.\" 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::PentSpiralSkewed 3pm" .TH Math::PlanePath::PentSpiralSkewed 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::PentSpiralSkewed \-\- integer points in a pentagonal shape .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::PentSpiralSkewed; \& my $path = Math::PlanePath::PentSpiralSkewed\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This path makes a pentagonal (five-sided) spiral with points skewed so as to fit a square grid and fully cover the plane. .PP .Vb 9 \& 10 ... 2 \& / \e \e \& 11 3 9 20 1 \& / / \e \e \e \& 12 4 1\-\-2 8 19 <\- Y=0 \& \e \e | | \& 13 5\-\-6\-\-7 18 \-1 \& \e | \& 14\-15\-16\-17 \-2 \& \& ^ ^ ^ ^ ^ ^ \& \-2 \-1 X=0 1 2 3 ... .Ve .PP The pattern is similar to the \f(CW\*(C`SquareSpiral\*(C'\fR but cuts three corners which makes each cycle is faster. Each cycle is just 5 steps longer than the previous (where it's 8 for a \f(CW\*(C`SquareSpiral\*(C'\fR). .SS "N Start" .IX Subsection "N Start" The default is to number points starting N=1 as shown above. An optional \&\f(CW\*(C`n_start\*(C'\fR can give a different start, in the same pattern. For example to start at 0, .PP .Vb 9 \& 38 n_start => 0 \& 39 21 37 ... \& 40 22 9 20 36 57 \& 41 23 10 2 8 19 35 56 \& 42 24 11 3 0 1 7 18 34 55 \& 43 25 12 4 5 6 17 33 54 \& 44 26 13 14 15 16 32 53 \& 45 27 28 29 30 31 52 \& 46 47 48 49 50 51 .Ve .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::PentSpiral\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::PentSpiral\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::PentSpiral->new ()" .PD 0 .ie n .IP """$path = Math::PlanePath::PentSpiral\->new (n_start => $n)""" 4 .el .IP "\f(CW$path = Math::PlanePath::PentSpiral\->new (n_start => $n)\fR" 4 .IX Item "$path = Math::PlanePath::PentSpiral->new (n_start => $n)" .PD Create and return a new path object. .ie n .IP """$n = $path\->xy_to_n ($x,$y)""" 4 .el .IP "\f(CW$n = $path\->xy_to_n ($x,$y)\fR" 4 .IX Item "$n = $path->xy_to_n ($x,$y)" Return the point number for coordinates \f(CW\*(C`$x,$y\*(C'\fR. \f(CW$x\fR and \f(CW$y\fR are each rounded to the nearest integer, which has the effect of treating each point in the path as a square of side 1. .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 7 \& n_start=1 (the default) \& A192136 N on X axis, (5*n^2 \- 3*n + 2)/2 \& A140066 N on Y axis \& A116668 N on X negative axis, (5n^2 + n + 2)/2 \& A134238 N on Y negative axis \& A158187 N on North\-West diagonal, 10*n^2 + 1 \& A005891 N on South\-East diagonal, centred pentagonals \& \& n_start=0 \& A000566 N on X axis, heptagonal numbers \& A005476 N on Y axis \& A005475 N on X negative axis \& A147875 N on Y negative axis, second heptagonals \& A033583 N on North\-West diagonal, 10*n^2 \& A028895 N on South\-East diagonal, 5*triangular .Ve .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::SquareSpiral, Math::PlanePath::DiamondSpiral, Math::PlanePath::HexSpiralSkewed .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 2010, 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 .