NAME

       Math::PlanePath::ComplexRevolving -- points in revolving complex base i+1

SYNOPSIS

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

DESCRIPTION

       This path traverses points by a complex number base i+1 with turn factor i (+90 degrees)
       at each 1 bit.  This is the "revolving binary representation" of Knuth's Seminumerical
       Algorithms section 4.1 exercise 28.

                    54 51       38 35            5
                 60 53       44 37               4
           39 46 43 58 23 30 27 42               3
              45  8 57  4 29 56 41 52            2
                 31  6  3  2 15 22 19 50         1
           16    12  5  0  1 28 21    49     <- Y=0
           55 62 59 10  7 14 11 26              -1
              61 24  9 20 13 40 25 36           -2
                 47       18 63       34        -3
           32          48 17          33        -4

                        ^
           -4 -3 -2 -1 X=0 1  2  3  4  5

       The 1 bits in N are exponents e0 to et, in increasing order,

           N = 2^e0 + 2^e1 + ... + 2^et        e0 < e1 < ... < et

       and are applied to a base b=i+1 as

           X+iY = b^e0 + i * b^e1 + i^2 * b^e2 + ... + i^t * b^et

       Each 2^ek has become b^ek base b=i+1.  The i^k is an extra factor i at each 1 bit of N,
       causing a rotation by +90 degrees for the bits above it.  Notice the factor is i^k not
       i^ek, ie. it increments only with the 1-bits of N, not the whole exponent.

       A single bit N=2^k is the simplest and is X+iY=(i+1)^k.  These N=1,2,4,8,16,etc are at
       successive angles 45, 90, 135, etc degrees (the same as in "ComplexPlus").  But points
       N=2^k+1 with two bits means X+iY=(i+1) + i*(i+1)^k and that factor "i*" is a rotation by
       90 degrees so points N=3,5,9,17,33,etc are in the next quadrant around from their
       preceding 2,4,8,16,32.

       As per the exercise in Knuth it's reasonably easy to show that this calculation is a one-
       to-one mapping between integer N and complex integer X+iY, so the path covers the plane
       and visits all points once each.

FUNCTIONS

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

       "$path = Math::PlanePath::ComplexRevolving->new ()"
           Create and return a new path object.

       "($x,$y) = $path->n_to_xy ($n)"
           Return the X,Y coordinates of point number $n on the path.  Points begin at 0 and if
           "$n < 0" then the return is an empty list.

   Level Methods
       "($n_lo, $n_hi) = $path->level_to_n_range($level)"
           Return "(0, 2**$level - 1)".

SEE ALSO

       Math::PlanePath, Math::PlanePath::ComplexMinus, Math::PlanePath::ComplexPlus,
       Math::PlanePath::DragonCurve

       Donald Knuth, "The Art of Computer Programming", volume 2 "Seminumerical Algorithms",
       section 4.1 exercise 28.

HOME PAGE

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

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/>.