/ 
Math-PlanePath-15
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::Corner

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 along y=0 is the perfect squares, because each applied row/column stripe makes a one-bigger square,

                        10 11 12 13
           5  6  7       5  6  7 14
2  3       2  3  8       2  3  8 15
1  4       1  4  9       1  4  9 16

The diagonal 2,6,12,20,etc upwards from x=0,y=1 is the pronic numbers s*(s+1), half way between those squares.

Each row/column stripe is 2 longer than the previous, similar to the Pyramid and SacksSpiral paths. The Corner and the PyramidSides are the same, just the PyramidSides stretched out to two quadrants instead of one for the Corner.

FORMULAS

For the rect_to_n_range calculation, within each row increasing X is increasing N so the smallest N is in the leftmost column and the biggest in the rightmost. Going up a column N values decrease until reaching the X=Y diagonal, and then increase. So if the bottom left corner is below the diagonal, ie. Y<X, then Y=X or the highest Y in the rectangle is the smallest N. Or if the top right corner is Y<X below the diagonal then the bottom right corner, ie. minimum Y, is the biggest N.

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 at y=0, then the start of that row N=1,2,5,10,17,etc 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

RemStart = 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 = RemStart - (d+0.5)
    = N - (d*(d+1) + 1)

And the x,y coordinates thus

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 x,y is on the horizontal part with d=y and in that case 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

Or otherwise if y<x then x,y is on the vertical and d=x. In that case 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, 2011 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/>.