NAME¶
Math::PlanePath::DragonMidpoint -- dragon curve midpoints
SYNOPSIS¶
use Math::PlanePath::DragonMidpoint;
my $path = Math::PlanePath::DragonMidpoint->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION¶
This is the midpoint of each segment of the dragon curve by Heighway, Harter, et
al, per Math::PlanePath::DragonCurve.
17--16 9---8 5
| | | |
18 15 10 7 4
| | | |
19 14--13--12--11 6---5---4 3
| |
20--21--22 3 2
| |
33--32 25--24--23 2 1
| | | |
34 31 26 0---1 <- Y=0
| | |
35 30--29--28--27 -1
|
36--37--38 43--44--45--46 -2
| | |
39 42 49--48--47 -3
| | |
40--41 50 -4
|
51 -5
|
52--53--54 -6
|
..--64 57--56--55 -7
| |
63 58 -8
| |
62--61--60--59 -9
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1
The dragon curve begins as follows. The midpoints of each segment are numbered
starting from 0,
+--8--+ +--4--+
| | | |
9 7 5 3
| | | | |
+-10--+--6--+ +--2--+ rotate 45 degrees |
| | v
11 1
| |
+-12--+ *--0--+ * = Origin
|
...
These midpoints are on fractions X=0.5,Y=0, X=1,Y=0.5, etc. For this
"DragonMidpoint" path they're turned clockwise 45 degrees and shrunk
by
sqrt(2) to be integer X,Y values a unit apart and initial direction
to the right.
The midpoints are distinct X,Y positions because the dragon curve traverses each
edge only once.
The dragon curve is self-similar in 2^level sections due to its unfolding. This
can be seen in the midpoints too as for example above N=0 to N=16 is the same
shape as N=16 to N=32, with the latter rotated 90 degrees and in reverse.
Arms¶
Like the "DragonCurve" the midpoints fill a quarter of the plane and
four copies mesh together perfectly when rotated by 90, 180 and 270 degrees.
The "arms" parameter can choose 1 to 4 curve arms, successively
advancing.
For example "arms => 4" begins as follows, with N=0,4,8,12,etc
being the first arm (the same as the plain curve above), N=1,5,9,13 the
second, N=2,6,10,14 the third and N=3,7,11,15 the fourth.
arms => 4
...-107-103 83--79--75--71 6
| | |
68--64 36--32 99 87 59--63--67 5
| | | | | | |
72 60 40 28 95--91 55 4
| | | | |
76 56--52--48--44 24--20--16 51 3
| | |
80--84--88 17--13---9---5 12 47--43--39 ... 2
| | | | | |
100--96--92 21 6---2 1 8 27--31--35 106 1
| | | | | |
104 33--29--25 10 3 0---4 23 94--98-102 <- Y=0
| | | | | |
... 37--41--45 14 7--11--15--19 90--86--82 -1
| | |
49 18--22--26 46--50--54--58 78 -2
| | | | |
53 89--93 30 42 62 74 -3
| | | | | | |
65--61--57 85 97 34--38 66--70 -4
| | |
69--73--77--81 101-105-... -5
^
-6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5
With four arms like this every X,Y point is visited exactly once, because four
arms of the "DragonCurve" traverse every edge exactly once.
Tiling¶
Taking pairs of adjacent points N=2k and N=2k+1 gives little rectangles with the
following tiling of the plane repeating in 4x4 blocks.
+---+---+---+-+-+---+-+-+---+
| | | | | | | | | | |
+---+ | +---+ | +---+ | +---+
| | | |9 8| | | | | | |
+-+-+---+-+-+-+-+-+-+-+-+-+-+
| | | | |7| | | | | | |
| | +---+ | +---+ | +---+ | |
| | | | |6|5 4| | | | | |
+---+-+-+-+-+-+-+-+-+-+-+-+-+
| | | | | |3| | | | |
+---+ | +---+ | +---+ | +---+
| | | | | |2| | | | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | | | | |0 1| | | | | | <- Y=0
| | +---+ | +---+ | +---+ | |
| | | | | | | | | | | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | | | | | | | | | |
+---+ | +---+ | +---+ | +---+
| | | | | | | | | | |
+---+-+-+---+-+-+---+-+-+---+
^
X=0
The pairs follow this pattern both for the main curve N=0 etc shown, and also
for the rotated copies per "Arms" above. This tiling is in the
tilingsearch database as
Taking pairs N=2k+1 and N=2k+2, being each odd N and its successor, gives a
regular pattern too, but this time repeating in blocks of 16x16.
