NAME
Math::PlanePath::KnightSpiral  integer points around a square, by chess knight moves
SYNOPSIS
use Math::PlanePath::KnightSpiral;
my $path = Math::PlanePath::KnightSpiral>new;
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
This path traverses the plane by an infinite "knight's tour" in the form of a square spiral.
...
21 4 9 14 19 2
10 15 20 3 8 28 1
5 22 1 18 13 < Y=0
16 11 24 7 2 27 1
23 6 17 12 25 2
26
^
2 1 X=0 1 2 3
Each step is a chess knight's move 1 across and 2 along, or vice versa. The pattern makes 4 cycles on a 2wide path around a square before stepping outwards to do the same again to a now bigger square. The above sample shows the first 4cycle around the central 1, then stepping out at 26 and beginning to go around the outside of the 5x5 square.
An attractive traced out picture of the path appeared in the past at www.borderschess.org
,
http://web.archive.org/web/1id_/http://www.borderschess.org/Infinite.gif
http://web.archive.org/web/1id_/http://www.borderschess.org/KTinfinity.gif
http://web.archive.org/web/20161028114643id_/http://www.borderschess.org/KTart.htm
(HTML colours might might make the text invisible. Try deleting, or browser option to ignore page colours, or a text browser.)
See mathimage to draw the path lines too.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::KnightSpiral>new ()

Create and return a new knight spiral object.
($x,$y) = $path>n_to_xy ($n)

Return the X,Y coordinates of point number
$n
on the path.For
$n < 1
the return is an empty list, it being considered the path starts at 1. $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.
OEIS
This Knight's tour is in Sloane's OEIS following the Knight spiral and giving the resulting X,Y location by the SquareSpiral
numbering. There's eight forms for 4 rotations and the two spirals same or opposite directions.
http://oeis.org/A068608 (etc)
permutations
A068608 same knight and square spiral directions
A068609 rotate 90 degrees
A068610 rotate 180 degrees
A068611 rotate 270 degrees
A068612 rotate 180 degrees, spiral opp dir (X negate)
A068613 rotate 270 degrees, spiral opp dir
A068614 spiral opposite direction (Y negate)
A068615 rotate 90 degrees, spiral opp dir (X,Y transpose)
See examples/knightsoeis.pl for a sample program printing the values of A068608.
SEE ALSO
Math::PlanePath, Math::PlanePath::SquareSpiral
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019 Kevin Ryde
This file is part of MathPlanePath.
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.