Math::PlanePath::ImaginaryHalf -- half-plane replications in three directions
use Math::PlanePath::ImaginaryBase; my $path = Math::PlanePath::ImaginaryBase->new (radix => 4); my ($x, $y) = $path->n_to_xy (123);
This is a half-plane variation on the ImaginaryBase
path.
54-55 50-51 62-63 58-59 22-23 18-19 30-31 26-27 3 \ \ \ \ \ \ \ \ 52-53 48-49 60-61 56-57 20-21 16-17 28-29 24-25 2 38-39 34-35 46-47 42-43 6--7 2--3 14-15 10-11 1 \ \ \ \ \ \ \ \ 36-37 32-33 44-45 40-41 4--5 0--1 12-13 8--9 <- Y=0 ------------------------------------------------- -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5
The pattern can be seen by dividing into blocks,
+---------------------------------+ | 22 23 18 19 30 31 26 27 | | | | 20 21 16 17 28 29 24 25 | +--------+-------+----------------+ | 6 7 | 2 3 | 14 15 10 11 | | +---+---+ | | 4 5 | 0 | 1 | 12 13 8 9 | <- Y=0 +--------+---+---+----------------+ ^ X=0
N=0 is at the origin, then N=1 replicates it to the right. Those two repeat above as N=2 and N=3. Then that 2x2 repeats to the left as N=4 to N=7, then 4x2 repeats to the right as N=8 to N=15, and 8x2 above as N=16 to N=31, etc. The replications are successively to the right, above, left. The relative layout within a replication is unchanged.
This is similar to the ImaginaryBase
, but where it repeats in 4 directions there's just 3 directions here. The ZOrderCurve
is a 2 direction replication.
The radix
parameter controls the radix used to break N into X,Y. For example radix => 4
gives 4x4 blocks, with radix-1 replications of the preceding level at each stage.
radix => 4 60 61 62 63 44 45 46 47 28 29 30 31 12 13 14 15 3 56 57 58 59 40 41 42 43 24 25 26 27 8 9 10 11 2 52 53 54 55 36 37 38 39 20 21 22 23 4 5 6 7 1 48 49 50 51 32 33 34 35 16 17 18 19 0 1 2 3 <- Y=0 --------------------------------------^----------- -12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3
Notice for X negative the parts replicate successively towards -infinity, so the block N=16 to N=31 is first at X=-4, then N=32 at X=-8, N=48 at X=-12, and N=64 at X=-16 (not shown).
The digit_order
parameter controls the order digits from N are applied to X and Y. The default above is "XYX" so the replications go X then Y then negative X.
"XXY" goes to negative X before Y, so N=2,N=3 goes to negative X before repeating N=4 to N=7 in the Y direction.
digit_order => "XXY" 38 39 36 37 46 47 44 45 34 35 32 33 42 43 40 41 6 7 4 5 14 15 12 13 2 3 0 1 10 11 8 9 ---------^-------------------- -2 -1 X=0 1 2 3 4 5
The further options are as follows, for six permutations of each 3 digits from N.
digit_order => "YXX" digit_order => "XnYX" 38 39 36 37 46 47 44 45 19 23 18 22 51 55 50 54 34 35 32 33 42 43 40 41 17 21 16 20 49 53 48 52 6 7 4 5 14 15 12 13 3 7 2 6 35 39 34 38 2 3 0 1 10 11 8 9 1 5 0 4 33 37 32 36 digit_order => "XnXY" digit_order => "YXnX" 37 39 36 38 53 55 52 54 11 15 9 13 43 47 41 45 33 35 32 34 49 51 48 50 10 14 8 12 42 46 40 44 5 7 4 6 21 23 20 22 3 7 1 5 35 39 33 37 1 3 0 2 17 19 16 18 2 6 0 4 34 38 32 36
"Xn" means the X negative direction. It's still spaced 2 apart (or whatever radix), so the result is not simply a -X,Y.
N=0,1,4,5,8,9,etc on the X axis (positive and negative) are those integers with a 0 at every third bit starting from the second least significant bit. This is simply demanding that the bits going to the Y coordinate must be 0.
X axis Ns = binary ...__0__0__0_ with _ either 0 or 1 in octal, digits 0,1,4,5 only
N=0,1,8,9,etc on the X positive axis have the highest 1-bit in the first slot of a 3-bit group. Or N=0,4,5,etc on the X negative axis have the high 1 bit in the third slot,
X pos Ns = binary 1_0__0__0...0__0__0_ X neg Ns = binary 10__0__0__0...0__0__0_ ^^^ three bit group X pos Ns in octal have high octal digit 1 X neg Ns in octal high octal digit 4 or 5
N=0,2,16,18,etc on the Y axis are conversely those integers with a 0 in two of each three bits, demanding the bits going to the X coordinate must be 0.
Y axis Ns = binary ..._00_00_00_0 with _ either 0 or 1 in octal has digits 0,2 only
For a radix other than binary the pattern is the same. Each "_" is any digit of the given radix, and each 0 must be 0. The high 1 bit for X positive and negative become a high non-zero digit.
Because the X direction replicates twice for each once in the Y direction the width grows at twice the rate, so after each 3 replications
width = height*height
For this reason N values for a given Y grow quite rapidly.
The Proth numbers, k*2^n+1 for k<2^n, fall in columns on the path.
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ----------------------------------------------------------------- -31 -23 -15 -7 -3-1 0 3 5 9 17 25 33
The height of the column is from the zeros in X ending binary ...1000..0001 since this limits the "k" part of the Proth numbers which can have N ending suitably. Or for X negative ending ...10111...11.
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::ImaginaryBase->new ()
$path = Math::PlanePath::ImaginaryBase->new (radix => $r, digit_order => $str)
Create and return a new path object. The choices for digit_order
are
"XYX" "XXY" "YXX" "XnYX" "XnXY" "YXnX"
($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) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)
The returned range is exact, meaning $n_lo
and $n_hi
are the smallest and biggest in the rectangle.
($n_lo, $n_hi) = $path->level_to_n_range($level)
Return (0, $radix**$level - 1)
.
Math::PlanePath, Math::PlanePath::ImaginaryBase, Math::PlanePath::ZOrderCurve
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2012, 2013, 2014, 2015, 2016, 2017 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.
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