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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::Diagonals \-\- points in diagonal stripes .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::Diagonals; \& my $path = Math::PlanePath::Diagonals\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This path follows successive diagonals going from the Y axis down to the X axis. .PP .Vb 9 \& 6 | 22 \& 5 | 16 23 \& 4 | 11 17 24 \& 3 | 7 12 18 ... \& 2 | 4 8 13 19 \& 1 | 2 5 9 14 20 \& Y=0 | 1 3 6 10 15 21 \& +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& X=0 1 2 3 4 5 .Ve .PP N=1,3,6,10,etc on the X axis is the triangular numbers. N=1,2,4,7,11,etc on the Y axis is the triangular plus 1, the next point visited after the X axis. .IX Xref "Triangular numbers" .SS "Direction" .IX Subsection "Direction" Option \f(CW\*(C`direction => \*(Aqup\*(Aq\*(C'\fR reverses the order within each diagonal to count upward from the X axis. .PP .Vb 1 \& direction => "up" \& \& 5 | 21 \& 4 | 15 20 \& 3 | 10 14 19 ... \& 2 | 6 9 13 18 24 \& 1 | 3 5 8 12 17 23 \& Y=0 | 1 2 4 7 11 16 22 \& +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& X=0 1 2 3 4 5 6 .Ve .PP This is merely a transpose changing X,Y to Y,X, but it's the same as in \&\f(CW\*(C`DiagonalsOctant\*(C'\fR and can be handy to control the direction when combining \&\f(CW\*(C`Diagonals\*(C'\fR with some other path or calculation. .SS "N Start" .IX Subsection "N Start" The default is to number points starting N=1 as shown above. An optional \&\f(CW\*(C`n_start\*(C'\fR can give a different start, in the same diagonals sequence. For example to start at 0, .PP .Vb 2 \& n_start => 0, n_start=>0 \& direction=>"down" direction=>"up" \& \& 4 | 10 | 14 \& 3 | 6 11 | 9 13 \& 2 | 3 7 12 | 5 8 12 \& 1 | 1 4 8 13 | 2 4 7 11 \& Y=0 | 0 2 5 9 14 | 0 1 3 6 10 \& +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& X=0 1 2 3 4 X=0 1 2 3 4 .Ve .PP N=0,1,3,6,10,etc on the Y axis of \*(L"down\*(R" or the X axis of \*(L"up\*(R" is the triangular numbers Y*(Y+1)/2. .IX Xref "Triangular numbers" .SS "X,Y Start" .IX Subsection "X,Y Start" Options \f(CW\*(C`x_start => $x\*(C'\fR and \f(CW\*(C`y_start => $y\*(C'\fR give a starting position for the diagonals. For example to start at X=1,Y=1 .PP .Vb 10 \& 7 | 22 x_start => 1, \& 6 | 16 23 y_start => 1 \& 5 | 11 17 24 \& 4 | 7 12 18 ... \& 3 | 4 8 13 19 \& 2 | 2 5 9 14 20 \& 1 | 1 3 6 10 15 21 \& Y=0 | \& +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& X=0 1 2 3 4 5 .Ve .PP The effect is merely to add a fixed offset to all X,Y values taken and returned, but it can be handy to have the path do that to step through non-negatives or similar. .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::Diagonals\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::Diagonals\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::Diagonals->new ()" .PD 0 .ie n .IP """$path = Math::PlanePath::Diagonals\->new (direction => $str, n_start => $n, x_start => $x, y_start => $y)""" 4 .el .IP "\f(CW$path = Math::PlanePath::Diagonals\->new (direction => $str, n_start => $n, x_start => $x, y_start => $y)\fR" 4 .IX Item "$path = Math::PlanePath::Diagonals->new (direction => $str, n_start => $n, x_start => $x, y_start => $y)" .PD Create and return a new path object. The \f(CW\*(C`direction\*(C'\fR option (a string) can be .Sp .Vb 2 \& direction => "down" the default \& direction => "up" number upwards from the X axis .Ve .ie n .IP """($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$n\fR on the path. .Sp For \f(CW\*(C`$n < 0.5\*(C'\fR the return is an empty list, it being considered the path begins at 1. .ie n .IP """$n = $path\->xy_to_n ($x,$y)""" 4 .el .IP "\f(CW$n = $path\->xy_to_n ($x,$y)\fR" 4 .IX Item "$n = $path->xy_to_n ($x,$y)" Return the point number for coordinates \f(CW\*(C`$x,$y\*(C'\fR. \f(CW$x\fR and \f(CW$y\fR are each rounded to the nearest integer, which has the effect of treating each point \f(CW$n\fR as a square of side 1, so the quadrant x>=\-0.5, y>=\-0.5 is entirely covered. .