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

           <http://oeis.org/A192136> (etc)

           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/>.