Math::PlanePath::FilledRings(3pm)

Usage

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

Overview

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

Home page

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

Releases

6 releases since Debian 8 (Jessie); the text changed 3 times.