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Kevin Ryde
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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.

Level Ranges

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 $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.


Math::PlanePath, Math::PlanePath::PeanoCurve, Math::PlanePath::HilbertCurve, Math::PlanePath::KochPeaks, Math::PlanePath::KochSnowflakes





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/>.