NAME¶
Math::PlanePath::FlowsnakeCentres -- self-similar path of hexagon centres
SYNOPSIS¶
use Math::PlanePath::FlowsnakeCentres;
my $path = Math::PlanePath::FlowsnakeCentres->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION¶
This path is a variation of the flowsnake curve by William Gosper which follows
the flowsnake tiling the same way but the centres of the hexagons instead of
corners across. The result is the same overall shape, but a symmetric base
figure.
39----40 8
/ \
32----33 38----37 41 7
/ \ \ \
31----30 34----35----36 42 47 6
\ / / \
28----29 16----15 43 46 48--... 5
/ / \ \ \
27 22 17----18 14 44----45 4
/ / \ \ \
26 23 21----20----19 13 10 3
\ \ / / \
25----24 4---- 5 12----11 9 2
/ \ /
3---- 2 6---- 7---- 8 1
\
0---- 1 <- Y=0
-5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9
The points are spread out on every second X coordinate to make little triangles
with integer coordinates, per "Triangular Lattice" in
Math::PlanePath.
The base pattern is the seven points 0 to 6,
4---- 5
/ \
3---- 2 6---
\
0---- 1
This repeats at 7-fold increasing scale, with sub-sections rotated according to
the edge direction, and the 1, 2 and 6 sub-sections in reverse. Eg. N=7 to
N=13 is the "1" part taking the base figure in reverse and rotated
so the end points towards the "2".
The next level can be seen at the midpoints of each such group, being
N=2,11,18,23,30,37,46.
---- 37
---- ---
30---- ---
| ---
| 46
|
| ----18
| ----- ---
23--- ---
---
--- 11
-----
2 ---
Arms¶
The optional "arms" parameter can give up to three copies of the
curve, each advancing successively. For example "arms=>3" is as
follows. Notice the N=3*k points are the plain curve, and N=3*k+1 and N=3*k+2
are rotated copies of it.
84---... 48----45 5
/ / \
81 66 51----54 42 4
/ / \ \ \
28----25 78 69 63----60----57 39 30 3
/ \ \ \ / / \
31----34 22 75----72 12----15 36----33 27 2
\ \ / \ /
40----37 19 4 9---- 6 18----21----24 1
/ / / \ \
43 58 16 7 1 0---- 3 77----80 <- Y=0
/ / \ \ \ / \
46 55 61 13----10 2 11 74----71 83 -1
\ \ \ / / \ \ \
49----52 64 73 5---- 8 14 65----68 86 -2
/ / \ / / /
... 67----70 76 20----17 62 53 ... -3
\ / / / / \
85----82----79 23 38 59----56 50 -4
/ / \ /
26 35 41----44----47 -5
\ \
29----32 -6
^
-9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9
As described in "Arms" in Math::PlanePath::Flowsnake the flowsnake
essentially fills a hexagonal shape with wiggly sides. For this Centres
variation the start of each arm corresponds to the centre of a little hexagon.
The N=0 little hexagon is at the origin, and the 1 and 2 beside and below,
^ / \ / \
\ \ / \
| \ | |
| 1 | 0--->
| | |
\ / \ /
\ / \ /
| |
| 2 |
| / |
/ /
v \ /
Like the main Flowsnake the sides of the arms mesh perfectly and three arms fill
the plane.
FUNCTIONS¶
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
classes.
- "$path = Math::PlanePath::FlowsnakeCentres->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.
Fractional positions give an X,Y position along a straight line between the
integer positions.
- "($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1,
$x2,$y2)"
- In the current code the returned range is exact, meaning $n_lo and $n_hi
are the smallest and biggest in the rectangle, but don't rely on that yet
since finding the exact range is a touch on the slow side. (The advantage
of which though is that it helps avoid very big ranges from a simple
over-estimate.)
Level Methods¶
- "($n_lo, $n_hi) = $path->level_to_n_range($level)"
- Return "(0, 7**$level - 1)", or for multiple arms return
"(0, $arms * 7**$level - 1)".
There are 7^level points in a level, or arms*7^level for multiple arms,
numbered starting from 0.
N to X,Y¶
The "n_to_xy()" calculation follows Ed Schouten's method
breaking N into base-7 digits, applying reversals from high to low according to
digits 1, 2, or 6, then applying rotation and position according to the
resulting digits.
Unlike Ed's code, the path here starts from N=0 at the edge of the Gosper island
shape and for that reason doesn't cover the plane. An offset of N-2*7^21 and
suitable X,Y offset can be applied to get the same result.
X,Y to N¶
The "xy_to_n()" calculation also follows Ed Schouten's method. It's
based on a nice observation that the seven cells of the base figure can be
identified from their X,Y coordinates, and the centre of those seven cell
figures then shrunk down a level to be a unit apart, thus generating digits of
N from low to high.
In triangular grid X,Y a remainder is formed
m = (x + 2*y) mod 7
Taking the base figure's N=0 at 0,0 the remainders are
4---- 6
/ \
1---- 3 5
\
0---- 2
The remainders are unchanged when the shape is moved by some multiple of the
next level X=5,Y=1 or the same at 120 degrees X=1,Y=3 or 240 degrees X=-4,Y=1.
Those vectors all have X+2*Y==0 mod 7.
From the m remainder an offset can be applied to move X,Y to the 0 position,
leaving X,Y a multiple of the next level vectors X=5,Y=1 etc. Those vectors
can then be shrunk down with
Xshrunk = (3*Y + 5*X) / 14
Yshrunk = (5*Y - X) / 14
This gives integers since 3*Y+5*X and 5*Y-X are always multiples of 14. For
example the N=35 point at X=2,Y=6 reduces to X = (3*6+5*2)/14 = 2 and Y =
(5*6-2)/14 = 2, which is then the "5" part of the base figure.
The remainders can be mapped to digits and then reversals and rotations applied,
from high to low, according to the edge orientation. Those steps can be
combined in a single lookup table with 6 states (three rotations, and each one
forward or reverse).
For the main curve the reduction ends at 0,0. For the multi-arm form the second
arm ends to the right at -2,0 and the third below at -1,-1. Notice the modulo
and shrink procedure maps those three points back to themselves unchanged. The
calculation can be done without paying attention to which arms are supposed to
be in use. On reaching one of the three ends the "arm" is determined
and the original X,Y can be rejected or accepted accordingly.
The key to this approach is that the base figure is symmetric around a central
point, so the tiling can be broken down first, and the rotations or reversals
in the path applied afterwards. Can it work on a non-symmetric base figure
like the "across" style of the main Flowsnake, or something like the
"DragonCurve" for that matter?
SEE ALSO¶
Math::PlanePath, Math::PlanePath::Flowsnake, Math::PlanePath::GosperIslands
Math::PlanePath::KochCurve, Math::PlanePath::HilbertCurve,
Math::PlanePath::PeanoCurve, Math::PlanePath::ZOrderCurve
<
http://80386.nl/projects/flowsnake/> -- Ed Schouten's code
HOME PAGE¶
<
http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE¶
Copyright 2011, 2012, 2013, 2014 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/>.