NAME
Math::PlanePath::PeanoDiagonals  3x3 selfsimilar quadrant traversal across squares
SYNOPSIS
use Math::PlanePath::PeanoDiagonals;
my $path = Math::PlanePath::PeanoDiagonals>new;
my ($x, $y) = $path>n_to_xy (123);
# or another radix digits ...
my $path5 = Math::PlanePath::PeanoDiagonals>new (radix => 5);
DESCRIPTION
This path is the Peano curve with segments going diagonally across unit squares.
Giuseppe Peano, "Sur Une Courbe, Qui Remplit Toute Une Aire Plane", Mathematische Annalen, volume 36, number 1, 1890, pages 157160. DOI 10.1007/BF01199438. https://link.springer.com/article/10.1007/BF01199438, https://eudml.org/doc/157489
Points N are at each corner of the squares, so even locations (X+Y even),
9  61,425 63,423 65,421 79,407 81,405
8  60 58,62 64,68 66,78 76,80
7  55,59 57,69 67,71 73,77 75,87
6  54 52,56 38,70 36,72 34,74
5  49,53 39,51 37,41 31,35 33,129
4  48 46,50 40,44 30,42 28,32
3  7,47 9,45 11,43 25,29 27,135
2  6 4,8 10,14 12,24 22,26
1  1,5 3,15 13,17 19,23 21,141
Y=0  0 2 16 18 20
+
X=0 1 2 3 4 5 6 7 8 9
Moore (figure 3) draws this form, though here is transposed so first unit squares go East.
E. H. Moore, "On Certain Crinkly Curves", Transactions of the American Mathematical Society, volume 1, number 1, 1900, pages 7290.
http://www.ams.org/journals/tran/190000101/S00029947190015005264/, http://www.ams.org/journals/tran/190000104/S00029947190015004283/
Segments between the initial points can be illustrated,
 \ \
+ 47,7 + 45,9 
 ^  \  ^  \
 /  \  /  v
 /  v  /  ...
6 + 4,8 +
 ^  /  ^ 
 \  /  \ 
 \  v  \ 
+5,1 + 3,15
 ^  \  ^ 
 /  \  / 
 /  v  / 
N=0+2+
Segment N=0 to N=1 goes from the origin X=0,Y=0 up to X=1,Y=1, then N=2 is down again to X=2,Y=0, and so on. The plain PeanoCurve is the middle of each square, so points N + 1/2 here (and reckoning the first such midpoint as the origin).
The rule for block reversals is described with PeanoCurve. N is split to an X and Y digit alternately. If the sum of Y digits above is odd then the X digit is reversed, and vice versa X odd is Y reversed.
A plain diagonal is NorthEast per N=0 to 1. Diagonals are mirrored according to the final sum of all digits. If sum of Y digits is odd then mirror horizontally. If sum of X digits is odd then mirror vertically. Such mirroring is X+1 and/or Y+1 as compared to the plain PeanoCurve.
An integer N is at the start of the segment with its final reversal. Fractional N follows the diagonal across its unit square.
As noted above all locations are even (X+Y even). Those on the axes are visited once and all others twice.
Diamond Shape
Some authors take this diagonals form and raw it rotated 45 degrees so that the segments are X,Y aligned, and the pattern fills a wedge shape between diagonals X=Y and X=Y (for X>=0).
67,47
 
 
01,54,89,45
  
  ...
23,15
In terms of the coordinates here, this is (X+Y)/2, (YX)/2.
Even Radix
In an even radix, the mirror rule for diagonals across unit squares is applied the same way. But in this case the end of one segment does not always coincide with the start of the next.
+15,125+13,127 16 +18,98
 /  ^  /  ^  \  ^  \
 /  \  /  \  \  /  \
 v  \  v  \  v  /  v
+ 9  14  11  12  17 + ...
 ^  \  ^  \ 
 /  \  /  \ 
 /  v  /  v 
8  7  10  5 +
 /  ^  /  ^ 
 /  \  /  \  radix => 4
 v  \  v  \ 
+ 1  6  3  4 
 ^  \  ^  \ 
 /  \  /  \ 
 /  v  /  v 
N=0+ 2 ++
The first row N=0 to N=3 goes left to right. The next row N=4 to N=7 is a horizontal mirror image to go right to left. N = 3.99.. < 4 follows its diagonal across its unit square, so approaches X=3.99,Y=0. There is then a discontinuity up to N=4 at X=4,Y=1.
Block N=0 to N=15 repeats starting N=16, with vertical mirror image. There is a bigger discontinuity between N=15 to N=16 (like there is in even radix PeanoCurve).
Some doublevisited points occur, such as N=15 and N=125 both at X=1,Y=4. This is when the 4x16 block N=0 to 64 is copied above, mirrored horizontally.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.
$path = Math::PlanePath::PeanoDiagonals>new ()
$path = Math::PlanePath::PeanoDiagonals>new (radix => $r)

Create and return a new path object.
The optional
radix
parameter gives the base for digit splitting. The default is ternary,radix => 3
. ($x,$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
$n
gives an X,Y position along the diagonals across unit squares. ($n_lo, $n_hi) = $path>rect_to_n_range ($x1,$y1, $x2,$y2)

Return a range of N values which covers the rectangle with corners at
$x1
,$y1
and$x2
,$y2
. If the X,Y values are not integers then the curve is treated as unit squares centred on each integer point and squares which are partly covered by the given rectangle are included.In the current implementation, the returned range is an overestimate, so that
$n_lo
might be smaller than the smallest actually in the rectangle, and$n_hi
bigger than the actual biggest.
Level Methods
FORMULAS
N to Turn
The curve turns left or right 90 degrees at each point N >= 1. The turn is 90 degrees
turn(N) = (1)^(N + number of low ternary 0s of N)
= 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
by 90 degrees (+1 left, 1 right)
The power of 1 means left or right flip for each low ternary 0 of N, and flip again if N is odd. Odd N is an odd number of ternary 1 digits.
This formula follows from the turns in a new low base9 digit. For a segment crossing a given unit square, the expanded segments have the same start and end directions, so existing turns, now 9*N, are unchanged. Then 9*N+r goes as r in the base figure, but flipped L<>R when N odd since blocks are mirrored alternately.
turn(9N) = turn(N)
turn(9N+r) = turn(r)*(1)^N for 1 <= r <= 8
Or in terms of base 3, a single new low ternary digit is a transpose of what's above, and the base figure turns r=1,2 are L<>R when N above is odd.
turn(3N) =  turn(N)
turn(3N+r) = turn(r)*(1)^N for r = 1 or 2
Similarly in any odd radix.
SEE ALSO
Math::PlanePath, Math::PlanePath::PeanoCurve, Math::PlanePath::HilbertSides
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2019, 2020 Kevin Ryde
This file is part of MathPlanePath.
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.