NAME
Math::PlanePath::OctagramSpiral  integer points drawn around an octagram
SYNOPSIS
use Math::PlanePath::OctagramSpiral;
my $path = Math::PlanePath::OctagramSpiral>new;
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
This path makes a spiral around an octagram (8pointed star),
29 25 4
 \ / 
30 28 26 24 ...5655 3
 \ /  /
333231 7 27 5 232221 54 2
\ \ /  / /
34 9 8 6 4 3 20 53 1
\ \ / / /
35 10 12 19 52 < Y=0
/ / \ \
36 1112 14 161718 51 1
/ / \  \
373839 13 43 15 47484950 2
 / \ 
40 42 44 46 3
/ \ 
41 45 4
^
4 3 2 1 X=0 1 2 3 4 5 ...
Each loop is 16 longer than the previous. The 18gonal numbers 18,51,100,etc fall on the horizontal at Y=1.
The inner corners like 23, 31, 39, 47 are similar to the SquareSpiral
path, but instead of going directly between them the octagram takes a detour out to make the points of the star. Those excursions make each loops 8 longer (1 per excursion), hence a step of 16 here as compared to 8 for the SquareSpiral
.
N Start
The default is to number points starting N=1 as shown above. An optional n_start
can give a different start, in the same pattern. For example to start at 0,
n_start => 0
28 24
29 27 25 23 ... 55 54
32 31 30 6 26 4 22 21 20 53
33 8 7 5 3 2 19 52
34 9 0 1 18 51
35 10 11 13 15 16 17 50
36 37 38 12 42 14 46 47 48 49
39 41 43 45
40 44
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::OctagramSpiral>new ()

Create and return a new octagram spiral object.
($x,$y) = $path>n_to_xy ($n)

Return the X,Y coordinates of point number
$n
on the path.For
$n < 1
the return is an empty list, it being considered the path starts at 1. $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 N in the path as centred in a square of side 1, so the entire plane is covered.
FORMULAS
X,Y to N
The symmetry of the octagram can be used by rotating a given X,Y back to the first star excursion such as N=19 to N=23. If Y is negative then rotate back by 180 degrees, then if X is negative rotate back by 90, and if Y>=X then by a further 45 degrees. Each such rotation, if needed, is counted as a multiple of the sidelength to be added to the final N. For example at N=19 the side length is 2. Rotating by 180 degrees is 8 side lengths, by 90 degrees 4 sides, and by 45 degrees is 2 sides.
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
http://oeis.org/A125201 (etc)
n_start=1 (the default)
A125201 N on X axis, from X=1 onwards, 18gonals + 1
A194268 N on diagonal SouthEast
n_start=0
A051870 N on X axis, 18gonal numbers
A139273 N on Y axis
A139275 N on X negative axis
A139277 N on Y negative axis
A139272 N on diagonal X=Y
A139274 N on diagonal NorthWest
A139276 N on diagonal SouthWest
A139278 N on diagonal SouthEast, second 18gonals
SEE ALSO
Math::PlanePath, Math::PlanePath::SquareSpiral, Math::PlanePath::PyramidSpiral
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014 Kevin Ryde
This file is part of MathPlanePath.
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.