.\" Automatically generated by Pod::Man 2.28 (Pod::Simple 3.28) .\" .\" 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 .\" .\" Accent mark definitions (@(#)ms.acc 1.5 88/02/08 SMI; from UCB 4.2). .\" Fear. Run. Save yourself. No user-serviceable parts. . \" fudge factors for nroff and troff .if n \{\ . ds #H 0 . ds #V .8m . ds #F .3m . ds #[ \f1 . ds #] \fP .\} .if t \{\ . ds #H ((1u-(\\\\n(.fu%2u))*.13m) . ds #V .6m . ds #F 0 . ds #[ \& . ds #] \& .\} . \" simple accents for nroff and troff .if n \{\ . ds ' \& . ds ` \& . ds ^ \& . ds , \& . ds ~ ~ . ds / .\} .if t \{\ . ds ' \\k:\h'-(\\n(.wu*8/10-\*(#H)'\'\h"|\\n:u" . ds ` \\k:\h'-(\\n(.wu*8/10-\*(#H)'\`\h'|\\n:u' . ds ^ \\k:\h'-(\\n(.wu*10/11-\*(#H)'^\h'|\\n:u' . ds , \\k:\h'-(\\n(.wu*8/10)',\h'|\\n:u' . ds ~ \\k:\h'-(\\n(.wu-\*(#H-.1m)'~\h'|\\n:u' . ds / \\k:\h'-(\\n(.wu*8/10-\*(#H)'\z\(sl\h'|\\n:u' .\} . \" troff and (daisy-wheel) nroff accents .ds : \\k:\h'-(\\n(.wu*8/10-\*(#H+.1m+\*(#F)'\v'-\*(#V'\z.\h'.2m+\*(#F'.\h'|\\n:u'\v'\*(#V' .ds 8 \h'\*(#H'\(*b\h'-\*(#H' .ds o \\k:\h'-(\\n(.wu+\w'\(de'u-\*(#H)/2u'\v'-.3n'\*(#[\z\(de\v'.3n'\h'|\\n:u'\*(#] .ds d- \h'\*(#H'\(pd\h'-\w'~'u'\v'-.25m'\f2\(hy\fP\v'.25m'\h'-\*(#H' .ds D- D\\k:\h'-\w'D'u'\v'-.11m'\z\(hy\v'.11m'\h'|\\n:u' .ds th \*(#[\v'.3m'\s+1I\s-1\v'-.3m'\h'-(\w'I'u*2/3)'\s-1o\s+1\*(#] .ds Th \*(#[\s+2I\s-2\h'-\w'I'u*3/5'\v'-.3m'o\v'.3m'\*(#] .ds ae a\h'-(\w'a'u*4/10)'e .ds Ae A\h'-(\w'A'u*4/10)'E . \" corrections for vroff .if v .ds ~ \\k:\h'-(\\n(.wu*9/10-\*(#H)'\s-2\u~\d\s+2\h'|\\n:u' .if v .ds ^ \\k:\h'-(\\n(.wu*10/11-\*(#H)'\v'-.4m'^\v'.4m'\h'|\\n:u' . \" for low resolution devices (crt and lpr) .if \n(.H>23 .if \n(.V>19 \ \{\ . ds : e . ds 8 ss . ds o a . ds d- d\h'-1'\(ga . ds D- D\h'-1'\(hy . ds th \o'bp' . ds Th \o'LP' . ds ae ae . ds Ae AE .\} .rm #[ #] #H #V #F C .\" ======================================================================== .\" .IX Title "Math::PlanePath::FilledRings 3pm" .TH Math::PlanePath::FilledRings 3pm "2014-06-14" "perl v5.20.1" "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::FilledRings \-\- concentric filled lattice rings .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::FilledRings; \& my $path = Math::PlanePath::FilledRings\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This path puts points on integer X,Y pixels of filled rings with radius 1 unit each ring. .PP .Vb 10 \& 110\-109\-108\-107\-106 6 \& / \e \& 112\-111 79\-\-78\-\-77\-\-76\-\-75 105\-104 5 \& | / \e | \& 114\-113 80 48\-\-47\-\-46\-\-45\-\-44 74 103\-102 4 \& | / | | \e | \& 115 81 50\-\-49 27\-\-26\-\-25 43\-\-42 73 101 3 \& / / | / \e | \e \e \& 116 82 52\-\-51 28 14\-\-13\-\-12 24 41\-\-40 72 100 2 \& | | | / / \e \e | | | \& 117 83 53 29 15 5\-\- 4\-\- 3 11 23 39 71 99 1 \& | | | | | | | | | | | | \& 118 84 54 30 16 6 1\-\- 2 10 22 38 70 98 <\- Y=0 \& | | | | | | / / / / / / \& 119 85 55 31 17 7\-\- 8\-\- 9 21 37 69 97 137 \-1 \& | | | \e \e / / | | | \& 120 86 56\-\-57 32 18\-\-19\-\-20 36 67\-\-68 96 136 \-2 \& \e \e | \e / | / / \& 121 87 58\-\-59 33\-\-34\-\-35 65\-\-66 95 135 \-3 \& | \e | | / | \& 122\-123 88 60\-\-61\-\-62\-\-63\-\-64 94 133\-134 \-4 \& | \e / | \& 124\-125 89\-\-90\-\-91\-\-92\-\-93 131\-132 \-5 \& \e / \& 126\-127\-128\-129\-130 \& \& ^ \& \-6 \-5 \-4 \-3 \-2 \-1 X=0 1 2 3 4 5 6 .Ve .PP For example the ring N=22 to N=37 is all the points .PP .Vb 2 \& 2.5 < hypot(X,Y) < 3.5 \& where hypot(X,Y) = sqrt(X^2+Y^2) .Ve .SS "N Start" .IX Subsection "N Start" The default is to number points starting N=1 as shown above. An optional \&\f(CW\*(C`n_start\*(C'\fR can give a different start with the same shape. For example to start at 0, .PP .Vb 10 \& 26\-25\-24 n_start => 0 \& / \e \& 27 13\-12\-11 23 \& / / \e \e \& 28 14 4\-\-3\-\-2 10 22 \& | | | | | | \& 29 15 5 0\-\-1 9 21 \& | | | / / / \& 30 16 6\-\-7\-\-8 20 36 \& \e \e / / \& 31 17\-18\-19 35 \& \e / \& 8 32\-33\-34 .Ve .PP The only effect is to push the N values by a constant amount but can help match N on the axes to counts of X,Y points < R or similar. .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for the behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::FilledRings\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::FilledRings\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::FilledRings->new ()" .PD 0 .ie n .IP """$path = Math::PlanePath::FilledRings\->new (n_start => $n)""" 4 .el .IP "\f(CW$path = Math::PlanePath::FilledRings\->new (n_start => $n)\fR" 4 .IX Item "$path = Math::PlanePath::FilledRings->new (n_start => $n)" .PD Create and return a new path object. .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 3 \& A036705 first diffs of N on X axis, \& being count of X,Y points n\-1/2 < norm <= n+1/2 \& A036706 1/4 of those diffs \& \& n_start=1 (the default) \& A036707 N/2+X\-1 along X axis, \& being count norm <= n+1/2 in half plane \& A036708 (N(X,0)\-N(X\-1,0))/2+1, \& first diffs of the half plane count \& \& n_start=0 \& A036704 N on X axis, from X=1 onwards \& count of X,Y points norm <= n+1/2 .Ve .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::PixelRings, Math::PlanePath::Hypot, Math::PlanePath::MultipleRings .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 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 .