Math::PlanePath::ComplexRevolving -- points in revolving complex base i+1
use Math::PlanePath::ComplexRevolving; my $path = Math::PlanePath::ComplexRevolving->new; my ($x, $y) = $path->n_to_xy (123);
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.
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 $n < 0
then the return is an empty list.
($n_lo, $n_hi) = $path->level_to_n_range($level)
Return (0, 2**$level - 1)
.
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.
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2012, 2013, 2014, 2015, 2016 Kevin Ryde
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