/ 
Math-PlanePath-39
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::KochPeaks

NAME

Math::PlanePath::KochPeaks -- Koch curve peaks

SYNOPSIS

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

DESCRIPTION

This path traces out concentric peaks made from integer versions of the self-similar Koch curve at successively greater iteration levels.

                           29                                 9
                          /  \
                  27----28    30----31                        8
                    \              /
         23          26          32          35               7
        /  \        /              \        /  \
21----22    24----25                33----34    36----37      6
  \                                                  /
   20                                              38         5
  /                                                  \
19----18                                        40----39      4
        \                                      /
         17                 8                41               3
        /                 /  \                 \
15----16           6---- 7     9----10          42----43      2
  \                 \              /                 /
   14                 5     2    11                44         1
  /                 /     /  \     \                 \
13                 4     1    3     12                45  <- Y=0

                            ^
-9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7  8  9 ...

The initial figure is the peak N=1,2,3 then for the next level each straight side expands to 3x longer with a notch like N=4 through N=8,

                            *
                           / \
*---*     becomes     *---*   *---*

The angle is maintained in each replacement,

                            *
                           /
                      *---*
                       \
  *                     *
 /        becomes      /
*                     *

So the segment N=1 to N=2 becomes N=4 to N=8, or in the next level N=5 to N=6 becomes N=17 to N=21.

The X,Y coordinates are arranged as integers on a square grid. The result is flattened triangular segments with diagonals at a 45 degree angle.

Unlike other triangular grid paths KochPeaks uses the "odd" squares, with one of X,Y odd and the other even. This means the rotation formulas etc described in "Triangular Lattice" in Math::PlanePath don't apply directly.

Level Ranges

Counting the innermost peak as level 0, each peak is

Nstart = level + (2*4^level + 1)/3
length = 2*4^level + 1       including endpoints

For example the outer ring shown above is level 2 starting at N=2+(2*4^2+1)/3=13 and having length=2*4^2+1=9 many points through to N=12 (inclusive). The X range at a given level is the endpoints at

Xlo = -(3^level)
Xhi = +(3^level)

For example the level 2 above runs from X=-9 to X=+9. The highest Y is the centre peak at

Ypeak = 3^level
Npeak = level + (5*4^level + 1)/3

Notice that for each level the extents grow by a factor of 3. But the new triangular notch in each segment is not big enough to go past the X start and end points. They can equal the ends, such as N=6 or N=19, but not beyond.

FUNCTIONS

$path = Math::PlanePath::KochPeaks->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 $n gives an X,Y position along a straight line between the integer positions.

FORMULAS

Rectangle to N Range

As noted above ("Level Ranges"), for a given level

-(3^level) <= X <= 3^level

So the maximum X in a rectangle gives a level,

level = ceil (log3 (max(x1,x2)))

and the endpoint in that level is simply 1 before the start of the next, so

Nlast = Nstart(level+1) - 1
      = (level+1) + (2*4^(level+1) + 1)/3 - 1
      = level + (8*4^level + 1)/3

Using this Nlast is an over-estimate of the N range needed, but an easy calculation. It's not too difficult to work down for an exact range.

SEE ALSO

Math::PlanePath, Math::PlanePath::KochCurve, Math::PlanePath::PeanoCurve, Math::PlanePath::HilbertCurve, Math::PlanePath::KochCurve, Math::PlanePath::KochSnowflakes

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/>.