NAME

Math::PlanePath::Corner -- points shaped in a corner

SYNOPSIS

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

DESCRIPTION

This path puts points in layers working outwards from the corner of the first quadrant.

    ...
      5  |  26 ................
      4  |  17  18  19  20  21 .
      3  |  10  11  12  13  22 .
      2  |   5   6   7  14  23 .
      1  |   2   3   8  15  24 .
    y=0  |   1   4   9  16  25 .
          ----------------------
           x=0,  1   2   3   4 ...

The horizontal 1,4,9,16,etc at y=0 is the perfect squares. The diagonal 2,6,12,20,etc starting x=0,y=1 is the pronic numbers s*(s+1), half way between those squares.

Each stripe across then down is 2 longer than the previous and in that respect the corner is the same as the Pyramid and SacksSpiral paths. The Corner and the PyramidSides are the same thing, just with a stretch from a single quadrant to two.

FUNCTIONS

$path = Math::PlanePath::Corner->new ()

Create and return a new path object.

($x,$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 there are no points before 1 in the corner.

$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 as a square of side 1, so the quadrant x>=-0.5 and y>=-0.5 is entirely covered.

FORMULAS

N to X,Y

Counting d=0 for the first row the N=1,2,5,10,17,etc which is the start of each row is

    StartN(d) = d^2 + 1

The current n_to_xy code extends to the left by an extra 0.5 for fractional N, so for example N=9.5 is at x=-0.5,y=3. With this the starting N for each d row is

    StartNfrac(d) = d^2 + 0.5

Inverting gives the row for an N,

    d = floor(sqrt(N - 0.5))

And subtracting that start gives an offset into the row

    RemBase = N - StartNfrac(d)

The corner point 1,3,7,13,etc where the row turns down is at d+0.5 into that remainder, and it's convenient to subtract that, giving a negative for the horizontal or positive for the vertical,

    Rem = N - StartNfrac(d) - (d+0.5)
        = N - (d*(d+1) + 1)

    if (Rem < 0)  then x=d+Rem, y=d
    if (Rem >= 0) then x=d, y=d-Rem

X,Y to N

For a given x,y the bigger of x or y determines the d row. If y>=x then it's the horizontal part with d=y. StartN(d) above is the N for x=0, and the given x can be added to that,

    N = StartN(d) + x
      = y^2 + 1 + x

If y<x then it's the vertical with d=x. The y=0 is the last point on the row and is one back from the start of the following row,

    LastN(d) = StartN(d+1) - 1
             = (d+1)^2

    N = LastN(d) - y
      = (x+1)^2 - y

SEE ALSO

Math::PlanePath, Math::PlanePath::PyramidRows, Math::PlanePath::PyramidSides, Math::PlanePath::SacksSpiral

HOME PAGE

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

LICENSE

Math-PlanePath is Copyright 2010 Kevin Ryde

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/>.

cpanm Math::PlanePath

perl -MCPAN -e shell
install Math::PlanePath