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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::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. .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 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 .