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 the following 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 that point 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 repetitions 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 only 3 here. The ZOrderCurve is a 2 direction replication.
The radix parameter controls the "r" used to break N into X,Y. For example radix => 4 gives 4x4 blocks, with r-1 copies of the preceding level at each stage.
radix
radix => 4
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).
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 0s in each two of three bits, again simply 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 becomes high non-zero digit 1 to radix-1.
Because the X direction replicates twice for each once in the Y direction the width grows at twice the rate, so width = height*height, after each 3 replications. For this reason N values for a given Y grow quite rapidly.
The Proth numbers fall in columns on the path.
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ----------------------------------------------------------------- -31 -23 -15 -7 -3-1 0 3 5 9 17 25 33
The height of the column follows the position of the number of zeros in X ending ...1000..0001 in binary as this limits the "k" part of the Proth numbers which can have N ending suitably. Or for X negative the 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 => $r)
Create and return a new path object.
($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
$n < 0
($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
Math::PlanePath, Math::PlanePath::ImaginaryBase, Math::PlanePath::ZOrderCurve
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2012 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/>.
To install Math::PlanePath, copy and paste the appropriate command in to your terminal.
cpanm
cpanm Math::PlanePath
CPAN shell
perl -MCPAN -e shell install Math::PlanePath
For more information on module installation, please visit the detailed CPAN module installation guide.