Provided by: libmath-planepath-perl_113-1_all 
NAME
Math::PlanePath::QuintetCurve -- self-similar "plus" shaped curve
SYNOPSIS
use Math::PlanePath::QuintetCurve;
my $path = Math::PlanePath::QuintetCurve->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This path is traces out a spiralling self-similar "+" shape,
125--... 93--92 11
| | |
123-124 94 91--90--89--88 10
| | |
122-121-120 103-102 95 82--83 86--87 9
| | | | | | |
115-116 119 104 101-100--99 96 81 84--85 8
| | | | | | |
113-114 117-118 105 32--33 98--97 80--79--78 7
| | | | |
112-111-110-109 106 31 34--35--36--37 76--77 6
| | | | |
108-107 30 43--42 39--38 75 5
| | | | |
25--26 29 44 41--40 73--74 4
| | | | |
23--24 27--28 45--46--47 72--71--70--69--68 3
| | |
22--21--20--19--18 49--48 55--56--57 66--67 2
| | | | |
5---6---7 16--17 50--51 54 59--58 65 1
| | | | | | |
0---1 4 9---8 15 52--53 60--61 64 <- Y=0
| | | | | |
2---3 10--11 14 62--63 -1
| |
12--13 -2
^
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ...
The base figure is the initial N=0 to N=4.
5
|
|
0---1 4 base figure
| |
| |
2---3
It corresponds to a traversal of the following "+" shape,
.... 5
. |
. <|
|
0----1 .. 4 ....
v | | .
. |> |> .
. | | .
.... 2----3 ....
. v .
. .
. .
. .. .
The "v" and ">" notches are the side the figure is directed at the higher replications.
The 0, 2 and 3 parts are the right hand side of the line and are a plain repetition of the
base figure. The 1 and 4 parts are to the left and are a reversal. The first such
reversal is seen above as N=5 to N=10.
.....
. .
5---6---7 ...
. . | .
. | . reversed figure
... 9---8 ...
| .
| .
10 ...
In the base figure it can be seen the N=5 endpoint is rotated up around from the N=0 to
N=1 direction. This makes successive higher levels slowly spiral around.
N = 5^level
angle = level * atan(1/2)
= level * 26.56 degrees
radius = sqrt(5) ^ level
In the sample shown above N=125 is level=3 and has spiralled around to angle 3*26.56=79.7
degrees. The next level goes into the second quadrant with X negative. A full circle
around the plane is around level 14.
Arms
The optional "arms => $a" parameter can give 1 to 4 copies of the curve, each advancing
successively. For example "arms=>4" is as follows. N=4*k points are the plain curve, and
N=4*k+1, N=4*k+2 and N=4*k+3 are rotated copies of it.
69--65 ...
| | |
..-117-113-109 73 61--57--53--49 120
| | | |
101-105 77 25--29 41--45 100-104 116
| | | | | | | |
97--93 81 21 33--37 92--96 108-112
| | | |
50--46 89--85 17--13-- 9 88--84--80--76--72
| | | |
54 42--38 10-- 6 1-- 5 20--24--28 64--68
| | | | | | |
58 30--34 14 2 0-- 4 16 36--32 60
| | | | | | |
66--62 26--22--18 7-- 3 8--12 40--44 56
| | | |
70--74--78--82--86 11--15--19 87--91 48--52
| | | |
110-106 94--90 39--35 23 83 95--99
| | | | | | | |
114 102--98 47--43 31--27 79 107-103
| | | |
118 51--55--59--63 75 111-115-119-..
| | |
... 67--71
The curve is essentially an ever expanding "+" shape with one corner at the origin. Four
such shapes can be packed as follows,
+---+
| |
+---+--- +---+
| | A |
+---+ +---+ +---+
| B | | |
+---+ +---O---+ +---+
| | | D |
+---+ +---+ +---+
| C | |
+---+ +---+---+
| |
+---+
At higher replication levels the sides are wiggly and spiralling and the centres of each
rotated around, but they sides are symmetric and mesh together perfectly to fill the
plane.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::QuintetCurve->new ()"
"$path = Math::PlanePath::QuintetCurve->new (arms => $a)"
Create and return a new path object.
"($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 = $path->n_start()"
Return 0, the first N in the path.
"($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.)
FORMULAS
X,Y to N
The current approach uses the "QuintetCentres" "xy_to_n()". Because the tiling in
"QuintetCurve" and "QuintetCentres" is the same, the X,Y coordinates for a given N are no
more than 1 away in the grid.
The way the two lowest shapes are arranged in fact means that for a "QuintetCurve" N at
X,Y then the same N on the "QuintetCentres" is at one of three locations
X, Y same
X, Y+1 up
X-1, Y+1 up and left
X-1, Y left
This is so even when the "arms" multiple paths are in use (the same arms in both
coordinates).
Is there an easy way to know which of the four offsets is right? The current approach is
to give each to "QuintetCentres" to make an N, put that N back through "n_to_xy()" to see
if it's the target $n.
SEE ALSO
Math::PlanePath, Math::PlanePath::QuintetCentres, Math::PlanePath::QuintetReplicate,
Math::PlanePath::Flowsnake
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2011, 2012, 2013 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/>.