-1 for an odd numbered level. .PP The parts of the curve in the X,Y negative parts are the same as the plain \&\f(CW\*(C`HilbertCurve\*(C'\fR, just mirrored along the anti-diagonal. For example. N=4 to N=15 .PP .Vb 1 \& HilbertSpiral HilbertCurve \& \& \e 5\-\-\-6 9\-\-10 \& \e | | | | \& \e 4 7\-\-\-8 11 \& \e | \& 5\-\- 4 \e 13\-\-12 \& | \e | \& 6\-\- 7 \e 14\-\-15 \& | \e \& 9\-\- 8 13\-\-14 \e \& | | | \e \& 10\-\-11\-\-12 15 .Ve .PP This mirroring has the effect of mapping .PP .Vb 1 \& HilbertCurve X,Y \-> \-Y,\-X for HilbertSpiral .Ve .PP Notice the coordinate difference (\-Y)\-(\-X) = X\-Y so that difference, representing a projection onto the X=\-Y opposite diagonal, is the same in both paths. .SS "Level Ranges" .IX Subsection "Level Ranges" Reckoning the initial N=0 to N=3 as level 1, a replication level extends to .PP .Vb 2 \& Nstart = 0 \& Nlevel = 4^level \- 1 (inclusive) \& \& Xmin = Ymin = \- (4^floor(level/2) \- 1) * 2 / 3 \& = binary 1010...10 \& Xmax = Ymax = (4^ceil(level/2) \- 1) / 3 \& = binary 10101...01 \& \& width = height = Xmax \- Xmin \& = Ymax \- Ymin \& = 2^level \- 1 .Ve .PP The X,Y range doubles alternately above and below, so the result is a 1 bit going alternately to the max or min, starting with the max for level 1. .PP .Vb 10 \& level X,Ymin binary X,Ymax binary \& \-\-\-\-\- \-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \-\-\-\-\-\-\-\-\-\-\-\-\-\- \& 0 0 0 \& 1 0 0 1 = 1 \& 2 \-2 = \-10 1 = 01 \& 3 \-2 = \-010 5 = 101 \& 4 \-10 = \-1010 5 = 0101 \& 5 \-10 = \-01010 21 = 10101 \& 6 \-42 = \-101010 21 = 010101 \& 7 \-42 = \-0101010 85 = 1010101 .Ve .PP The power\-of\-4 formulas above for Ymin/Ymax have the effect of producing alternating bit patterns like this. .PP This is the same sort of level range as \f(CW\*(C`BetaOmega\*(C'\fR has on its Y coordinate, but on this \f(CW\*(C`HilbertSpiral\*(C'\fR it applies to both X and Y. .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::HilbertSpiral\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::HilbertSpiral\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::HilbertSpiral->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. .ie n .IP """($n_lo, $n_hi) = $path\->rect_to_n_range ($x1,$y1, $x2,$y2)""" 4 .el .IP "\f(CW($n_lo, $n_hi) = $path\->rect_to_n_range ($x1,$y1, $x2,$y2)\fR" 4 .IX Item "($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)" The returned range is exact, meaning \f(CW$n_lo\fR and \f(CW$n_hi\fR are the smallest and biggest in the rectangle. .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, 4**$level \- 1)\*(C'\fR. .SH "OEIS" .IX Header "OEIS" Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include .Sp .RS 4 (etc) .RE .PP .Vb 1 \& A059285 X\-Y coordinate diff .Ve .PP The difference X\-Y is the same as the \f(CW\*(C`HilbertCurve\*(C'\fR, since the \*(L"negative\*(R" spiral parts are mirrored across the X=\-Y anti-diagonal, which means coordinates (\-Y,\-X) and \-Y\-(\-X) = X\-Y. .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::HilbertCurve, Math::PlanePath::BetaOmega .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 .