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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::HeptSpiralSkewed \-\- integer points around a skewed seven sided spiral .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::HeptSpiralSkewed; \& my $path = Math::PlanePath::HeptSpiralSkewed\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This path makes a seven-sided spiral by cutting one corner of a square .PP .Vb 10 \& 31\-30\-29\-28 3 \& | \e \& 32 14\-13\-12 27 2 \& | | \e \e \& 33 15 4\-\-3 11 26 1 \& | | | \e \e \e \& 34 16 5 1\-\-2 10 25 <\- Y=0 \& | | | | | \& 35 17 6\-\-7\-\-8\-\-9 24 \-1 \& | | | \& 36 18\-19\-20\-21\-22\-23 \-2 \& | \& 37\-38\-39\-40\-41\-... \-3 \& \& ^ \& \-3 \-2 \-1 X=0 1 2 3 .Ve .PP The path is as if around a heptagon, with the left and bottom here as two sides of the heptagon straightened out, and the flat top here skewed across to fit a square grid. .SS "N Start" .IX Subsection "N Start" The default is to number points starting N=1 as shown above. An optional \&\f(CW\*(C`n_start\*(C'\fR can give a different start, in the same pattern. For example to start at 0, .PP .Vb 7 \& 30 29 28 27 n_start => 0 \& 31 13 12 11 26 \& 32 14 3 2 10 25 \& 33 15 4 0 1 9 24 \& 34 16 5 6 7 8 23 \& 35 17 18 19 20 21 22 \& 36 37 38 39 40 ... .Ve .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::HeptSpiralSkewed\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::HeptSpiralSkewed\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::HeptSpiralSkewed->new ()" .PD 0 .ie n .IP """$path = Math::PlanePath::HeptSpiralSkewed\->new (n_start => $n)""" 4 .el .IP "\f(CW$path = Math::PlanePath::HeptSpiralSkewed\->new (n_start => $n)\fR" 4 .IX Item "$path = Math::PlanePath::HeptSpiralSkewed->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 N in the path as centred in a square of side 1, so the entire plane is covered. .SH "FORMULAS" .IX Header "FORMULAS" .SS "N to X,Y" .IX Subsection "N to X,Y" It's convenient to work in terms of Nstart=0 and to take each loop as beginning on the South-West diagonal, .PP .Vb 1 \& top length = d \& \& 30\-29\-28\-27 \& | \e \& 31 26 diagonal length = d \& left | \e \& length 32 25 \& = 2*d | \e \& 33 0 24 \& | | right \& 34 . 23 length = d\-1 \& | | \& 35 17\-18\-19\-20\-21\-22 \& | \& . bottom length = 2*d\-1 .Ve .PP The \s-1SW\s0 diagonal is N=0,5,17,36,etc which is .PP .Vb 1 \& N = (7d\-11)*d/2 + 2 # starting d=1 first loop .Ve .PP This can be inverted to get d from N .PP .Vb 1 \& d = floor( (sqrt(56*N+9)+11)/14 ) .Ve .PP The side lengths are as shown above. The first loop is d=1 and for it the \&\*(L"right\*(R" vertical length is zero, so no such side on that first loop 0 <= N < 5. .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 2 \& n_start=1 \& A140065 N on Y axis \& \& n_start=0 \& A001106 N on X axis, 9\-gonal numbers \& A218471 N on Y axis \& A022265 N on X negative axis \& A179986 N on Y negative axis, second 9\-gonals \& A195023 N on X=Y diagonal \& A022264 N on North\-West diagonal \& A186029 N on South\-West diagonal \& A024966 N on South\-East diagonal .Ve .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::SquareSpiral, Math::PlanePath::PentSpiralSkewed, Math::PlanePath::HexSpiralSkewed .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 2010, 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 .