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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::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 the 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 (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 sequence 1=left,0=straight, being 1 at N=k*(k+3)/2 \& \& 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" .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 .