.\" Automatically generated by Pod::Man 4.14 (Pod::Simple 3.40) .\" .\" 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 .. .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::DiamondArms 3pm" .TH Math::PlanePath::DiamondArms 3pm "2021-01-23" "perl v5.32.0" "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::DiamondArms \-\- four spiral arms .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::DiamondArms; \& my $path = Math::PlanePath::DiamondArms\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This path follows four spiral arms, each advancing successively in a diamond pattern, .PP .Vb 8 \& 25 ... 4 \& 29 14 21 36 3 \& 33 18 7 10 17 32 2 \& ... 22 11 4 3 6 13 28 1 \& 26 15 8 1 2 9 24 ... <\- Y=0 \& 30 19 12 5 20 35 \-1 \& 34 23 16 31 \-2 \& ... 27 \-3 \& \& ^ \& \-3 \-2 \-1 X=0 1 2 3 4 .Ve .PP Each arm makes a spiral widening out by 4 each time around, thus leaving room for four such arms. Each arm loop is 64 longer than the preceding loop. For example N=13 to N=85 below is 84\-13=72 points, and the next loop N=85 to N=221 is 221\-85=136 which is an extra 64, ie. 72+64=136. .PP .Vb 10 \& 25 ... \& / \e \e \& 29 . 21 . . . 93 \& / \e \e \& 33 . . . 17 . . . 89 \& / \e \e \& 37 . . . . . 13 . . . 85 \& / / / \& 41 . . . 1 . 9 . . . 81 \& \e \e / / \& 45 . . . 5 . . . 77 \& \e / \& 49 . . . . . 73 \& \e / \& 53 . . . 69 \& \e / \& 57 . 65 \& \e / \& 61 .Ve .PP Each arm is N=4*k+rem for a remainder rem=0,1,2,3, so sequences related to multiples of 4 or with a modulo 4 pattern may fall on particular arms. .PP The starts of each arm N=1,2,3,4 are at X=0 or 1 and Y=0 or 1, .PP .Vb 9 \& .. \& \e \& 4 3 .. Y=1 \& / / \& .. 1 2 <\- Y=0 \& \e \& .. \& ^ ^ \& X=0 X=1 .Ve .PP They could be centred around the origin by taking X\-1/2,Y\-1/2 so for example N=1 would be at \-1/2,\-1/2. But the it's done as N=1 at 0,0 to stay in integers. .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::DiamondArms\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::DiamondArms\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::DiamondArms->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$n\fR on the path. For \f(CW\*(C`$n < 1\*(C'\fR the return is an empty list, as the path starts at 1. .Sp Fractional \f(CW$n\fR gives a point on the line between \f(CW$n\fR and \f(CW\*(C`$n+4\*(C'\fR, that \&\f(CW\*(C`$n+4\*(C'\fR being the next point on the same spiralling arm. This is probably of limited use, but arises fairly naturally from the calculation. .SS "Descriptive Methods" .IX Subsection "Descriptive Methods" .ie n .IP """$arms = $path\->arms_count()""" 4 .el .IP "\f(CW$arms = $path\->arms_count()\fR" 4 .IX Item "$arms = $path->arms_count()" Return 4. .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::SquareArms, Math::PlanePath::DiamondSpiral .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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 .