--- name: "Math::PlanePath::HexSpiralSkewed" section: "3pm" platform: "linux" package: "libmath-planepath-perl" version: "129-1" release: "Debian 13 (Trixie)" platforms: ["linux"] url: "https://new.carta.tech/Math::PlanePath::HexSpiralSkewed" --- # Math::PlanePath::HexSpiralSkewed(3pm) ## NAME Math::PlanePath::HexSpiralSkewed -- integer points around a skewed hexagonal spiral ## SYNOPSIS ``` use Math::PlanePath::HexSpiralSkewed; my $path = Math::PlanePath::HexSpiralSkewed->new; my ($x, $y) = $path->n_to_xy (123); ``` ## DESCRIPTION This path makes a hexagonal spiral with points skewed so as to fit a square grid and fully cover the plane. ``` 13--12--11 ... 2 | \ \ 14 4---3 10 23 1 | | \ \ \ 15 5 1---2 9 22 <- Y=0 \ \ | | 16 6---7---8 21 -1 \ | 17--18--19--20 -2 ^ ^ ^ ^ ^ ^ -2 -1 X=0 1 2 3 ... ``` The kinds of N=3\*k^2 numbers which fall on straight lines in the plain `"HexSpiral"` also fall on straight lines when skewed. See Math::PlanePath::HexSpiral for notes on this. ### Skew The skewed path is the same shape as the plain `"HexSpiral"`, 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. ``` HexSpiralSkewed HexSpiral 13--12--11 13--12--11 | \ / \ 14 10 14 10 | \ / \ 15 9 15 9 -2 -1 X=0 1 2 -4 -3 -2 X=0 2 3 4 ``` In general the coordinates can be converted each way by ``` plain X,Y -> skewed (X-Y)/2, Y skewed X,Y -> plain 2*X+Y, Y ``` ## Corners `"HexSpiralSkewed"` is similar to the `"SquareSpiral"` but cuts off the top-right and bottom-left corners so that each loop is 6 steps longer than the previous, whereas for the `"SquareSpiral"` it's 8. See "Corners" in Math::PlanePath::SquareSpiral for other corner cutting. ### Wider An optional `"wider"` parameter makes the path wider, stretched along the top and bottom horizontals. For example ``` $path = Math::PlanePath::HexSpiralSkewed->new (wider => 2); ``` gives ``` 21--20--19--18--17 2 | \ 22 8---7---6---5 16 1 | | \ \ 23 9 1---2---3---4 15 <- Y=0 \ \ | 24 10--11--12--13--14 ... -1 \ | 25--26--27--28--29--30 -2 ^ ^ ^ ^ ^ ^ ^ ^ -4 -3 -2 -1 X=0 1 2 3 ... ``` The centre horizontal from N=1 is extended by `"wider"` many further places, then the path loops around that shape. The starting point 1 is shifted to the left by wider/2 places (rounded up to an integer) to keep the spiral centred on the origin X=0,Y=0. Each loop is still 6 longer than the previous, since the widening is basically a constant amount added into each loop. The result is the same as the plain `"HexSpiral"` of the same widening too. The effect looks better in the plain `"HexSpiral"`. ### N Start The default is to number points starting N=1 as shown above. An optional `"n_start"` can give a different start with the same shape etc. For example to start at 0, ``` n_start => 0 27 26 25 24 3 28 12 11 10 23 2 29 13 3 2 9 22 1 30 14 4 0 1 8 21 ... <- Y=0 31 15 5 6 7 20 39 -1 32 16 17 18 19 38 -2 33 34 35 36 37 -3 -3 -2 -1 X=0 1 2 3 4 ``` In this numbering the X axis N=0,1,8,21,etc is the octagonal numbers 3\*X\*(X+1). ## FUNCTIONS See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes. - "$path = Math::PlanePath::HexSpiralSkewed-\>new ()" - "$path = Math::PlanePath::HexSpiralSkewed-\>new (wider =\> $w)" Create and return a new hexagon spiral object. An optional `"wider"` parameter widens the spiral path, it defaults to 0 which is no widening. - "$n = $path-\>xy\_to\_n ($x,$y)" Return the point number for coordinates `"$x,$y"`. `$x` and `$y` are each rounded to the nearest integer, which has the effect of treating each point in the path as a square of side 1. ## OEIS Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include \ (etc) ``` 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 ``` ## SEE ALSO Math::PlanePath, Math::PlanePath::HexSpiral, Math::PlanePath::HeptSpiralSkewed, Math::PlanePath::PentSpiralSkewed, Math::PlanePath::DiamondSpiral ## HOME PAGE \ ## LICENSE Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde This file is part of Math-PlanePath. Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version. Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see \.