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. .SH "FORMULAS" .IX Header "FORMULAS" .SS "X,Y to N" .IX Subsection "X,Y to N" The sum d=X+Y numbers each diagonal from d=0 upwards, corresponding to the Y coordinate where the diagonal starts (or X if direction=up). .PP .Vb 6 \& d=2 \& \e \& d=1 \e \& \e \e \& d=0 \e \e \& \e \e \e .Ve .PP N is then given by .PP .Vb 2 \& d = X+Y \& N = d*(d+1)/2 + X + Nstart .Ve .PP The d*(d+1)/2 shows how the triangular numbers fall on the Y axis when X=0 and Nstart=0. For the default Nstart=1 it's 1 more than the triangulars, as noted above. .PP d can be expanded out to the following quite symmetric form. This almost suggests something parabolic but is still the straight line diagonals. .PP .Vb 3 \& X^2 + 3X + 2XY + Y + Y^2 \& N = \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- + Nstart \& 2 .Ve .SS "N to X,Y" .IX Subsection "N to X,Y" The above formula N=d*(d+1)/2 can be solved for d as .PP .Vb 2 \& d = floor( (sqrt(8*N+1) \- 1)/2 ) \& # with n_start=0 .Ve .PP For example N=12 is d=floor((sqrt(8*12+1)\-1)/2)=4 as that N falls in the fifth diagonal. Then the offset from the Y axis NY=d*(d\-1)/2 is the X position, .PP .Vb 2 \& X = N \- d*(d\-1)/2 \& Y = d \- X .Ve .PP In the code fractional N is handled by imagining each diagonal beginning 0.5 back from the Y axis. That's handled by adding 0.5 into the sqrt, which is +4 onto the 8*N. .PP .Vb 2 \& d = floor( (sqrt(8*N+5) \- 1)/2 ) \& # N>=\-0.5 .Ve .PP The X and Y formulas are unchanged, since N=d*(d\-1)/2 is still the Y axis. But each diagonal d begins up to 0.5 before that and therefor X extends back to \-0.5. .SS "Rectangle to N Range" .IX Subsection "Rectangle to N Range" Within each row increasing X is increasing N, and in each column increasing Y is increasing N. So in a rectangle the lower left corner is the minimum N and the upper right is the maximum N. .PP .Vb 8 \& | \e \e N max \& | \e \-\-\-\-\-\-\-\-\-\-+ \& | | \e |\e \& | |\e \e | \& | \e| \e \e | \& | +\-\-\-\-\-\-\-\-\-\- \& | N min \e \e \e \& +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- .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 6 \& direction=down (the default) \& A002262 X coordinate, runs 0 to k \& A025581 Y coordinate, runs k to 0 \& A003056 X+Y coordinate sum, k repeated k+1 times \& A114327 Y\-X coordinate diff \& A101080 HammingDist(X,Y) \& \& A127949 dY, change in Y coordinate \& \& A000124 N on Y axis, triangular numbers + 1 \& A001844 N on X=Y diagonal \& \& A185787 total N in row to X=Y diagonal \& A185788 total N in row to X=Y\-1 \& A100182 total N in column to Y=X diagonal \& A101165 total N in column to Y=X\-1 \& A185506 total N in rectangle 0,0 to X,Y \& \& direction=down, n_start=0 \& A023531 dSum = dX+dY, being 1 at N=triangular+1 (and 0) \& A000096 N on X axis, X*(X+3)/2 \& A000217 N on Y axis, the triangular numbers \& A129184 turn 1=left,0=right \& A103451 turn 1=left or right,0=straight, but extra initial 1 \& A103452 turn 1=left,0=straight,\-1=right, but extra initial 1 \& direction=up, n_start=0 \& A129184 turn 0=left,1=right \& \& direction=up, n_start=\-1 \& A023531 turn 1=left,0=right \& direction=down, n_start=\-1 \& A023531 turn 0=left,1=right \& \& in direction=up the X,Y coordinate forms are the same but swap X,Y \& \& either direction, n_start=1 \& A038722 permutation N at transpose Y,X \& which is direction=down <\-> direction=up \& \& n_start=1, x_start=1, y_start=1, either direction \& A003991 X*Y coordinate product \& A003989 GCD(X,Y) greatest common divisor starting (1,1) \& A003983 min(X,Y) \& A051125 max(X,Y) \& n_start=1, x_start=1, y_start=1, direction=down \& A057046 X for N=2^k \& A057047 Y for N=2^k \& \& n_start=0 (either direction) \& A049581 abs(X\-Y) coordinate diff \& A004197 min(X,Y) \& A003984 max(X,Y) \& A004247 X*Y coordinate product \& A048147 X^2+Y^2 \& A109004 GCD(X,Y) greatest common divisor starting (0,0) \& A004198 X bit\-and Y \& A003986 X bit\-or Y \& A003987 X bit\-xor Y \& A156319 turn 0=straight,1=left,2=right \& \& A061579 permutation N at transpose Y,X \& which is direction=down <\-> direction=up .Ve .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::DiagonalsAlternating, Math::PlanePath::DiagonalsOctant, Math::PlanePath::Corner, Math::PlanePath::Rows, Math::PlanePath::Columns .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 2010, 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 .