# Copyright 2019, 2020 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 . package Math::PlanePath::PeanoDiagonals; use 5.004; use strict; use vars '$VERSION', '@ISA'; $VERSION = 129; use Math::PlanePath; @ISA = ('Math::PlanePath'); use Math::PlanePath; *max = \&Math::PlanePath::_max; use Math::PlanePath::PeanoCurve; *_n_to_xykk = \&Math::PlanePath::PeanoCurve::_n_to_xykk; *_xykk_to_n = \&Math::PlanePath::PeanoCurve::_xykk_to_n; use Math::PlanePath::Base::Generic 'is_infinite', 'round_nearest'; use Math::PlanePath::Base::Digits 'round_up_pow', 'round_down_pow', 'digit_split_lowtohigh', 'digit_join_lowtohigh'; # uncomment this to run the ### lines # use Smart::Comments; use constant n_start => 0; use constant class_x_negative => 0; use constant class_y_negative => 0; use constant turn_any_straight => 0; # never straight use constant dx_minimum => -1; use constant dx_maximum => 1; use constant dy_minimum => -1; use constant dy_maximum => 1; use constant parameter_info_array => [ { name => 'radix', share_key => 'radix_3', display => 'Radix', type => 'integer', minimum => 2, default => 3, width => 3, } ]; # odd radix is unit steps diagonally, # even radix unlimited sub _UNDOCUMENTED__dxdy_list { my ($self) = @_; return ($self->{'radix'} % 2 ? (1,1, -1,1, -1,-1, 1,-1) : ()); # even, unlimited } sub new { my $self = shift->SUPER::new(@_); if (! $self->{'radix'} || $self->{'radix'} < 2) { $self->{'radix'} = 3; } return $self; } sub n_to_xy { my ($self, $n) = @_; ### PeanoDiagonals n_to_xy(): "$n" if ($n < 0) { # negative return; } if (is_infinite($n)) { return ($n,$n); } my $frac; { my $int = int($n); $frac = $n - $int; # inherit possible BigFloat $n = $int; } my ($x,$y, $xk,$yk) = _n_to_xykk($self,$n); ### xykk: "$x,$y $xk,$yk" return ($x + ($xk&1 ? 1-$frac : $frac), $y + ($yk&1 ? 1-$frac : $frac)); } sub xy_to_n { return scalar((shift->xy_to_n_list(@_))[0]); } sub xy_to_n_list { my ($self, $x, $y) = @_; ### PeanoDiagonals xy_to_n(): "$x, $y" # For odd radix, if X is even then segments are NE or SW, so offset 0,0 or # 1,1 to go to "middle" points. Conversely if X is odd then segments are # NW or SE so offset 0,1 or 1,0. # # ENHANCE-ME: For odd radix, the two offsets are exactly the two visits. # Should be able to pay attention to the low 0s or 2s and so have the # digits of both N in one look. # # ENHANCE-ME: Is the offset rule for even radix found as easily? $x = round_nearest ($x); $y = round_nearest ($y); if ($x < 0 || $y < 0) { return; } if (is_infinite($x)) { return $x; } if (is_infinite($y)) { return $y; } return sort {$a<=>$b} map {_xykk_to_n($self, $x,$y, @$_)} ($self->{'radix'}&1 ? ($x&1 ? ([0,1],[1,0]) : ([0,0],[1,1])) : ([0,0],[1,1], [0,1],[1,0])); } #------------------------------------------------------------------------------ # not exact # block 0 .. 3^k-1 contains all sub rect_to_n_range { my ($self, $x1,$y1, $x2,$y2) = @_; $x1 = round_nearest ($x1); $y1 = round_nearest ($y1); $x2 = round_nearest ($x2); $y2 = round_nearest ($y2); ($x1,$x2) = ($x2,$x1) if $x1 > $x2; ($y1,$y2) = ($y2,$y1) if $y1 > $y2; ### rect_to_n_range(): "$x1,$y1 to $x2,$y2" if ($x2 < 0 || $y2 < 0) { return (1, 0); } my $radix = $self->{'radix'}; my ($power, $level) = round_down_pow (max($x2,$y2)*$radix, $radix); if (is_infinite($level)) { return (0, $level); } return (0, $power*$power - 1); } #------------------------------------------------------------------------------ # levels sub level_to_n_range { my ($self, $level) = @_; return (0, $self->{'radix'}**(2*$level)); } sub n_to_level { my ($self, $n) = @_; if ($n < 0) { return undef; } $n = round_nearest($n); my ($pow, $exp) = round_up_pow ($n, $self->{'radix'}*$self->{'radix'}); return $exp; } #------------------------------------------------------------------------------ # num low ternary 0s, and whether odd or even above there which is parity of # how many 1-digits # sub _UNDOCUMENTED__n_to_turn_LSR { my ($self, $n) = @_; if ($n < 1 || is_infinite($n)) { return undef; } my $radix = $self->{'radix'}; if ($radix & 1) { my $turn = 1; until ($n % $radix) { # parity of low 0s $turn = -$turn; $n /= $radix; } return ($n % 2 ? -$turn : $turn); # and flip again if odd } return $self->SUPER::_UNDOCUMENTED__n_to_turn_LSR($n); } #------------------------------------------------------------------------------ 1; __END__ =for stopwords Giuseppe Peano Peano's eg Sur Une Courbe Qui Remplit Toute Aire Mathematische Annalen Ryde OEIS ZOrderCurve ie Math-PlanePath versa Online Radix radix HilbertCurve PeanoCurve DOI =head1 NAME Math::PlanePath::PeanoDiagonals -- 3x3 self-similar quadrant traversal across squares =head1 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); =head1 