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.IX Title "Math::PlanePath::TriangleSpiralSkewed 3pm"
.TH Math::PlanePath::TriangleSpiralSkewed 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::TriangleSpiralSkewed \-\- integer points drawn around a skewed equilateral triangle
.SH "SYNOPSIS"
.IX Header "SYNOPSIS"
.Vb 3
\& use Math::PlanePath::TriangleSpiralSkewed;
\& my $path = Math::PlanePath::TriangleSpiralSkewed\->new;
\& my ($x, $y) = $path\->n_to_xy (123);
.Ve
.SH "DESCRIPTION"
.IX Header "DESCRIPTION"
This path makes an spiral shaped as an equilateral triangle (each side the
same length), but skewed to the left to fit on a square grid,
.PP
.Vb 10
\&    16                              4
\&     |\e
\&    17 15                           3
\&     |   \e
\&    18  4 14                        2
\&     |  |\e  \e
\&    19  5  3 13                     1
\&     |  |   \e  \e
\&    20  6  1\-\-2 12 ...         <\- Y=0
\&     |  |         \e  \e
\&    21  7\-\-8\-\-9\-10\-11 30           \-1
\&     |                  \e
\&    22\-23\-24\-25\-26\-27\-28\-29        \-2
\&
\&           ^
\&    \-2 \-1 X=0 1  2  3  4  5
.Ve
.PP
The properties are the same as the spread-out \f(CW\*(C`TriangleSpiral\*(C'\fR.  The
triangle numbers fall on straight lines as they do in the \f(CW\*(C`TriangleSpiral\*(C'\fR
but the skew means the top corner goes up at an angle to the vertical and
the left and right downwards are different angles plotted (but are symmetric
by N count).
.SS "Skew Right"
.IX Subsection "Skew Right"
Option \f(CW\*(C`skew => \*(Aqright\*(Aq\*(C'\fR directs the skew towards the right, giving
.PP
.Vb 11
\&      4                  16      skew="right"
\&                        / |
\&      3               17 15
\&                     /    |
\&      2            18  4 14
\&                  /  / |  |
\&      1        ...  5  3 13
\&                  /    |  |
\&    Y=0 \->       6  1\-\-2 12
\&               /          |
\&     \-1       7\-\-8\-\-9\-10\-11
\&
\&                    ^
\&             \-2 \-1 X=0 1  2
.Ve
.PP
This is a shear \*(L"X \-> X+Y\*(R" of the default skew=\*(L"left\*(R" shown above.  The
coordinates are related by
.PP
.Vb 2
\&    Xright = Xleft + Yleft         Xleft = Xright \- Yright
\&    Yright = Yleft                 Yleft = Yright
.Ve
.SS "Skew Up"
.IX Subsection "Skew Up"
.Vb 9
\&      2       16\-15\-14\-13\-12\-11      skew="up"
\&               |            /   
\&      1       17  4\-\-3\-\-2 10
\&               |  |   /  /  
\&    Y=0 \->    18  5  1  9 
\&               |  |   /  
\&     \-1      ...  6  8 
\&                  |/  
\&     \-2           7 
\&
\&                    ^
\&             \-2 \-1 X=0 1  2
.Ve
.PP
This is a shear \*(L"Y \-> X+Y\*(R" of the default skew=\*(L"left\*(R" shown above.  The
coordinates are related by
.PP
.Vb 2
\&    Xup = Xleft                 Xleft = Xup
\&    Yup = Yleft + Xleft         Yleft = Yup \- Xup
.Ve
.SS "Skew Down"
.IX Subsection "Skew Down"
.Vb 11
\&      2          ..\-18\-17\-16       skew="down"
\&                           |  
\&      1        7\-\-6\-\-5\-\-4 15 
\&                \e       |  | 
\&    Y=0 \->        8  1  3 14 
\&                   \e  \e |  | 
\&     \-1              9  2 13 
\&                      \e    | 
\&     \-2                10 12 
\&                         \e | 
\&                          11 
\&
\&                     ^
\&              \-2 \-1 X=0 1  2
.Ve
.PP
This is a rotate by \-90 degrees of the skew=\*(L"up\*(R" above.  The coordinates are
related
.PP
.Vb 2
\&    Xdown = Yup          Xup = \- Ydown
\&    Ydown = \- Xup        Yup = Xdown
.Ve
.PP
Or related to the default skew=\*(L"left\*(R" by
.PP
.Vb 2
\&    Xdown = Yleft + Xleft        Xleft = \- Ydown
\&    Ydown = \- Xleft              Yleft = Xdown + Ydown
.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 etc.  For example
to start at 0,
.PP
.Vb 10
\&    15        n_start => 0
\&     |\e
\&    16 14
\&     |   \e
\&    17  3 13 ...