|||--||||||--||--||--||||||--||||||--||||||--||||||--||||||--|||
|||--||||||--||--||--||||||--||||||--||||||--||||||--||||||--|||
-||------||------||------||------||------||------||------||-----
-||------||------||------||------||------||------||------||-----
|||--||||||||||||||--||||||||||||||--||||||||||||||--|||||||||||
|||--||||||||||||||--||||||||||||||--||||||||||||||--|||||||||||
-----||------||------||------||------||------||------||------||-
-----||------||------||------||------||------||------||------||-
-||--||--||--||--||--||||||--||--||--||--||--||--||--||||||--||-
-||--||--||--||--||--||||||--||--||--||--||--||--||--||||||--||-
-||------||------||------||------||------||------||------||-----
-||------||------||------||------||------||------||------||-----
|||||||||||--||||||||||||||--||||||||||||||--||||||||||||||--|||
|||||||||||--||||||||||||||--||||||||||||||--||||||||||||||--|||
-----||------||------||------||------||------||------||------||-
-----||------||------||------||------||------||------||------||-
|||--||||||--||--||--||||||--|| ||--||||||--||--||--||||||--|||
|||--||||||--||--||--||||||--|| ||--||||||--||--||--||||||--|||
-||------||------||------||------||------||------||------||-----
-||------||------||------||------||------||------||------||-----
|||--||||||||||||||--||||||||||||||--||||||||||||||--|||||||||||
|||--||||||||||||||--||||||||||||||--||||||||||||||--|||||||||||
-----||------||------||------||------||------||------||------||-
-----||------||------||------||------||------||------||------||-
-||--||||||--||--||--||--||--||--||--||||||--||--||--||--||--||-
-||--||||||--||--||--||--||--||--||--||||||--||--||--||--||--||-
-||------||------||------||------||------||------||------||-----
-||------||------||------||------||------||------||------||-----
|||||||||||--||||||||||||||--||||||||||||||--||||||||||||||--|||
|||||||||||--||||||||||||||--||||||||||||||--||||||||||||||--|||
-----||------||------||------||------||------||------||------||-
-----||------||------||------||------||------||------||------||-
Curve from Midpoints¶
Since the dragon curve always turns left or right, never straight ahead or
reverse, its segments are alternately horizontal and vertical. Rotated -45
degrees for the midpoints here this means alternately "opposite
diagonal" and "leading diagonal". They fall on X,Y alternately
even or odd. So the original dragon curve can be recovered from the midpoints
by choosing leading diagonal or opposite diagonal segment according to X,Y
even or odd, which is the same as N even or odd.
DragonMidpoint dragon segment
-------------- -----------------
"even" N==0 mod 2 opposite diagonal
which is X+Y==0 mod 2 too
"odd" N==1 mod 2 leading diagonal
which is X+Y==1 mod 2 too
/
3 0 at X=0,Y=0 "even", opposite diagonal
/ 1 at X=1,Y=0 "odd", leading diagonal
\ etc
2
\
\ /
0 1
\ /
FUNCTIONS¶
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
classes.
- "$path = Math::PlanePath::DragonMidpoint->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 = $path->n_start()"
- Return 0, the first N in the path.
Level Methods¶
- "($n_lo, $n_hi) = $path->level_to_n_range($level)"
- Return "(0, 2**$level - 1)", or for multiple arms return
"(0, $arms * 2**$level - 1)".
There are 2^level segments comprising the dragon, or arms*2^level when
multiple arms, numbered starting from 0.
X,Y to N¶
An X,Y point is turned into N by dividing out digits of a complex base i+1. This
base is per the doubling of the "DragonCurve" at each level. In
midpoint coordinates an adjustment subtracting 0 or 1 must be applied to move
an X,Y which is either N=2k or N=2k+1 to the position where dividing out i+1
gives the N=k X,Y.
The adjustment is in a repeating pattern of 4x4 blocks. Points N=2k and N=2k+1
both move to the same place corresponding to N=k multiplied by i+1. The
adjustment pattern is a little like the pair tiling shown above, but for some
pairs both the N=2k and N=2k+1 positions must move, it's not enough just to
shift the N=2k+1 to the N=2k.
Xadj Yadj
Ymod4 Ymod4
3 | 0 1 1 0 3 | 1 1 0 0
2 | 1 0 0 1 2 | 1 1 0 0
1 | 1 0 0 1 1 | 0 0 1 1
0 | 0 1 1 0 0 | 0 0 1 1
+-------- +--------
0 1 2 3 0 1 2 3
Xmod4 Xmod4
The same tables work for both the main curve and for the rotated copies per
"Arms" above.
until -1<=X<=0 and 0<=Y<=1
Xm = X - Xadj(X mod 4, Y mod 4)
Ym = Y - Yadj(X mod 4, Y mod 4)
new X,Y = (Xm+i*Ym) / (i+1)
= (Xm+i*Ym) * (1-i)/2
= (Xm+Ym)/2, (Ym-Xm)/2 # Xm+Ym and Ym-Xm are both even
Nbit = Xadj xor Yadj # bits of N low to high
The X,Y reduction stops at one of the start points for the four arms
X,Y endpoint Arm +---+---+
------------ --- | 2 | 1 | Y=1
0, 0 0 +---+---+
0, 1 1 | 3 | 0 | Y=0
-1, 1 2 +---+---+
-1, 0 3 X=-1 X=0
For arms 1 and 3 the N bits must be flipped 0<->1. The arm number and
hence whether this flip is needed is not known until reaching the endpoint.
