trusty (3) Math::PlanePath::CCurve.3pm.gz
NAME
Math::PlanePath::CCurve -- Levy C curve
SYNOPSIS
use Math::PlanePath::CCurve;
my $path = Math::PlanePath::CCurve->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This is an integer version of the Levy "C" curve.
11-----10-----9,7-----6------5 3
| | |
13-----12 8 4------3 2
| |
19---14,18----17 2 1
| | | |
21-----20 15-----16 0------1 <- Y=0
|
22 -1
|
25,23---24 -2
|
26 35-----34-----33 -3
| | |
27,37--28,36 32 -4
| | |
38 29-----30-----31 -5
|
39,41---40 -6
|
42 ... -7
| |
43-----44 49-----48 64-----63 -8
| | | |
45---46,50----47 62 -9
| |
51-----52 56 60-----61 -10
| | |
53-----54----55,57---58-----59 -11
^
-7 -6 -5 -4 -3 -2 -1 X=0 1
The initial segment N=0 to N=1 is repeated with a turn +90 degrees left to give N=1 to N=2. Then N=0to2
is repeated likewise turned +90 degrees to make N=2to4. And so on doubling each time.
The 90 degree rotation is always relative to the initial N=0to1 direction along the X axis. So at any
N=2^level the turn is +90 making the direction upwards at each of N=1,2,4,8,16,etc.
The curve crosses itself and repeats some X,Y positions. The first doubled point is X=-2,Y=3 which is
both N=7 and N=9. The first tripled point is X=18,Y=-7 which is N=189, N=279 and N=281. The number of
repeats at a given point is always finite but as N increases there's points where that number of repeats
becomes ever bigger (is that right?).
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.
"$path = Math::PlanePath::CCurve->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 = $path->xy_to_n ($x,$y)"
Return the point number for coordinates "$x,$y". If there's nothing at "$x,$y" then return "undef".
"$n = $path->n_start()"
Return 0, the first N in the path.
FORMULAS
Direction
The direction or net turn of the curve is the count of 1 bits in N,
direction = count_1_bits(N) * 90degrees
For example N=11 is binary 1011 has three 1 bits, so direction 3*90=270 degrees, ie. to the south.
This bit count is because at each power-of-2 position the curve is a copy of the lower bits but turned
+90 degrees, so +90 for each 1-bit.
For powers-of-2 N=2,4,8,16, etc, there's only a single 1-bit so the direction is always +90 degrees
there, ie. always upwards.
Turn
At each point N the curve can turn in any direction: left, right, straight, or 180 degrees back. The
turn is given by the number of low 0-bits of N,
turn right = (count_low_0_bits(N) - 1) * 90degrees
For example N=8 is binary 0b100 which is 2 low 0-bits for turn=(2-1)*90=90 degrees to the right.
When N is odd there's no low zero bits and the turn is always (0-1)*90=-90 to the right in that case,
which means every second turn is 90 degrees to the left.
Next Turn
The turn at the point following N, ie. at N+1, can be calculated from the bits of N by counting the low
1-bits,
next turn right = (count_low_1_bits(N) - 1) * 90degrees
For example N=11 is binary 0b1011 which is 2 low one bits for nextturn=(2-1)*90=90 degrees to the right
at the following point, ie. at N=12.
This works simply because low 1-bits like ..0111 increment to low 0-bits ..1000 to become N+1. The low
1-bits at N are thus the low 0-bits at N+1.
N to dX,dY
"n_to_dxdy()" is implemented using the direction described above. If N is an integer then count mod 4
gives the direction for dX,dY.
dir = count_1_bits(N) mod 4
dx = dir_to_dx[dir] # table 0 to 3
dy = dir_to_dy[dir]
For fractional N the direction at int(N)+1 can be obtained from the direction at int(N) by applying the
turn at int(N)+1, that being the low 1-bits of N per "Next Turn" above. Those two directions can then be
combined per "N to dX,dY -- Fractional" in Math::PlanePath.
# apply turn to make direction at Nint+1
turn = count_low_1_bits(N) - 1 # N integer part
dir = (dir - turn) mod 4 # direction at N+1
# adjust dx,dy by fractional amount in this direction
dx += Nfrac * (dir_to_dx[dir] - dx)
dy += Nfrac * (dir_to_dy[dir] - dy)
A tiny optimization can be made by working the "-1" of the turn formula into a +90 degree rotation of the
"dir_to_dx[]" and "dir_to_dy[]" parts by a swap and sign change,
turn_plus_1 = count_low_1_bits(N) # on N integer part
dir = (dir - turn_plus_1) mod 4 # direction-1 at N+1
# adjustment including extra +90 degrees on dir
dx -= $n*(dir_to_dy[dir] + dx)
dy += $n*(dir_to_dx[dir] - dy)
X,Y to N
The N values at a given X,Y can be found by traversing the curve. At a given digit position if X,Y is
within the curve extents at that level and position then descend to consider the next lower digit
position, otherwise step to the next digit at the current digit position.
It's convenient to work in base-4 digits since that keeps the digit steps straight rather than diagonals.
The maximum extent of the curve at a given even numbered level is
k = level/2
Lmax(level) = 2^k + floor(2^(k-1) - 1);
For example k=2 is level=4, N=0 to N=2^4=16 has extent Lmax=2^2+2^1-1=5. That extent can be seen at
points N=13,N=14,N=15.
The extents width-ways and backwards are shorter and using them would tighten the traversal, cutting off
some unnecessary descending. But the calculations are then a little trickier.
The first N found by this traversal is the smallest. Continuing the search gives all the N which are the
target X,Y.
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
<http://oeis.org/A179868> (etc)
A010059 abs(dX), count1bits(N) mod 2
A010060 abs(dY), count1bits(N)+1 mod 2, being Thue-Morse
A000120 total turn, being count 1-bits
A179868 direction 0to3, count 1-bits mod 4
A096268 turn 1=straight,0=left or right
A007814 turn-1 to the right, being count low 0-bits
A003159 N positions of left or right turn, ends even num 0 bits
A036554 N positions of straight or 180 turn, ends odd num 0 bits
A146559 X at N=2^k, being Re((i+1)^k)
A009545 Y at N=2^k, being Im((i+1)^k)
SEE ALSO
Math::PlanePath, Math::PlanePath::DragonCurve, Math::PlanePath::AlternatePaper,
Math::PlanePath::KochCurve
ccurve(6x) back end of xscreensaver(1) displaying the C curve (and various other dragon curve and Koch
curves).
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2011, 2012, 2013 Kevin Ryde
This file is part of Math-PlanePath.
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/>.