/ 
Math-PlanePath-129
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::HeptSpiralSkewed

NAME

Math::PlanePath::HeptSpiralSkewed -- integer points around a skewed seven sided spiral

SYNOPSIS

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

DESCRIPTION

This path makes a seven-sided spiral by cutting one corner of a square

31-30-29-28                       3
 |         \
32 14-13-12 27                    2
 |  |      \  \
33 15  4--3 11 26                 1
 |  |  |   \  \  \
34 16  5  1--2 10 25         <- Y=0
 |  |  |        |  |
35 17  6--7--8--9 24             -1
 |  |              |
36 18-19-20-21-22-23             -2
 |
37-38-39-40-41-...               -3

          ^
-3 -2 -1 X=0 1  2  3

The path is as if around a heptagon, with the left and bottom here as two sides of the heptagon straightened out, and the flat top here skewed across to fit a square grid.

N Start

The default is to number points starting N=1 as shown above. An optional n_start can give a different start, in the same pattern. For example to start at 0,

30 29 28 27              n_start => 0
31 13 12 11 26
32 14  3  2 10 25
33 15  4  0  1  9 24
34 16  5  6  7  8 23
35 17 18 19 20 21 22
36 37 38 39 40 ...

FUNCTIONS

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

$path = Math::PlanePath::HeptSpiralSkewed->new ()
$path = Math::PlanePath::HeptSpiralSkewed->new (n_start => $n)

Create and return a new path object.

$n = $path->xy_to_n ($x,$y)

Return the point number for coordinates $x,$y. $x and $y are each rounded to the nearest integer, which has the effect of treating each N in the path as centred in a square of side 1, so the entire plane is covered.

FORMULAS

N to X,Y

It's convenient to work in terms of Nstart=0 and to take each loop as beginning on the South-West diagonal,

           top length = d

           30-29-28-27
            |         \
           31          26    diagonal length = d
left        |            \
length     32             25
 = 2*d      |               \
           33        0       24
            |                 |    right
           34     .          23    length = d-1
            |                 |
           35 17-18-19-20-21-22
            |
            .    bottom length = 2*d-1

The SW diagonal is N=0,5,17,36,etc which is

N = (7d-11)*d/2 + 2           # starting d=1 first loop

This can be inverted to get d from N

d = floor( (sqrt(56*N+9)+11)/14 )

The side lengths are as shown above. The first loop is d=1 and for it the "right" vertical length is zero, so no such side on that first loop 0 <= N < 5.

OEIS

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

n_start=1
  A140065    N on Y axis

n_start=0
  A001106    N on X axis, 9-gonal numbers
  A218471    N on Y axis
  A022265    N on X negative axis
  A179986    N on Y negative axis, second 9-gonals
  A195023    N on X=Y diagonal
  A022264    N on North-West diagonal
  A186029    N on South-West diagonal
  A024966    N on South-East diagonal

SEE ALSO

Math::PlanePath, Math::PlanePath::SquareSpiral, Math::PlanePath::PentSpiralSkewed, Math::PlanePath::HexSpiralSkewed

HOME PAGE

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

LICENSE

Copyright 2010, 2011, 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/>.