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.IX Title "Math::PlanePath::HexSpiralSkewed 3pm"
.TH Math::PlanePath::HexSpiralSkewed 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::HexSpiralSkewed \-\- integer points around a skewed hexagonal spiral
.SH "SYNOPSIS"
.IX Header "SYNOPSIS"
.Vb 3
\& use Math::PlanePath::HexSpiralSkewed;
\& my $path = Math::PlanePath::HexSpiralSkewed\->new;
\& my ($x, $y) = $path\->n_to_xy (123);
.Ve
.SH "DESCRIPTION"
.IX Header "DESCRIPTION"
This path makes a hexagonal spiral with points skewed so as to fit a square
grid and fully cover the plane.
.PP
.Vb 9
\&    13\-\-12\-\-11   ...              2
\&     |         \e   \e
\&    14   4\-\-\-3  10  23            1
\&     |   |     \e   \e   \e
\&    15   5   1\-\-\-2   9  22    <\- Y=0
\&      \e   \e          |   | 
\&        16   6\-\-\-7\-\-\-8  21       \-1
\&          \e              |    
\&            17\-\-18\-\-19\-\-20       \-2
\&
\&     ^   ^   ^   ^   ^   ^ 
\&    \-2  \-1  X=0  1   2   3  ...
.Ve
.PP
The kinds of N=3*k^2 numbers which fall on straight lines in the plain
\&\f(CW\*(C`HexSpiral\*(C'\fR also fall on straight lines when skewed.  See
Math::PlanePath::HexSpiral for notes on this.
.SS "Skew"
.IX Subsection "Skew"
The skewed path is the same shape as the plain \f(CW\*(C`HexSpiral\*(C'\fR, but fits more
points on a square grid.  The skew pushes the top horizontal to the left, as
shown by the following parts, and the bottom horizontal is similarly skewed
but to the right.
.PP
.Vb 1
\&    HexSpiralSkewed               HexSpiral
\&
\&    13\-\-12\-\-11                   13\-\-12\-\-11       
\&     |         \e                /          \e      
\&    14          10            14            10    
\&     |             \e         /                \e  
\&    15               9     15                   9
\&
\&    \-2  \-1  X=0  1   2     \-4 \-3 \-2  X=0  2  3  4
.Ve
.PP
In general the coordinates can be converted each way by
.PP
.Vb 1
\&    plain X,Y \-> skewed (X\-Y)/2, Y
\&
\&    skewed X,Y \-> plain 2*X+Y, Y
.Ve
.SH "Corners"
.IX Header "Corners"
\&\f(CW\*(C`HexSpiralSkewed\*(C'\fR is similar to the \f(CW\*(C`SquareSpiral\*(C'\fR but cuts off the
top-right and bottom-left corners so that each loop is 6 steps longer than
the previous, whereas for the \f(CW\*(C`SquareSpiral\*(C'\fR it's 8.  See
\&\*(L"Corners\*(R" in Math::PlanePath::SquareSpiral for other corner cutting.
.SS "Wider"
.IX Subsection "Wider"
An optional \f(CW\*(C`wider\*(C'\fR parameter makes the path wider, stretched along the top
and bottom horizontals.  For example
.PP
.Vb 1
\&    $path = Math::PlanePath::HexSpiralSkewed\->new (wider => 2);
.Ve
.PP
gives
.PP
.Vb 9
\&    21\-\-20\-\-19\-\-18\-\-17                    2
\&     |                 \e    
\&    22   8\-\-\-7\-\-\-6\-\-\-5  16                1
\&     |   |             \e   \e    
\&    23   9   1\-\-\-2\-\-\-3\-\-\-4  15        <\- Y=0
\&      \e   \e                  |     
\&       24   10\-\-11\-\-12\-\-13\-\-14  ...      \-1
\&          \e                      |    
\&            25\-\-26\-\-27\-\-28\-\-29\-\-30       \-2
\&
\&     ^   ^   ^   ^   ^   ^   ^   ^ 
\&    \-4  \-3  \-2  \-1  X=0  1   2   3  ...
.Ve
.PP
The centre horizontal from N=1 is extended by \f(CW\*(C`wider\*(C'\fR many further places,
then the path loops around that shape.  The starting point 1 is shifted to
the left by wider/2 places (rounded up to an integer) to keep the spiral
centred on the origin X=0,Y=0.
.PP
Each loop is still 6 longer than the previous, since the widening is
basically a constant amount added into each loop.  The result is the same as
the plain \f(CW\*(C`HexSpiral\*(C'\fR of the same widening too.  The effect looks better in
the plain \f(CW\*(C`HexSpiral\*(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 with the same shape etc.  For example
to start at 0,
.PP
.Vb 1
\&    n_start => 0
\&
\&    27  26  25  24                            3
\&    28  12  11  10  23                        2
\&    29  13   3   2   9  22                    1
\&    30  14   4   0   1   8  21 ...       <\- Y=0
\&        31  15   5   6   7  20  39           \-1
\&            32  16  17  18  19  38           \-2
\&                33  34  35  36  37           \-3
\&                 
\&    \-3  \-2  \-1  X=0  1   2   3   4
.Ve
.PP
In this numbering the X axis N=0,1,8,21,etc is the octagonal numbers
3*X*(X+1).
.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::HexSpiralSkewed\->new ()""" 4
.el .IP "\f(CW$path = Math::PlanePath::HexSpiralSkewed\->new ()\fR" 4
.IX Item "$path = Math::PlanePath::HexSpiralSkewed->new ()"
.PD 0
.ie n .IP """$path = Math::PlanePath::HexSpiralSkewed\->new (wider => $w)""" 4
.el .IP "\f(CW$path = Math::PlanePath::HexSpiralSkewed\->new (wider => $w)\fR" 4
.IX Item "$path = Math::PlanePath::HexSpiralSkewed->new (wider => $w)"
.PD
Create and return a new hexagon spiral object.  An optional \f(CW\*(C`wider\*(C'\fR
parameter widens the spiral path, it defaults to 0 which is no widening.
.ie n .IP """$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
<http://oeis.org/A056105> (etc)
.RE
.PP
.Vb 6
\&    A056105    N on X axis, 3n^2\-2n+1
\&    A056106    N on Y axis, 3n^2\-n+1
\&    A056107    N on North\-West diagonal, 3n^2+1
\&    A056108    N on X negative axis, 3n^2+n+1
\&    A056109    N on Y negative axis, 3n^2+2n+1
\&    A003215    N on South\-East diagonal, centred hexagonals
\&
\&    n_start=0
\&      A000567    N on X axis, octagonal numbers
\&      A049450    N on Y axis
\&      A049451    N on X negative axis
\&      A045944    N on Y negative axis, octagonal numbers second kind
\&      A062783    N on X=Y diagonal north\-east
\&      A033428    N on north\-west diagonal, 3*k^2
\&      A063436    N on south\-west diagonal
\&      A028896    N on south\-east diagonal
.Ve
.SH "SEE ALSO"
.IX Header "SEE ALSO"
Math::PlanePath,
Math::PlanePath::HexSpiral,
Math::PlanePath::HeptSpiralSkewed,
Math::PlanePath::PentSpiralSkewed,
Math::PlanePath::DiamondSpiral
.SH "HOME PAGE"
.IX Header "HOME PAGE"
<http://user42.tuxfamily.org/math\-planepath/index.html>
.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 <http://www.gnu.org/licenses/>.