\&     |  |\e  \e  \e
\&    18  4  2 12 31
\&     |  |   \e  \e  \e
\&    19  5  0\-\-1 11 30
\&     |  |         \e  \e
\&    20  6\-\-7\-\-8\-\-9\-10 29
\&     |                  \e
\&    21\-22\-23\-24\-25\-26\-27\-28
.Ve
.PP
With this adjustment for example the X axis N=0,1,11,30,etc is (9X\-7)*X/2,
the hendecagonal numbers (11\-gonals).  And South-East N=0,8,25,etc is the
hendecagonals of the second kind, (9Y\-7)*Y/2 with Y negative.
.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::TriangleSpiralSkewed\->new ()""" 4
.el .IP "\f(CW$path = Math::PlanePath::TriangleSpiralSkewed\->new ()\fR" 4
.IX Item "$path = Math::PlanePath::TriangleSpiralSkewed->new ()"
.PD 0
.ie n .IP """$path = Math::PlanePath::TriangleSpiralSkewed\->new (skew => $str, n_start => $n)""" 4
.el .IP "\f(CW$path = Math::PlanePath::TriangleSpiralSkewed\->new (skew => $str, n_start => $n)\fR" 4
.IX Item "$path = Math::PlanePath::TriangleSpiralSkewed->new (skew => $str, n_start => $n)"
.PD
Create and return a new skewed triangle spiral object.  The \f(CW\*(C`skew\*(C'\fR
parameter can be
.Sp
.Vb 4
\&    "left"    (the default)
\&    "right"
\&    "up"
\&    "down"
.Ve
.ie n .IP """$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 "Rectangle to N Range"
.IX Subsection "Rectangle to N Range"
Within each row there's a minimum N and the N values then increase
monotonically away from that minimum point.  Likewise in each column.  This
means in a rectangle the maximum N is at one of the four corners of the
rectangle.
.PP
.Vb 8
\&              |
\&    x1,y2 M\-\-\-|\-\-\-\-M x2,y2        maximum N at one of
\&          |   |    |              the four corners
\&       \-\-\-\-\-\-\-O\-\-\-\-\-\-\-\-\-          of the rectangle
\&          |   |    |
\&          |   |    |
\&    x1,y1 M\-\-\-|\-\-\-\-M x1,y1
\&              |
.Ve
.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/A117625> (etc)
.RE
.PP
.Vb 4
\&    n_start=1, skew="left" (the defaults)
\&      A204439     abs(dX)
\&      A204437     abs(dY)
\&      A010054     turn 1=left,0=straight, extra initial 1
\&
\&      A117625     N on X axis
\&      A064226     N on Y axis, but without initial value=1
\&      A006137     N on X negative
\&      A064225     N on Y negative
\&      A081589     N on X=Y leading diagonal
\&      A038764     N on X=Y negative South\-West diagonal
\&      A081267     N on X=\-Y negative South\-East diagonal
\&      A060544     N on ESE slope dX=+2,dY=\-1
\&      A081272     N on SSE slope dX=+1,dY=\-2
\&
\&      A217010     permutation N values of points in SquareSpiral order
\&      A217291      inverse
\&      A214230     sum of 8 surrounding N
\&      A214231     sum of 4 surrounding N
\&
\&    n_start=0
\&      A051682     N on X axis (11\-gonal numbers)
\&      A081268     N on X=1 vertical (next to Y axis)
\&      A062708     N on Y axis
\&      A062725     N on Y negative axis
\&      A081275     N on X=Y+1 North\-East diagonal
\&      A062728     N on South\-East diagonal (11\-gonal second kind)
\&      A081266     N on X=Y negative South\-West diagonal
\&      A081270     N on X=1\-Y North\-West diagonal, starting N=3
\&      A081271     N on dX=\-1,dY=2 NNW slope up from N=1 at X=1,Y=0
\&
\&    n_start=\-1
\&      A023531     turn 1=left,0=straight, being 1 at N=k*(k+3)/2
\&      A023532     turn 1=straight,0=left
\&
\&    n_start=1, skew="right"
\&      A204435     abs(dX)
\&      A204437     abs(dY)
\&      A217011     permutation N values of points in SquareSpiral order
\&                    but with 90\-degree rotation
\&      A217292     inverse
\&      A214251     sum of 8 surrounding N
\&
\&    n_start=1, skew="up"
\&      A204439     abs(dX)
\&      A204435     abs(dY)
\&      A217012     permutation N values of points in SquareSpiral order
\&                    but with 90\-degree rotation
\&      A217293     inverse
\&      A214252     sum of 8 surrounding N
\&
\&    n_start=1, skew="down"
\&      A204435     abs(dX)
\&      A204439     abs(dY)
.Ve
.PP
The square spiral order in A217011,A217012 and their inverses has first step
at 90\-degrees to the first step of the triangle spiral, hence the rotation
by 90 degrees when relating to the \f(CW\*(C`SquareSpiral\*(C'\fR path.  A217010 on the
other hand has no such rotation since it reckons the square and triangle
spirals starting in the same direction.
.SH "SEE ALSO"
.IX Header "SEE ALSO"
Math::PlanePath,
Math::PlanePath::TriangleSpiral,
Math::PlanePath::PyramidSpiral,
Math::PlanePath::SquareSpiral
.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/>.