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Always turn off hyphenation; it makes .\" way too many mistakes in technical documents. .if n .ad l .nh .SH "NAME" Math::PlanePath::CornerReplicate \-\- replicating U parts .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::CornerReplicate; \& my $path = Math::PlanePath::CornerReplicate\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This path is a self-similar replicating corner fill with 2x2 blocks. It's sometimes called a \*(L"U order\*(R" since the base N=0 to N=3 is like a \*(L"U\*(R" (sideways). .IX Xref "U Order" .PP .Vb 10 \& 7 | 63\-\-62 59\-\-58 47\-\-46 43\-\-42 \& | | | | | \& 6 | 60\-\-61 56\-\-57 44\-\-45 40\-\-41 \& | | | \& 5 | 51\-\-50 55\-\-54 35\-\-34 39\-\-38 \& | | | | | \& 4 | 48\-\-49 52\-\-53 32\-\-33 36\-\-37 \& | | \& 3 | 15\-\-14 11\-\-10 31\-\-30 27\-\-26 \& | | | | | \& 2 | 12\-\-13 8\-\- 9 28\-\-29 24\-\-25 \& | | | \& 1 | 3\-\- 2 7\-\- 6 19\-\-18 23\-\-22 \& | | | | | \& Y=0 | 0\-\- 1 4\-\- 5 16\-\-17 20\-\-21 \& +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& X=0 1 2 3 4 5 6 7 .Ve .PP The pattern is the initial N=0 to N=3 section, .PP .Vb 9 \& +\-\-\-\-\-\-\-+\-\-\-\-\-\-\-+ \& | | | \& | 3 | 2 | \& | | | \& +\-\-\-\-\-\-\-+\-\-\-\-\-\-\-+ \& | | | \& | 0 | 1 | \& | | | \& +\-\-\-\-\-\-\-+\-\-\-\-\-\-\-+ .Ve .PP It repeats as 2x2 blocks arranged in the same pattern, then 4x4 blocks, etc. There's no rotations or reflections within sub-parts. .PP X axis N=0,1,4,5,16,17,etc is all the integers which use only digits 0 and 1 in base 4. For example N=17 is 101 in base 4. .PP Y axis N=0,3,12,15,48,etc is all the integers which use only digits 0 and 3 in base 4. For example N=51 is 303 in base 4. .PP The X=Y diagonal N=0,2,8,10,32,34,etc is all the integers which use only digits 0 and 2 in base 4. .PP The X axis is the same as the \f(CW\*(C`ZOrderCurve\*(C'\fR. The Y axis here is the X=Y diagonal of the \f(CW\*(C`ZOrderCurve\*(C'\fR, and conversely the X=Y diagonal here is the Y axis of the \f(CW\*(C`ZOrderCurve\*(C'\fR. .PP The N value at a given X,Y is converted to or from the \f(CW\*(C`ZOrderCurve\*(C'\fR by transforming base\-4 digit values 2<\->3. This can be done by a bitwise \*(L"X xor Y\*(R". When Y has a 1\-bit the xor swaps 2<\->3 in N. .PP .Vb 2 \& ZOrder X = CRep X xor CRep Y \& ZOrder Y = CRep Y \& \& CRep X = ZOrder X xor ZOrder Y \& CRep Y = ZOrder Y .Ve .SS "Level Ranges" .IX Subsection "Level Ranges" A given replication extends to .PP .Vb 3 \& Nlevel = 4^level \- 1 \& 0 <= X < 2^level \& 0 <= Y < 2^level .Ve .SS "Hamming Distance" .IX Subsection "Hamming Distance" The Hamming distance between two integers X and Y is the number of bit positions where the two values differ when written in binary. In this corner replicate each bit-pair of N becomes a bit of X and a bit of Y, .PP .Vb 6 \& N X Y \& \-\-\-\-\-\- \-\-\- \-\-\- \& 0 = 00 0 0 \& 1 = 01 1 0 <\- difference 1 bit \& 2 = 10 1 1 \& 3 = 11 0 1 <\- difference 1 bit .Ve .PP So the Hamming distance is the number of base4 bit-pairs of N which are 01 or 11. Counting bit positions from 0 for the least significant bit then this is the 1\-bits in even positions, .PP .Vb 1 \& HammingDist(X,Y) = count 1\-bits at even bit positions in N .Ve .