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 N=0 to N=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 X=0 to X=2*3^level inclusive. The highest Y is 3 times greater at each level similarly, for peak Y=3^level in the middle of the level at X=3^level and 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.

Koch conceived the curve as having infinitely fine structure making it everywhere continuous but nowhere differentiable. The code here can be pressed into service for that sort of construction at a given level by scaling X/3^level and Y/3^level to make it a fixed 2 wide by 1 high (and perhaps 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/>.