NAME

       Math::PlanePath::FilledRings -- concentric filled lattice rings

SYNOPSIS

        use Math::PlanePath::FilledRings;
        my $path = Math::PlanePath::FilledRings->new;
        my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

       This path puts points on integer X,Y pixels of filled rings with radius 1 unit each ring.

                           110-109-108-107-106                        6
                          /                   \
                   112-111  79--78--77--76--75 105-104                5
                     |    /                   \      |
               114-113  80  48--47--46--45--44  74 103-102            4
                 |    /      |               |    \      |
               115  81  50--49  27--26--25  43--42  73 101            3
              /   /      |    /           \      |    \   \
           116  82  52--51  28  14--13--12  24  41--40  72 100        2
             |   |   |    /   /           \   \      |   |   |
           117  83  53  29  15   5-- 4-- 3  11  23  39  71  99        1
             |   |   |   |   |   |       |   |   |   |   |   |
           118  84  54  30  16   6   1-- 2  10  22  38  70  98   <- Y=0
             |   |   |   |   |   |        /   /   /   /   /   /
           119  85  55  31  17   7-- 8-- 9  21  37  69  97 137       -1
             |   |   |    \   \           /   /      |   |   |
           120  86  56--57  32  18--19--20  36  67--68  96 136       -2
              \   \      |    \           /      |    /   /
               121  87  58--59  33--34--35  65--66  95 135           -3
                 |    \      |               |    /      |
               122-123  88  60--61--62--63--64  94 133-134           -4
                     |    \                   /      |
                   124-125  89--90--91--92--93 131-132               -5
                          \                   /
                           126-127-128-129-130

                                     ^
            -6  -5  -4  -3  -2  -1  X=0  1   2   3   4   5   6

       For example the ring N=22 to N=37 is all the points

           2.5 < hypot(X,Y) < 3.5
           where hypot(X,Y) = sqrt(X^2+Y^2)

   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.  For example to start at 0,

                 26-25-24          n_start => 0
                /        \
              27 13-12-11 23
             /  /        \  \
           28 14  4--3--2 10 22
            |  |  |     |  |  |
           29 15  5  0--1  9 21
            |  |  |      /  /  /
           30 16  6--7--8 20 36
             \  \        /  /
              31 17-18-19 35
                \        /
               8 32-33-34

       The only effect is to push the N values by a constant amount but can help match N on the
       axes to counts of X,Y points < R or similar.

FUNCTIONS

       See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.

       "$path = Math::PlanePath::FilledRings->new ()"
       "$path = Math::PlanePath::FilledRings->new (n_start => $n)"
           Create and return a new path object.

OEIS

       Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include,

           <http://oeis.org/A036704> (etc)

           A036705  first diffs of N on X axis,
                      being count of X,Y points n-1/2 < norm <= n+1/2
           A036706  1/4 of those diffs

           n_start=1 (the default)
             A036707  N/2+X-1 along X axis,
                        being count norm <= n+1/2 in half plane
             A036708  (N(X,0)-N(X-1,0))/2+1,
                        first diffs of the half plane count

           n_start=0
             A036704  N on X axis, from X=1 onwards
                        count of X,Y points norm <= n+1/2

SEE ALSO

       Math::PlanePath, Math::PlanePath::PixelRings, Math::PlanePath::Hypot,
       Math::PlanePath::MultipleRings

HOME PAGE

       <http://user42.tuxfamily.org/math-planepath/index.html>

LICENSE

       Copyright 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 <http://www.gnu.org/licenses/>.