Math::PlanePath::KochCurve -- horizontal Koch curve
use Math::PlanePath::KochCurve; my $path = Math::PlanePath::KochCurve->new; my ($x, $y) = $path->n_to_xy (123);
This path is an integer version of the self-similar curve by Helge von Koch going along the X axis and making triangular excursions.
8 3 / \ 6---- 7 9----10 19-... 2 \ / \ 2 5 11 14 18 1 / \ / \ / \ / 0----1 3---- 4 12----13 15----16 <- Y=0 ^ X=0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
The replicating shape is the initial N=0 to N=4,
* / \ *---* *---*
which is rotated and repeated 3 times in the same shape to give sections N=4 to N=8, N=8 to N=12, and N=12 to N=16. Then that N=0 to N=16 is itself replicated three times at the angles of the base pattern, and so on infinitely.
The X,Y coordinates are arranged on a square grid using every second point, per "Triangular Lattice" in Math::PlanePath. The result is flattened triangular segments with diagonals at a 45 degree angle.
Each replication adds 3 copies of the existing points and is thus 4 times bigger, so if N=0 to N=4 is reckoned as level 1 then a given replication level goes from
Nstart = 0 Nlevel = 4^level (inclusive)
Each replication is 3 times the width. The initial N=0 to N=4 figure is 6 wide, so in general a level runs from
Xstart = 0 Xlevel = 2*3^level (at Nlevel)
The highest Y is 3 times greater at each level similarly. The peak is at the midpoint of each level,
Npeak = (4^level)/2 Ypeak = 3^level Xpeak = 3^level
It can be seen that the N=6 point backtracks horizontally to the same X as the start of its section N=4 to N=8. This happens in the further replications too and is the maximum extent of the backtracking.
The Nlevel is multiplied by 4 to get the end of the next higher level. The same 4*N can be applied to all points N=0 to N=Nlevel to get the same shape but a factor of 3 bigger X,Y coordinates. The in-between points 4*N+1, 4*N+2 and 4*N+3 are then new finer structure in the higher level.
Koch conceived the curve as having a fixed length and infinitely fine structure, making it continuous everywhere but differentiable nowhere. The code here can be pressed into use for that sort of construction to a given level of granularity by scaling
to make it a fixed 2 wide by 1 high. Or for unit-side equilateral triangles then further factors 1/2 and sqrt(3)/2, as noted in "Triangular Lattice" in Math::PlanePath.
(X/2) / 3^level (Y*sqrt(3)/2) / 3^level
See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.
$path = Math::PlanePath::KochCurve->new ()
Create and return a new path object.
($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.
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.
Copyright 2011 Kevin Ryde
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/>.