Math::PlanePath::QuintetCurve -- self-similar "plus" shaped curve
use Math::PlanePath::QuintetCurve; my $path = Math::PlanePath::QuintetCurve->new; my ($x, $y) = $path->n_to_xy (123);
This path is traces out a spiralling self-similar "+" shape,
... 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 a "+" 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
The optional arms => $a parameter can give 1 to 4 copies of the curve, each advancing successively. For example arms=>4 is as follows. Notice the 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.
arms => $a
arms=>4
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
Essentially the curve is an ever expanding "+" shape with one corner at the origin. This means four of them back be packed as follows,
+---+ | | +---+--- +---+ | | A | +---+ +---+ +---+ | B | | | +---+ +---O---+ +---+ | | | D | +---+ +---+ +---+ | C | | +---+ +---+---+ | | +---+
At higher replication levels the sides become wiggly and spiralling, but they're symmetric and mesh to fill the plane.
See "FUNCTIONS" in Math::PlanePath for the 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.
$n
$n < 0
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.)
$n_lo
$n_hi
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.
xy_to_n()
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.
n_to_xy()
Math::PlanePath, Math::PlanePath::QuintetCentres, Math::PlanePath::QuintetReplicate, Math::PlanePath::Flowsnake
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011 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.