.\" Automatically generated by Pod::Man 2.28 (Pod::Simple 3.29) .\" .\" 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 turned on, 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::QuadricCurve 3pm" .TH Math::PlanePath::QuadricCurve 3pm "2016-05-03" "perl v5.22.2" "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::QuadricCurve \-\- eight segment zig\-zag .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::QuadricCurve; \& my $path = Math::PlanePath::QuadricCurve\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This is a self-similar zig-zag of eight segments, .PP .Vb 10 \& 18\-19 5 \& | | \& 16\-17 20 23\-24 4 \& | | | | \& 15\-14 21\-22 25\-26 3 \& | | \& 11\-12\-13 29\-28\-27 2 \& | | \& 2\-\-3 10\-\-9 30\-31 58\-59 ... 1 \& | | | | | | | \& 0\-\-1 4 7\-\-8 32 56\-57 60 63\-64 <\- Y=0 \& | | | | | | \& 5\-\-6 33\-34 55\-54 61\-62 \-1 \& | | \& 37\-36\-35 51\-52\-53 \-2 \& | | \& 38\-39 42\-43 50\-49 \-3 \& | | | | \& 40\-41 44 47\-48 \-4 \& | | \& 45\-46 \-5 \& ^ \& X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 .Ve .PP The base figure is the initial N=0 to N=8, .PP .Vb 5 \& 2\-\-\-3 \& | | \& 0\-\-\-1 4 7\-\-\-8 \& | | \& 5\-\-\-6 .Ve .PP It then repeats, turned to follow edge directions, so N=8 to N=16 is the same shape going upwards, then N=16 to N=24 across, N=24 to N=32 downwards, etc. .PP The result is the base at ever greater scale extending to the right and with wiggly lines making up the segments. The wiggles don't overlap. .PP The name \f(CW\*(C`QuadricCurve\*(C'\fR here is a slight mistake. Mandelbrot (\*(L"Fractal Geometry of Nature\*(R" 1982 page 50) calls any islands initiated from a square \&\*(L"quadric\*(R", only one of which is with sides by this eight segment expansion. This curve expansion also appears (unnamed) in Mandelbrot's \*(L"How Long is the Coast of Britain\*(R", 1967. .SS "Level Ranges" .IX Subsection "Level Ranges" A given replication extends to .PP .Vb 3 \& Nlevel = 8^level \& X = 4^level \& Y = 0 \& \& Ymax = 4^0 + 4^1 + ... + 4^level # 11...11 in base 4 \& = (4^(level+1) \- 1) / 3 \& Ymin = \- Ymax .Ve .SS "Turn" .IX Subsection "Turn" The sequence of turns made by the curve is straightforward. In the base 8 (octal) representation of N, the lowest non-zero digit gives the turn .PP .Vb 9 \& low digit turn (degrees) \& \-\-\-\-\-\-\-\-\- \-\-\-\-\-\-\-\-\-\-\-\-\-\- \& 1 +90 \& 2 \-90 \& 3 \-90 \& 4 0 \& 5 +90 \& 6 +90 \& 7 \-90 .Ve .PP When the least significant digit is non-zero it determines the turn, to make the base N=0 to N=8 shape. When the low digit is zero it's instead the next level up, the N=0,8,16,24,etc shape which is in control, applying a turn for the subsequent base part. So for example at N=16 = 20 octal 20 is a turn \&\-90 degrees. .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::QuadricCurve\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::QuadricCurve\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::QuadricCurve->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, 8**$level)\*(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 \& A133851 Y at N=2^k, being successive powers 2^j at k=1mod4 .Ve .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::QuadricIslands, Math::PlanePath::KochCurve .PP Math::Fractal::Curve \*(-- its \fIexamples/generator4.pl\fR is this curve .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 2011, 2012, 2013, 2014, 2015, 2016 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 .