DESCRIPTION This path is the Peano curve with segments going diagonally across unit squares. =over Giuseppe Peano, "Sur Une Courbe, Qui Remplit Toute Une Aire Plane", Mathematische Annalen, volume 36, number 1, 1890, pages 157-160. DOI 10.1007/BF01199438. L, L =back Points N are at each corner of the squares, so even locations (X+Y even), =cut # generated by: # math-image --path=PeanoDiagonals --all --output=numbers --size=45x10 =pod 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. =over E. H. Moore, "On Certain Crinkly Curves", Transactions of the American Mathematical Society, volume 1, number 1, 1900, pages 72-90. L, L =back =cut # Eliakim Hastings =pod 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 North-East 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. =cut # Peano's conception for a space-filling curve is ternary digits below the # radix point to X and Y ... of a fractional f which fills a unit square going from f=0 # at X=0,Y=0 up to f=1 at X=1,Y=1. The integer form here does this with # digits above the ternary point. =pod =head2 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 XE=0). 6----7,47 | | | | 0---1,5---4,8---9,45 | | | | | ... 2----3,15 In terms of the coordinates here, this is (X+Y)/2, (Y-X)/2. =for GP-Test ('x+I*'y)/(1+I) == ('x+'y)/2 + ('y-'x)/2 * I =head2 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. =cut # compare # math-image --path=PeanoDiagonals,radix=4 --all --output=numbers --size=30x9 =pod +---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 double-visited 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. =head1 FUNCTIONS See L for the behaviour common to all path classes. =over 4 =item C<$path = Math::PlanePath::PeanoDiagonals-Enew ()> =item C<$path = Math::PlanePath::PeanoDiagonals-Enew (radix =E $r)> Create and return a new path object. The optional C parameter gives the base for digit splitting. The default is ternary, C 3>. =item C<($x,$y) = $path-En_to_xy ($n)> Return the X,Y coordinates of point number C<$n> on the path. Points begin at 0 and if C<$n E 0> then the return is an empty list. Fractional C<$n> gives an X,Y position along the diagonals across unit squares. =item C<($n_lo, $n_hi) = $path-Erect_to_n_range ($x1,$y1, $x2,$y2)> Return a range of N values which covers the rectangle with corners at C<$x1>,C<$y1> and C<$x2>,C<$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 over-estimate, so that C<$n_lo> might be smaller than the smallest actually in the rectangle, and C<$n_hi> bigger than the actual biggest. =back =head2 Level Methods =over =item C<($n_lo, $n_hi) = $path-Elevel_to_n_range($level)> Return C<(0, $radix**(2*$level) - 1)>. =back =head1 FORMULAS =head2 N to Turn The curve turns left or right 90 degrees at each point N E= 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) =cut # checking in xt/PeanoDiagonals-seq.t too. # # GP-DEFINE turn(n) = (-1)^(n + valuation(n,3)); # GP-Test vector(18,n, turn(n)) == \ # GP-Test [-1,1, 1, 1,-1, -1, -1,1,-1,1,-1, -1, -1,1,1,1,-1,1] # not in OEIS: -1,1,1,1,-1,-1,-1,1,-1,1,-1,-1,-1,1,1,1,-1,1 # not in OEIS: 1,-1,-1,-1,1,1,1,-1,1,-1,1,1,1,-1,-1,-1,1,-1 \\ negated # not in OEIS: 0,1,1,1,0,0,0,1,0,1,0,0,0,1,1,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0 \\ ones # not in OEIS: 1,0,0,0,1,1,1,0,1,0,1,1,1,0,0,0,1,0 \\ zeros # vector(25,n, (-1)^valuation(n,3)) # not in OEIS: 1,1,-1,1,1,-1,1,1,1,1,1,-1,1,1,-1,1,1,1,1,1,-1,1,1,-1,1,1,-1,1 # vector(100,n, valuation(n,3)%2) # A182581 num ternary low 0s mod 2 =pod 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 base-9 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 LE-ER when N odd since blocks are mirrored alternately. turn(9N) = turn(N) turn(9N+r) = turn(r)*(-1)^N for 1 <= r <= 8 =cut # GP-Test vector(900,n, turn(9*n)) == \ # GP-Test vector(900,n, turn(n)) # GP-Test matrix(90,8,n,r, turn(9*n+r)) == \ # GP-Test matrix(90,8,n,r, turn(r)*(-1)^n) =pod 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 LE-ER when N above is odd. turn(3N) = - turn(N) turn(3N+r) = turn(r)*(-1)^N for r = 1 or 2 =cut # GP-Test vector(900,n, turn(3*n)) == \ # GP-Test vector(900,n, - turn(n)) # GP-Test matrix(900,2,n,r, turn(3*n+r)) == \ # GP-Test matrix(900,2,n,r, turn(r)*(-1)^n) # GP-Test vector(900,n, turn(3*n)) == \ # GP-Test vector(900,n, -turn(n)) # GP-Test vector(900,n, turn(3*n+1)) == \ # GP-Test vector(900,n, -(-1)^n) # GP-Test vector(900,n, turn(3*n+2)) == \ # GP-Test vector(900,n, (-1)^n) =pod Similarly in any odd radix. =head1 SEE ALSO L, L, L =head1 HOME PAGE L =head1 LICENSE Copyright 2019, 2020 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 . =cut