Math::...dRings(3pm)

💖 Math::PlanePath::FilledRings 💖

okay bestie, gather round, because this is a 3pm manual page and it's literally about drawing concentric filled lattice rings and I'm obsessed 🎀

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✨ NAME ✨

Math::PlanePath::FilledRings — concentric filled lattice rings. that's it, that's the vibe 🦋

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✨ SYNOPSIS ✨

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

three lines. three. and you're already out here mapping integers to coordinates like the main character you are 💅

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✨ DESCRIPTION ✨

so this path drops points onto integer X,Y pixels, arranged in filled rings, and every single ring has radius 1 unit. one unit per ring, no cap 🫶

                       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

look at her. the way she spirals outward ring by ring?? iconic. 👑

for example, the ring from N=22 to N=37 is literally all the points where

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

so it's a ring-shaped annulus, mathematically speaking, and it's giving geometry class but make it cute 🌸

🌈 N Start 🌈

by default, numbering starts at N=1 like in the picture up there ^^ but bestie, you can pass an optional "n_start" to kick things off somewhere else — same shape, different starting number, lowkey the same girl with a new haircut 💇

so 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

and the only effect is that it pushes all the N values by a constant amount. that's it, that's the whole thing. but it can help match N on the axes to counts of X,Y points < R or similar — so it's a little convenience queen, fr fr ✨

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✨ FUNCTIONS ✨

for the behaviour common to all path classes, go check "FUNCTIONS" in Math::PlanePath. we're just covering the local faves here 💜

• $path = Math::PlanePath::FilledRings->new ()
• $path = Math::PlanePath::FilledRings->new (n_start => $n) — create and return a brand new path object. fresh out the box, no notes. 🎀

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✨ 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

sequence numbers, bestie!! this path is literally published. we love a documented queen 😭💖

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✨ SEE ALSO ✨

• Math::PlanePath
• Math::PlanePath::PixelRings
• Math::PlanePath::Hypot
• Math::PlanePath::MultipleRings

the whole friend group, all linked up 🫶

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✨ HOME PAGE ✨

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

go say hi 🌈

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✨ 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/>.

ngl the all-caps warranty disclaimer is highkey the scariest part of any manpage and this one is no exception 💀 no warranty bestie, we're all just out here redistributing software on a hope and a prayer 🙈

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and that's the page!! brought to you by perl v5.32.0, dated 2021-01-23 — slay, thanks for reading, go draw some rings 💅✨🎀