For bignum calculations there's no need to apply the "/2" shift in
newX=(Xm+Ym)/2 and newY=(Ym-Xm)/2. Instead keep a bit position which is the
logical low end and pick out two bits from there for the Xadj,Yadj lookup. A
whole word can be dropped when the bit position becomes a multiple of 32 or 64
or whatever.
Boundary¶
Taking unit squares at each point, the boundary MB[k] of the resulting shape
from 0 to N=2^k-1 inclusive can be had from the boundary B[k] of the plain
dragon curve. Taking points N=0 to N=2^k-1 inclusive is the midpoints of the
dragon curve line segments N=0 to N=2^k inclusive.
MB[k] = B[k] + 2
= 4, 6, 10, 18, 30, 50, 86, 146, 246, 418, 710, 1202, ...
2 + x + 2*x^2
generating function 2 * -------------
1 - x - 2*x^3
A unit square at the midpoint is a diamond on a dragon line segment
/ \
/ \ midpoint m
*--m--* diamond on dragon curve line segment
\ /
\ /
A boundary segment of the dragon curve has two sides of the diamond which are
boundary. But when the boundary makes a right hand turn two such sides touch
and are therefore not midpoint boundary.
/^\
/ | \ right turn
\ | //\ two diamond sides touch
\|// \
*<----*
\ /
\ /
The dragon curve at N=0 points East and the last segment N=2^k-1 to N=2^k is
North. Since the curve never overlaps itself this means that when going around
the right side of the curve there is 1 more left turn than right turn,
lefts - rights = 1
The total line segments on the right is the dragon curve R[k] and there are
R[k]-1 turns, so the total turns lefts+rights is
lefts + rights + 1 = R[k]
So the lefts and rights are obtained separately
2*lefts = R[k] adding the two equations
2*rights = R[k] - 2 subtracting the two equations
The result is then
MR[k] = 2*R[k] - 2*rights
= 2*R[k] - 2*(R[k]-2)/2
= R[k] + 2
A similar calculation is made on the left side of the curve. The net turn is the
same and so the same lefts-rights=1 and thus from the dragon curve L[k] left
boundary
ML[k] = 2*L[k] - 2*lefts
= 2*L[k] - 2*(L[k]/2)
= L[k]
The total is then
MB[k] = MR[k] + ML[k]
= R[k]+2 + L[k]
= B[k] + 2 since B[k]=R[k]+L[k]
The generating function can be had from the partial fractions form of the dragon
curve boundary. B[k]+2 means adding 2/(1-x) which cancels out the -2/(1-x) in
gB(x).
OEIS¶
The "DragonMidpoint" is in Sloane's Online Encyclopedia of Integer
Sequences as
A073089 abs(dY) of n-1 to n, so 0=horizontal,1=vertical
(extra initial 0)
A077860 Y at N=2^k, being Re(-(i+1)^k + i-1)
A203175 boundary of unit squares N=0 to N=2^k-1, value 4 onwards
A073089¶
For A073089=abs(dY), the midpoint curve is vertical when the
"DragonCurve" has a vertical followed by a left turn, or horizontal
followed by a right turn. "DragonCurve" verticals are whenever N is
odd, and the turn is the bit above the lowest 0 in N (per "Turn" in
Math::PlanePath::DragonCurve). So
abs(dY) = lowbit(N) XOR bit-above-lowest-zero(N)
The n in A073089 is offset by 2 from the N numbering of the path here, so n=N+2.
The initial value at n=1 in A073089 has no corresponding N (it would be N=-1).
The mod-16 definitions in A073089 express combinations of N odd/even and
bit-above-low-0 which are the vertical midpoint segments. The recurrence
a(8n+1)=a(4n+1) acts to strip zeros above a low 1 bit,
n = 0b..uu0001
-> 0b...uu001
In terms of N=n-2 this reduces
N = 0b..vv1111
-> 0b...vv111
which has the effect of seeking a lowest 0 in the range of the mod-16
conditions.
SEE ALSO¶
Math::PlanePath, Math::PlanePath::DragonCurve, Math::PlanePath::DragonRounded
Math::PlanePath::AlternatePaperMidpoint, Math::PlanePath::R5DragonMidpoint,
Math::PlanePath::TerdragonMidpoint
HOME PAGE¶
<
http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE¶
Copyright 2011, 2012, 2013, 2014 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/>.