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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::HIndexing \-\- self\-similar right\-triangle traversal .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::HIndexing; \& my $path = Math::PlanePath::HIndexing\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This is an infinite integer version of H\-indexing per .IX Xref "Niedermeier, Rolf Reinhardt, Klaus Sanders, Peter" .Sp .RS 4 Rolf Niedermeier, Klaus Reinhardt and Peter Sanders, \*(L"Towards Optimal Locality In Mesh Indexings\*(R", Discrete Applied Mathematics, volume 117, March 2002, pages 211\-237. .RE .PP It traverses an eighth of the plane by self-similar right triangles. Notice the \*(L"H\*(R" shapes that arise from the backtracking, for example N=8 to N=23, and repeating above it. .PP .Vb 10 \& | | \& 15 | 63\-\-64 67\-\-68 75\-\-76 79\-\-80 111\-112 115\-116 123\-124 127 \& | | | | | | | | | | | | | | | | \& 14 | 62 65\-\-66 69 74 77\-\-78 81 110 113\-114 117 122 125\-126 \& | | | | | | | | \& 13 | 61 58\-\-57 70 73 86\-\-85 82 109 106\-105 118 121 \& | | | | | | | | | | | | | | \& 12 | 60\-\-59 56 71\-\-72 87 84\-\-83 108\-107 104 119\-120 \& | | | | \& 11 | 51\-\-52 55 40\-\-39 88 91\-\-92 99\-100 103 \& | | | | | | | | | | | | \& 10 | 50 53\-\-54 41 38 89\-\-90 93 98 101\-102 \& | | | | | | \& 9 | 49 46\-\-45 42 37 34\-\-33 94 97 \& | | | | | | | | | | \& 8 | 48\-\-47 44\-\-43 36\-\-35 32 95\-\-96 \& | | \& 7 | 15\-\-16 19\-\-20 27\-\-28 31 \& | | | | | | | | \& 6 | 14 17\-\-18 21 26 29\-\-30 \& | | | | \& 5 | 13 10\-\- 9 22 25 \& | | | | | | \& 4 | 12\-\-11 8 23\-\-24 \& | | \& 3 | 3\-\- 4 7 \& | | | | \& 2 | 2 5\-\- 6 \& | | \& 1 | 1 \& | | \& Y=0 | 0 \& +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 .Ve .PP The tiling is essentially the same as the Sierpinski curve (see Math::PlanePath::SierpinskiCurve). The following is with two points per triangle. Or equally well it could be thought of with those triangles further divided to have one point each, a little skewed. .PP .Vb 10 \& +\-\-\-\-\-\-\-\-\-+\-\-\-\-\-\-\-\-\-+\-\-\-\-\-\-\-\-+\-\-\-\-\-\-\-\-/ \& | \e | / | \e | / \& | 15 \e 16| 19 /20 |27\e 28 |31 / \& | | \e || | / | | | \e | | | / \& | 14 \e17| 18/ 21 |26 \e29 |30 / \& | \e | / | \e | / \& +\-\-\-\-\-\-\-\-\-+\-\-\-\-\-\-\-\-\-+\-\-\-\-\-\-\-\-\-/ \& | / | \e | / \& | 13 /10 | 9 \e 22 | 25 / \& | | / | | | \e | | | / \& | 12/ 11 | 8 \e23 | 24/ \& | / | \e | / \& +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-/ \& | \e | / \& | 3 \e 4 | 7 / \& | | \e | | | / \& | 2 \e 5 | 6 / \& | \e | / \& +\-\-\-\-\-\-\-\-\-\-/ \& | / \& | 1 / \& | | / \& | 0 / \& | / \& +/ .Ve .PP The correspondence to the \f(CW\*(C`SierpinskiCurve\*(C'\fR path is as follows. The 4\-point verticals like N=0 to N=3 are a Sierpinski horizontal, and the 4\-point \*(L"U\*(R" parts like N=4 to N=7 are a Sierpinski vertical. In both cases there's an X,Y transpose and bit of stretching. .PP .Vb 7 \& 3 7 \& | / \& 2 1\-\-2 5\-\-6 6 \& | <=> / \e | | <=> | \& 1 0 3 4 7 5 \& | \e \& 0 4 .Ve .SS "Level Ranges" .IX Subsection "Level Ranges" Counting the initial N=0 to N=7 section as level 1, the X,Y ranges for a given level is .PP .Vb 3 \& Nlevel = 2*4^level \- 1 \& Xmax = 2*2^level \- 2 \& Ymax = 2*2^level \- 1 .Ve .PP For example level=3 is N through to Nlevel=2*4^3\-1=127 and X,Y ranging up to Xmax=2*2^3\-2=14 and Xmax=2*2^3\-1=15. .PP On even Y rows, the N on the X=Y diagonal is found by duplicating each bit in Y except the low zero (which is unchanged). For example Y=10 decimal is 1010 binary, duplicate to binary 1100110 is N=102. .PP It would be possible to take a level as N=0 to N=4^k\-1 too, which would be a triangle against the Y axis. The 2*4^level \- 1 is per the paper above. .