Math::PlanePath::MultipleRings -- rings of multiples
use Math::PlanePath::MultipleRings; my $path = Math::PlanePath::MultipleRings->new (step => 6); my ($x, $y) = $path->n_to_xy (123);
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 1-wide 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*(k-1)/2 + 1 (for k=1 upwards) and in general it's step*k*(k-1)/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*(k-1)/2 + 1 + 2*k. Multiples of k bigger than the step give lines in between those of the innermost ring.
For step==3 the pentagonal numbers 1,5,12,22,etc, P(k) = (3k-1)*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.
See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.
$path = Math::PlanePath::MultipleRings->new (step => $integer)
Create and return a new path object.
stepparameter controls how many points are added in each circle. It defaults to 6 which is an arbitrary choice and the suggestion is to always pass in a desired count.
($x,$y) = $path->n_to_xy ($n)
Return the X,Y coordinates of point number
$non the path.
$ncan be any value
$n >= 0and fractions give positions on the rings in between the integer points. For
$n < 1the return is an empty list since points begin at 1.
$ncurrently 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
$x,$ywithin 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
$x,$yis not within one then the return is
Copyright 2010, 2011 Kevin Ryde
This file is part of Math-PlanePath.
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