NAME
Math::PlanePath::ComplexPlus -- points in complex base i+r
SYNOPSIS
use Math::PlanePath::ComplexPlus;
my $path = Math::PlanePath::ComplexPlus->new;
my ($x, $y) = $path->n_to_xy (123);DESCRIPTION
This path traverses points by a complex number base i+r for integer r>=1. The default is base i+1
30 31 14 15 5
28 29 12 13 4
26 27 22 23 10 11 6 7 3
24 25 20 21 8 9 4 5 2
62 63 46 47 18 19 2 3 1
60 61 44 45 16 17 0 1 <- Y=0
58 59 54 55 42 43 38 39 -1
56 57 52 53 40 41 36 37 -2
50 51 94 95 34 35 78 79 -3
48 49 92 93 32 33 76 77 -4
90 91 86 87 74 75 70 71 -5
88 89 84 85 72 73 68 69 -6
126 127 110 111 82 83 66 67 -7
124 125 108 109 80 81 64 65 -8
122 123 118 119 106 107 102 103 -9
120 121 116 117 104 105 100 101 -10
114 115 98 99 -11
112 113 96 97 -12
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2The shape of these points N=0 to N=2^k-1 inclusive is equivalent to the twindragon turned 135 degrees. Each complex base point corresponds to a unit square inside the twindragon curve (two DragonCurve back-to-back).
Real Part
Option realpart => $r selects another r for complex base b=i+r. For example
realpart => 2
45 46 47 48 49 8
40 41 42 43 44 7
35 36 37 38 39 6
30 31 32 33 34 5
25 26 27 28 29 20 21 22 23 24 4
15 16 17 18 19 3
10 11 12 13 14 2
5 6 7 8 9 1
0 1 2 3 4 <- Y=0
^
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15N is broken into digits of a base norm=r*r+1, ie. digits 0 to r*r inclusive.
norm = r*r + 1
Nstart = 0
Nlevel = norm^level - 1The low digit of N makes horizontal runs of r*r+1 many points, such as N=0 to N=4, then N=5 to N=9, etc shown above. In the default r=1 these runs are 2 long. For r=2 shown above they're 2*2+1=5 long, or r=3 would be 3*3+1=10, etc.
The offset for each successive run is i+r, ie. Y=1,X=r such as at N=5 shown above. Then the offset for the next level is (i+r)^2 = (2r*i + r^2-1) so N=25 begins at Y=2*r=4, X=r*r-1=3. In general each level adds an angle
angle = atan(1/r)
Nlevel_angle = level * angleSo the points spiral around anti-clockwise. For r=1 the angle is atan(1/1)=45 degrees, so that for example level=4 is angle 4*45=180 degrees, putting N=2^4=16 on the negative X axis as shown in the first sample above.
As r becomes bigger the angle becomes smaller, making it spiral more slowly. The points never fill the plane, but the set of points N=0 to Nlevel are all touching.
Arms
For realpart => 1, an optional arms => 2 adds a second copy of the curve rotated 180 degrees and starting from X=0,Y=1. It meshes perfectly to fill the plane. Each arm advances successively so N=0,2,4,etc is the plain path and N=1,3,5,7,etc is the copy
arms=>2
60 62 28 30 5
56 58 24 26 4
52 54 44 46 20 22 12 14 3
48 50 40 42 16 18 8 10 2
36 38 3 1 4 6 35 33 1
32 34 7 5 0 2 39 37 <- Y=0
11 9 19 17 43 41 51 49 -1
15 13 23 21 47 45 55 53 -2
27 25 59 57 -3
31 29 63 61 -4
^
-6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6There's no arms parameter for other realpart values as yet, only for i+1. Is there a good rotated arrangement for others? Do "norm" many copies fill the plane in general?
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::ComplexPlus->new ()-
Create and return a new path object.
($x,$y) = $path->n_to_xy ($n)-
Return the X,Y coordinates of point number
$non the path. Points begin at 0 and if$n < 0then the return is an empty list.
Level Methods
($n_lo, $n_hi) = $path->level_to_n_range($level)-
Return
(0, 2**$level - 1), or for 2 arms return(0, 2 * 2**$level - 1). With therealpartoption return(0, $norm**$level - 1)where norm=realpart^2+1.
FORMULAS
Various formulas and pictures etc for the i+1 case can be found in the author's long mathematical write-up (section "Complex Base i+1")
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
http://oeis.org/A290885 (etc)
realpart=1 (i+1, the default)
A290885 -X
A290884 Y
A290886 norm X^2 + Y^2
A146559 dX at N=2^k-1 (step to next replication level)
A077950,A077870
location of ComplexMinus origin in ComplexPlus
(mirror horizontal even level, vertical odd level)SEE ALSO
Math::PlanePath, Math::PlanePath::ComplexMinus, Math::PlanePath::ComplexRevolving, Math::PlanePath::DragonCurve
HOME PAGE
http://user42.tuxfamily.org/math-planepath/index.html
LICENSE
Copyright 2011, 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/>.