NAME
Math::PlanePath::MultipleRings  rings of multiples
SYNOPSIS
use Math::PlanePath::MultipleRings;
my $path = Math::PlanePath::MultipleRings>new (step => 6);
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
This path puts points on concentric rings. Each ring is "step" many points more then the previous, and the first is also "step" so each has a successively increasing multiple of that many points. For example with the default step==6,
24 23
25 22
10
26 11 9 21 ...
27 12 3 2 8 20 38
28 13 4 1 7 19 37
29 14 5 6 18 36
30 15 17 35
16
31 24
32 33
X,Y positions returned are fractional. The innermost ring like the 1,2,...,6 above has points 1 unit apart. Subsequent rings are either packed similarly or spread out to ensure the X axis points like 1,7,19,37 above are 1 unit apart. The latter happens for step <= 6 and for step >= 7 the rings are big enough to separate those X points.
The layout is similar to the spiral paths of corresponding step. For example step==6 is like the HexSpiral, only rounded out to circles instead of a hexagonal grid. Similarly step==4 the DiamondSpiral or step==8 the SquareSpiral.
The step parameter is similar to the PyramidRows with the rows stretched around circles, though PyramidRows starts from a 1wide initial row and increases by the step, whereas for MultipleRings there's no initial.
The starting radial 1,7,19,37 etc for step==6 is 6*k*(k1)/2 + 1 (for k=1 upwards) and in general it's step*k*(k1)/2 + 1 which is basically a step multiple of the triangular numbers. Straight line radials further around have arise from adding multiples of k, so for example for step==6 above the line 3,11,25 is 6*k*(k1)/2 + 1 + 2*k. Multiples of k bigger than the step give lines in between those of the innermost ring.
Step 3 Pentagonals
For step==3 the pentagonal numbers 1,5,12,22,etc, P(k) = (3k1)*k/2, are a radial going up to the left, and the second pentagonal numbers 2,7,15,26, S(k) = (3k+1)*k/2 are a radial going down to the left, respectively 1/3 and 2/3 the way around the circles.
As described in "Step 3 Pentagonals" in Math::PlanePath::PyramidRows, those numbers and the preceding P(k)1, P(k)2, and S(k)1, S(k)2 are all composites, so plotting the primes on a step==3 MultipleRings has these values as two radial gaps where there's no primes.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.
$path = Math::PlanePath::MultipleRings>new (step => $integer)

Create and return a new path object.
The
step
parameter controls how many points are added in each circle. It defaults to 6 which is an arbitrary choice and the suggestion is to always pass in a desired count. ($x,$y) = $path>n_to_xy ($n)

Return the X,Y coordinates of point number
$n
on the path.$n
can be any value$n >= 0
and fractions give positions on the rings in between the integer points. For$n < 1
the return is an empty list since points begin at 1.Fractional
$n
currently ends up on the circle arc between the integer points. Would straight line chords between them be better, reflecting the unit spacing of the points? Neither seems particularly important. $n = $path>xy_to_n ($x,$y)

Return an integer point number for coordinates
$x,$y
. Each integer N is considered the centre of a circle of diameter 1 and an$x,$y
within that circle returns N.The unit spacing of the points means those circles don't overlap, but they also don't cover the plane and if
$x,$y
is not within one then the return isundef
.
SEE ALSO
Math::PlanePath, Math::PlanePath::SacksSpiral, Math::PlanePath::TheodorusSpiral, Math::PlanePath::PixelRings
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2010, 2011 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/>.