NAME

Math::PlanePath::KochCurve -- horizontal Koch curve

SYNOPSIS

 use Math::PlanePath::KochCurve;
 my $path = Math::PlanePath::KochCurve->new;
 my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

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
                            /  \
                     6---- 7     9----10                19-...
                      \              /                    \
            2           5          11          14          18
          /  \        /              \        /  \        /
    0----1     3---- 4                12----13    15----16
    ^
    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 section 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 -side pattern, and so on infinitely.

The X,Y coordinates are arranged on a square grid using every second point, see "Triangular Lattice" in Math::PlanePath. The result is flattened triangular segments with diagonals at a 45 degree angle.

Level Ranges

Each replication in adds 3 times 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, for peak

    X=3^level
    Y=3^level
    at N=(4^level)/2

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 replications too and is the maximum extent of the backtracking.

The Nlevel value 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 on the X,Y coordinates. The in-between points 4*N+1, 4*N+2 and 4*N+3 are the new finer structure in the higher level.

Fractal

Koch conceived the curve as having a fixed length and infinitely fine structure, so it's continuous everywhere but differentiable nowhere. The code here can be pressed into service for that sort of construction of a given level by scaling

    X/3^level
    Y/3^level

to make it a fixed 2 wide by 1 high. Or apply factors 1/2 and sqrt(3)/2 as above for unit-side equilateral triangles.

FUNCTIONS

$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.

SEE ALSO

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

Math::Fractal::Curve

HOME PAGE

http://user42.tuxfamily.org/math-planepath/index.html

LICENSE

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/>.

cpanm Math::PlanePath

perl -MCPAN -e shell
install Math::PlanePath