/ 
Math-PlanePath-127
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::ImaginaryHalf

NAME

Math::PlanePath::ImaginaryHalf -- half-plane replications in three directions

SYNOPSIS

use Math::PlanePath::ImaginaryBase;
my $path = Math::PlanePath::ImaginaryBase->new (radix => 4);
my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

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.

Radix

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).

Digit Order

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.

Axis Values

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.

Level Ranges

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.

Proth Numbers

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.

FUNCTIONS

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.

Level Methods

($n_lo, $n_hi) = $path->level_to_n_range($level)

Return (0, $radix**$level - 1).

SEE ALSO

Math::PlanePath, Math::PlanePath::ImaginaryBase, Math::PlanePath::ZOrderCurve

HOME PAGE

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

LICENSE

Copyright 2012, 2013, 2014, 2015, 2016, 2017, 2018 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/>.