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::CornerReplicate\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::CornerReplicate\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::CornerReplicate->new ()" Create and return a new path object. .ie n .IP """($x,$y) = $path\->n_to_xy ($n)""" 4 .el .IP "\f(CW($x,$y) = $path\->n_to_xy ($n)\fR" 4 .IX Item "($x,$y) = $path->n_to_xy ($n)" Return the X,Y coordinates of point number \f(CW$n\fR on the path. Points begin at 0 and if \f(CW\*(C`$n < 0\*(C'\fR then the return is an empty list. .ie n .IP """($n_lo, $n_hi) = $path\->rect_to_n_range ($x1,$y1, $x2,$y2)""" 4 .el .IP "\f(CW($n_lo, $n_hi) = $path\->rect_to_n_range ($x1,$y1, $x2,$y2)\fR" 4 .IX Item "($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)" The returned range is exact, meaning \f(CW$n_lo\fR and \f(CW$n_hi\fR are the smallest and biggest in the rectangle. .SS "Level Methods" .IX Subsection "Level Methods" .ie n .IP """($n_lo, $n_hi) = $path\->level_to_n_range($level)""" 4 .el .IP "\f(CW($n_lo, $n_hi) = $path\->level_to_n_range($level)\fR" 4 .IX Item "($n_lo, $n_hi) = $path->level_to_n_range($level)" Return \f(CW\*(C`(0, 4**$level \- 1)\*(C'\fR. .SH "FORMULAS" .IX Header "FORMULAS" .SS "N to dX,dY" .IX Subsection "N to dX,dY" The change dX,dY is given by N in base 4 count trailing 3s and the digit above those trailing 3s. .PP .Vb 2 \& N = ...[d]333...333 base 4 \& \e\-\-exp\-\-/ .Ve .PP When N to N+1 crosses between 4^k blocks it goes as follows. Within a block the pattern is the same, since there's no rotations or transposes etc. .PP .Vb 4 \& N, N+1 X Y dX dY dSum dDiffXY \& \-\-\-\-\-\-\-\- \-\-\-\-\- \-\-\-\-\-\-\- \-\-\-\-\- \-\-\-\-\-\-\-\- \-\-\-\-\-\- \-\-\-\-\-\-\- \& 033..33 0 2^k\-1 2^k \-(2^k\-1) +1 2*2^k\-1 \& 100..00 2^k 0 \& \& 133..33 2^k 2^k\-1 0 +1 +1 \-1 \& 200..00 2^k 2^k \& \& 133..33 2^k 2*2^k\-1 \-2^k 1\-2^k \-(2^k\-1) \-1 \& 200..00 0 2^k \& \& 133..33 0 2*2^k\-1 2*2^k \-(2*2^k\-1) +1 4*2^k\-1 \& 200..00 2*2^k 0 .Ve .PP It can be noted dSum=dX+dY the change in X+Y is at most +1, taking values 1, \&\-1, \-3, \-7, \-15, etc. The crossing from block 2 to 3 drops back, such as at N=47=\*(L"233\*(R" to N=48=\*(L"300\*(R". Everywhere else it advances by +1 anti-diagonal. .PP The difference dDiffXY=dX\-dY the change in X\-Y decreases at most \-1, taking similar values \-1, 1, 3, 7, 15, etc but in a different order to dSum. .SH "OEIS" .IX Header "OEIS" This path is in Sloane's Online Encyclopedia of Integer Sequences as .Sp .RS 4 (etc) .RE .PP .Vb 1 \& A059906 Y coordinate \& \& A059905 X xor Y, being ZOrderCurve X \& A139351 HammingDist(X,Y), count 1\-bits at even positions in N \& \& A000695 N on X axis, base 4 digits 0,1 only \& A001196 N on Y axis, base 4 digits 0,3 only \& A062880 N on diagonal, base 4 digits 0,2 only \& A163241 permutation base\-4 flip 2<\->3, \& converts N to ZOrderCurve N, and back \& \& A048647 permutation N at transpose Y,X \& base4 digits 1<\->3 .Ve .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::LTiling, Math::PlanePath::SquareReplicate, Math::PlanePath::GosperReplicate, Math::PlanePath::ZOrderCurve, Math::PlanePath::GrayCode .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 2011, 2012, 2013, 2014 Kevin Ryde .PP This file is part of Math-PlanePath. .PP Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the \s-1GNU\s0 General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version. .PP Math-PlanePath is distributed in the hope that it will be useful, but \&\s-1WITHOUT ANY WARRANTY\s0; without even the implied warranty of \s-1MERCHANTABILITY\s0 or \s-1FITNESS FOR A PARTICULAR PURPOSE. \s0 See the \s-1GNU\s0 General Public License for more details. .PP You should have received a copy of the \s-1GNU\s0 General Public License along with Math-PlanePath. If not, see .