# 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)`.

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

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

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/>.