Math::PlanePath::R5DragonCurve -- radix 5 dragon curve
use Math::PlanePath::R5DragonCurve; my $path = Math::PlanePath::R5DragonCurve->new; my ($x, $y) = $path->n_to_xy (123);
This is the R5 dragon curve by Jorg Arndt,
31-----30 27-----26 5 | | | | 32---29/33--28/24----25 4 | | 35---34/38--39/23----22 11-----10 7------6 3 | | | | | | | 36---37/41--20/40--21/17--16/12---13/9----8/4-----5 2 | | | | | | --50 47---42/46--19/43----18 15-----14 3------2 1 | | | | | 49/53--48/64 45/65--44/68 69 0------1 <-Y=0 ^ ^ ^ ^ ^ ^ ^ ^ ^ -7 -6 -5 -4 -3 -2 -1 X=0 1
The base figure is an "S" shape
4----5 | 3----2 | 0----1
which then repeats in self-similar style, so N=5 to N=10 is a copy rotated +90 degrees, as per the direction of the N=1 to N=2 segment.
10 7----6 | | | <- repeat rotated +90 9---8,4---5 | 3----2 | 0----1
This replication is similar to the TerdragonCurve in that there's no reversals or mirroring. Each replication is the plain base curve.
TerdragonCurve
The shape of N=0,5,10,15,20,25 repeats the initial N=0 to N=5,
25 4 / / 10__ 3 / / ----___ 20__ / 5 2 ----__ / / 15 / 1 / 0 <-Y=0 ^ ^ ^ ^ ^ ^ -4 -3 -2 -1 X=0 1
The curve never crosses itself. The vertices touch at corners like N=4 and N=8 above, but no edges repeat.
The first step N=1 is to the right along the X axis and the path then slowly spirals anti-clockwise and progressively fatter. The end of each replication is
Nlevel = 5^level
That point is at arctan(2/1)=63.43 degrees further around for each level,
Nlevel X,Y angle (degrees) ------ ----- ----- 1 1,0 0 5 2,1 63.4 25 -3,4 126.8 125 -11,-2 190.3
The curve fills a quarter of the plane and four copies mesh together perfectly rotated by 90, 180 and 270 degrees. The arms parameter can choose 1 to 4 such curve arms successively advancing.
arms
arms => 4 begins as follows. N=0,4,8,12,16,etc is the first arm (the same shape as the plain curve above), then N=1,5,9,13,17 the second, N=2,6,10,14 the third, etc.
arms => 4
arms => 4 16/32---20/63 | 21/60 9/56----5/12----8/59 | | | | 17/33--- 6/13--0/1/2/3---4/15---19/35 | | | | 10/57----7/14---11/58 23/62 | 22/61---18/34
With four arms every X,Y point is visited twice, except the origin 0,0 where all four begin. Every edge between the points is traversed once.
The little "S" shapes of the N=0to5 base shape tile the plane with 2x1 bricks and 1x1 holes in the following pattern,
| | | | | | | | | |---------+---------| |---------+---------| |- | | | | | | | | | | | | | | | | | | ------| |---------+---------| |---------+------ | | | | | | | | | | | | | | | | ------+---------| |---------+---------| |------ | | | | | | | | | | | | | | | | | | -| |---------+---------| |---------+---------| | | | | | | | | | | | | | | | | | | -+---------| |---------o---------| |---------+- | | | | | | | | | | | | | | | | | | |---------+---------| |---------+---------| |- | | | | | | | | | | | | | | | | | | ------| |---------+---------| |---------+------ | | | | | | | | | | | | | | | | ------+---------| |---------+---------| |------ | | | | | | | | | | | | | | | | | | -| |---------+---------| |---------+---------| | | | | | | | | |
This is simply the curve with segment N=2mod5 to N=3mod5 omitted from each mod5 block. In each 2x1 block the "S" traverses 4 of the 6 edges and the way the curve meshes together traverses the other 2 edges in another brick, possibly a brick on another arm of the curve.
This tiling is also for example
http://tilingsearch.org/HTML/data182/AL04.html
Or with enlarged square part, http://tilingsearch.org/HTML/data149/L3010.html
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::R5DragonCurve->new ()
$path = Math::PlanePath::R5DragonCurve->new (arms => 4)
Create and return a new path object.
The optional arms parameter can make 1 to 4 copies of the curve, each arm successively advancing.
($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.
$n
$n < 0
Fractional $n gives 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.
$x,$y
undef
The curve can visit an $x,$y twice. In the current code the smallest of the these N values is returned. Is that the best way?
@n_list = $path->xy_to_n_list ($x,$y)
Return a list of N point numbers for coordinates $x,$y. There can be none, one or two N's for a given $x,$y.
$n = $path->n_start()
Return 0, the first N in the path.
At each point N the curve always turns 90 degrees either to the left or right, it never goes straight ahead. As per the code in Jorg Arndt's fxtbook, if N is written in base 5 then the lowest non-zero digit gives the turn
lowest non-0 digit turn ------------------ ---- 1 left 2 left 3 right 4 right
At a point N=digit*5^level for digit=1,2,3,4 the turn follows the shape at that digit, so two lefts then two rights,
4*5^k----5^(k+1) | | 2*5^k----2*5^k | | 0------1*5^k
The first and last unit segments in each level are the same direction, so at those endpoints it's the next level up which gives the turn.
The turn at N+1 can be calculated in a similar way but from the lowest non-4 digit.
lowest non-4 digit turn ------------------ ---- 0 left 1 left 2 right 3 right
This works simply because in N=...z444 becomes N+1=...(z+1)000 and the turn at N+1 is given by digit z+1.
The direction at N, ie. the total cumulative turn, is given by the direction of each digit when N is written in base 5,
digit direction 0 0 1 1 2 2 3 1 4 0 direction = (sum direction for each digit) * 90 degrees
For example N=13 is base5 23 so direction=(2+1)*90 = 270 degrees, ie. south.
Because there's no reversals etc in the replications there's no state to maintain when considering the digits, just a plain sum of direction for each digit.
The R5 dragon is in Sloane's Online Encyclopedia of Integer Sequences as,
http://oeis.org/A175337 (etc)
A175337 next turn 0=left,1=right (n=0 is the turn at N=1) arms=1 and arms=3 A059841 abs(dX), being 1,0 repeating A000035 abs(dY), being 0,1 repeating arms=4 A165211 abs(dY), being 0,1,0,1,1,0,1,0 repeating
Math::PlanePath, Math::PlanePath::DragonCurve, Math::PlanePath::TerdragonCurve
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2012, 2013 Kevin Ryde
This file is part of Math-PlanePath.
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