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::HIndexing\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::HIndexing\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::HIndexing->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. .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, 2*4**$level \- 1)\*(C'\fR. .SH "FORMULAS" .IX Header "FORMULAS" .SS "Area" .IX Subsection "Area" The area enclosed by curve in its triangular level k is .PP .Vb 2 \& A[k] = (2^k\-1)^2 \& = 0, 1, 9, 49, 225, 961, 3969, 16129, ... (A060867) .Ve .PP For example level k=2 enclosed area marked by \*(L"@\*(R" signs, .PP .Vb 10 \& 7 | *\-\-\-*\-\-\-*\-\-\-*\-\-\-*\-\-\-*\-\-\-31 \& | | | @ | | @ | | @ | \& 6 | * *\-\-\-* * * *\-\-\-* \& | | | @ | \& 5 | * *\-\-\-* * * \& | | | @ | | @ | \& 4 | *\-\-\-* * *\-\-\-* level k=2 \& | | @ @ | N=0 to N=31 \& 3 | *\-\- * * \& | | | @ | A[2] = 9 \& 2 | * *\-\- * \& | | \& 1 | * \& | | \& Y=0 | 0 \& +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& X=0 1 2 3 4 5 6 .Ve .PP The block breakdowns are .PP .Vb 10 \& +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-+ ^ \& | \e ^ | | ^ / | \& |\e \e 2 | | 3 / | = 2^k \- 1 \& | \e \e | | / | \& | 1\e \e | | / | \& | v \e \e+\-\-+/ v \& +\-\-\-\-+ \& | | \& +\-\-\-\-+ \& | ^ / \& | 0 / \& | / \& | / \& +/ \& \& <\-\-\-\-> = 2^k \- 2 .Ve .PP Parts 0 and 3 are identical. Parts 1 and 2 are mirror images of 0 and 3 respectively. Parts 0 and 1 have an area in between 1 high and 2^k\-2 wide (eg. 2^2\-2=2 wide in the k=2 above). Parts 2 and 3 have an area in between 1 wide 2^k\-1 high (eg. 2^2\-1=3 high in the k=2 above). So the total area is .PP .Vb 9 \& A[k] = 4*A[k\-1] + 2^k\-2 + 2^k\-1 starting A[0] = 0 \& = 4^0 * (2*2^k \- 3) \& + 4^1 * (2*2^(k\-1) \- 3) \& + 4^2 * (2*2^(k\-2) \- 3) \& + ... \& + 4^(k\-1) * (2*2^1 \- 3) \& + 4^k * A[0] \& = 2*2*(4^k \- 2^k)/(4\-2) \- 3*(4^k \- 1)/(4\-1) \& = (2^k \- 1)^2 .Ve .SS "Half Level Areas" .IX Subsection "Half Level Areas" Block 1 ends at the top-left corner and block 2 start there. The area before that midpoint enclosed to the Y axis can be calculated. Likewise the area after that midpoint to the top line. Both are two blocks, and with either 2^k\-2 or 2^k\-1 in between. They're therefore half the total area A[k], with the extra unit square going to the top AT[k]. .PP .Vb 2 \& AY[k] = floor(A[k]/2) \& = 0, 0, 4, 24, 112, 480, 1984, 8064, 32512, ... (A059153) \& \& AT[k] = ceil(A[k]/2) \& = 0, 1, 5, 25, 113, 481, 1985, 8065, 32513, ... (A092440) .Ve .PP .Vb 10 \& 15 \& | \& 14 \& | \& 13 10\-\- 9 \& | | @ | \& 12\-\-11 8 \& @ @ | \& 3 3\-\- 4 7 \& | | | @ | \& 2 2 5\-\- 6 \& | | \& 1 1 \& | | \& 0 0 0 \& \& AY[0] = 0 AY[1] = 0 AY[2] = 4 .Ve .PP .Vb 7 \& 1 3\-\- 4 7 15\-\-16 19\-\-20 27\-\-28 31 \& | @ | | @ | | @ | | @ | \& 5\-\- 6 17\-\-18 21 26 29\-\-30 \& | @ | \& 22 25 \& | @ | \& 23\-\-24 \& \& AT[0] = 0 AT[1] = 1 AT[2] = 5 .Ve .SH "OEIS" .IX Header "OEIS" Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include .Sp .RS 4 (etc) .RE .PP .Vb 1 \& A097110 Y at N=2^k, being successively 2^j\-1, 2^j \& \& A060867 area of level \& A059153 area of level first half \& A092440 area of level second half .Ve .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::SierpinskiCurve .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 .