Provided by: libmath-planepath-perl_122-1_all 
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
"KochCurve" at successively greater replication 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 in the middle like N=4 through N=8,
*
/ \
*---* becomes *---* *---*
The angle is maintained in each replacement so
*
/
*---*
\
* *
/ becomes /
* *
For example 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 N=1 to N=3 peak as level 0, each peak is
Nstart = level + (2*4^level + 1)/3
Nend = level + (8*4^level + 1)/3
points = Nend-Nstart+1 = 2*4^level + 1
For example the outer peak shown above is level 2 starting at Nstart=2+(2*4^2+1)/3=13
through to Nend=2+(8*4^2+1)/3=45 with points=2*4^2+1=33 inclusive (45-13+1=33). The X
width at a given level is the endpoints at
Xlo = -(3^level)
Xhi = +(3^level)
For example the level 2 above runs from Xlo=-9 to Xhi=+9. The highest Y is the centre
peak half-way through the level at
Ypeak = 3^level
Npeak = level + (5*4^level + 1)/3
For example the level 2 outer peak above is Ypeak=3^2=9 at N=2+(5*4^2+1)/3=29. For each
level the Xlo,Xhi and Ypeak extents grow by a factor of 3.
The triangular notches in each segment are not big enough to go past the Xlo and Xhi end
points. The new triangular part can equal the ends, such as N=6 or N=19, but not go
beyond.
In general a segment like N=5 to N=6 which is at the Xlo end will expand to give two such
segments and two points at the limit in the next level, as for example N=5 to N=6 expands
to N=19,20 and N=20,21. So the count of points at Xlo doubles each time,
CountLo = 2^level
CountHi = 2^level same at Xhi
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$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 $n gives an X,Y position along a straight line between the integer
positions.
Level Methods
"($n_lo, $n_hi) = $path->level_to_n_range($level)"
Return per "Level Ranges" above,
((2 * 4**$level + 1)/3 + $level,
(8 * 4**$level + 1)/3 + $level)
FORMULAS
Rectangle to N Range
The baseline for a given level is along a diagonal X+Y=3^level or -X+Y=3^level. The
containing level can thus be found as
level = floor(log3( Xmax + Ymax ))
with Xmax as maximum absolute value, max(abs(X))
The endpoint in a 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, by considering which parts of the
curve might intersect a rectangle. But some backtracking and level descending is
necessary because a rectangle might extend into the empty part of a notch and so be past
its baseline but not intersect any. There's plenty of room for a rectangle to intersect
nothing at all too.
SEE ALSO
Math::PlanePath, Math::PlanePath::KochCurve, Math::PlanePath::KochSnowflakes,
Math::PlanePath::PeanoCurve, Math::PlanePath::HilbertCurve
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015 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/>.