Kevin Ryde > Math-PlanePath > Math::PlanePath::Diagonals

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NAME ^

Math::PlanePath::Diagonals -- points in diagonal stripes

SYNOPSIS ^

 use Math::PlanePath::Diagonals;
 my $path = Math::PlanePath::Diagonals->new;
 my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION ^

This path follows successive diagonals going from the Y axis down to the X axis.

      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

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.

Direction

Option direction => 'up' reverses the order within each diagonal to count upward from the X axis.

    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

This is merely a transpose changing X,Y to Y,X, but it's the same as in DiagonalsOctant and can be handy to control the direction when combining Diagonals with some other path or calculation.

N Start

The default is to number points starting N=1 as shown above. An optional n_start can give a different start, in the same diagonals sequence. For example to start at 0,

    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

N=0,1,3,6,10,etc on the Y axis of "down" or the X axis of "up" is the triangular numbers Y*(Y+1)/2.

X,Y Start

Options x_start => $x and y_start => $y give a starting position for the diagonals. For example to start at X=1,Y=1

      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

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.

FUNCTIONS ^

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

$path = Math::PlanePath::Diagonals->new ()
$path = Math::PlanePath::Diagonals->new (direction => $str, n_start => $n, x_start => $x, y_start => $y)

Create and return a new path object. The direction option (a string) can be

    direction => "down"       the default
    direction => "up"         number upwards from the X axis
($x,$y) = $path->n_to_xy ($n)

Return the X,Y coordinates of point number $n on the path.

For $n < 0.5 the return is an empty list, it being considered the path begins at 1.

$n = $path->xy_to_n ($x,$y)

Return the point number for coordinates $x,$y. $x and $y are each rounded to the nearest integer, which has the effect of treating each point $n as a square of side 1, so the quadrant x>=-0.5, y>=-0.5 is entirely covered.

($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)

The returned range is exact, meaning $n_lo and $n_hi are the smallest and biggest in the rectangle.

FORMULAS ^

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).

    d=2
        \
    d=1  \
        \ \
    d=0  \ \
        \ \ \

N is then given by

    d = X+Y
    N = d*(d+1)/2 + X + Nstart

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.

d can be expanded out to the following quite symmetric form. This almost suggests something parabolic but is still the straight line diagonals.

        X^2 + 3X + 2XY + Y + Y^2
    N = ------------------------ + Nstart
                   2

N to X,Y

The above formula N=d*(d+1)/2 can be solved for d as

    d = floor( (sqrt(8*N+1) - 1)/2 )
    # with n_start=0

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,

    X = N - d*(d-1)/2
    Y = d - X

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.

    d = floor( (sqrt(8*N+5) - 1)/2 )
    # N>=-0.5

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.

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.

    |            \     \ N max
    |       \ ----------+
    |        |     \    |\
    |        |\     \   |
    |       \| \     \  |
    |        +----------
    |  N min  \  \     \
    +-------------------------

OEIS ^

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

http://oeis.org/A002262 (etc)

    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

SEE ALSO ^

Math::PlanePath, Math::PlanePath::DiagonalsAlternating, Math::PlanePath::DiagonalsOctant, Math::PlanePath::Corner, Math::PlanePath::Rows, Math::PlanePath::Columns

HOME PAGE ^

http://user42.tuxfamily.org/math-planepath/index.html

LICENSE ^

Copyright 2010, 2011, 2012, 2013, 2014 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 <http://www.gnu.org/licenses/>.

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