# Copyright 2010, 2011, 2012, 2013 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/>.


# cf Matthew Szudzik ElegantPairing.pdf going inwardly.
#
#   5 | 25--...
#     |
#   4 | 16--17--18--19  24
#     |                  |
#   3 |  9--10--11  15  23
#     |              |   |
#   2 |  4-- 5   8  14  22
#     |          |   |   |
#   1 |  1   4   7  13  21
#     |      |   |   |   |
# Y=0 |  0   2   6  12  20
#     +---------------------
#      X=0   1   2   3   4
#
# cf A185728 where diagonal is last in each gnomon
#    A185725 gnomon sides alternately starting from ends
#    A185726 gnomon sides alternately starting from diagonal
#
# corner going alternately up and down
#    A081344,A194280 by diagonals
#    A081345 X axis, A081346 Y axis
#
# corner alternately up and down, starting with 3-wide
#   A080335  N on diagonal
#   A081347  N on axis
#   A081348  N on axis
#
# cf A004120 ??
#

package Math::PlanePath::Corner;
use 5.004;
use strict;

use vars '$VERSION', '@ISA';
$VERSION = 100;
use Math::PlanePath;
@ISA = ('Math::PlanePath');

use Math::PlanePath::Base::Generic
  'round_nearest';

# uncomment this to run the ### lines
#use Devel::Comments;


use constant class_x_negative => 0;
use constant class_y_negative => 0;
use constant n_frac_discontinuity => .5;
*xy_is_visited = \&Math::PlanePath::Base::Generic::xy_is_visited_quad1;

use Math::PlanePath::SquareSpiral;
*parameter_info_array = \&Math::PlanePath::SquareSpiral::parameter_info_array;

use constant dx_maximum => 1;
use constant dy_minimum => -1;
use constant dir_maximum_dxdy => (0,-1); # South
# use constant dir4_maximum  => 3; # South
# use constant dir_maximum_360  => 270;    # South


#------------------------------------------------------------------------------

# same as PyramidSides, just 45 degress around

sub new {
  my $self = shift->SUPER::new (@_);
  if (! defined $self->{'n_start'}) {
    $self->{'n_start'} = $self->default_n_start;
  }
  $self->{'wider'} ||= 0;  # default
  return $self;
}

sub n_to_xy {
  my ($self, $n) = @_;
  ### Corner n_to_xy: $n

  # adjust to N=1 at origin X=0,Y=0
  $n = $n - $self->{'n_start'} + 1;

  # $n<0.5 no good for Math::BigInt circa Perl 5.12, compare in integers
  if (2*$n < 1) {
    return;
  }

  my $wider = $self->{'wider'};

  # wider==0
  #   vertical at X=0 has N=1, 2, 5, 10, 17, 26
  #   but start 0.5 back so at X=-0.5 have N=0.5, 1.5, 4.5, 9.5, 16.5, 25.5
  #   N = (Y^2 + 1/2)
  #   Y = floor sqrt(N - 1/2)
  #     = floor sqrt(4*N - 2)/2   staying in integers for the sqrt()
  #
  # wider==1
  #   0.5 back so at X=-0.5 have N=0.5, 2.5, 6.5, 12.5
  #   N = (Y^2 + Y + 1/2)
  #   Y = floor -1/2 + sqrt(N - 1/4)
  #     = floor (-1 + sqrt(4*N - 1))/2
  #
  # wider==2
  #   0.5 back so at X=-0.5 have N=0.5, 3.5, 8.5, 15.5
  #   N = (Y^2 + 2 Y + 1/2)
  #   Y = floor -1 + sqrt(N + 1/2)
  #     = floor (-2 + sqrt(4*N + 2))/2
  #
  # wider==3
  #   0.5 back so at X=-0.5 have N=0.5, 4.5, 10.5, 18.5
  #   N = (Y^2 + 3 Y + 1/2)
  #   Y = floor -3/2 + sqrt(N + 7/4)
  #     = floor (-3 + sqrt(4*N + 7))/2
  #
  # 0,1,4,9
  # my $y = int((sqrt(4*$n + -1) - $wider) / 2);
  # ### y frac: (sqrt(4*$n + -1) - $wider) / 2

  my $y = int((sqrt(int(4*$n) + $wider*$wider - 2) - $wider) / 2);
  ### y frac: (sqrt(int(4*$n) + $wider*$wider - 2) - $wider) / 2
  ### $y

  # diagonal at X=Y has N=1, 3, 7, 13, 21
  # N = ((Y + 1)*Y + (Y+1)*wider + 1)
  #   = ((Y + 1 + wider)*Y + wider + 1)
  # so subtract that leaving N negative on the horizontal part, or positive
  # for the downward vertical part
  #
  $n -= $y*($y+1+$wider) + $wider + 1;
  ### corner n: $y*($y+1+$wider) + $wider + 1
  ### rem: $n
  ### assert: $n!=$n || $n >= -($y+$wider+0.5)
  # ### assert: $n <= ($y+0.5)

  if ($n < 0) {
    # top horizontal
    return ($n + $y+$wider,
            $y);
  } else {
    # right vertical
    return ($y+$wider,
            -$n + $y);
  }
}

sub xy_to_n {
  my ($self, $x, $y) = @_;
  ### Corner xy_to_n(): "$x,$y"

  $x = round_nearest ($x);
  $y = round_nearest ($y);
  if ($x < 0 || $y < 0) {
    return undef;
  }

  my $wider = $self->{'wider'};
  my $xw = $x - $wider;
  if ($y >= $xw) {
    ### top edge, N left is: $y*$y + $wider*$y + 1
    return ($y*$y + $wider*$y      # Y axis N value
            + $x                   # plus X offset across
            + $self->{'n_start'});
  } else {
    ### right vertical, N diag is: $xw*$xw + $xw*$wider
    ### $xw
    # Ndiag = Nleft + Y+w
    # N = xw*xw + w*xw + 1 + xw+w + (xw - y)
    #   = xw*xw + w*xw + 1 + xw+w + xw - y
    #   = xw*xw + xw*(w+2) + 1 + w - y
    #   = xw*(xw + w+2) + w+1 - y
    return ($xw*($xw+$wider+2) + $wider
            - $y
            +  $self->{'n_start'});
  }
}

# exact
sub rect_to_n_range {
  my ($self, $x1,$y1, $x2,$y2) = @_;
  ### Corner rect_to_n_range(): "$x1,$y1, $x2,$y2"

  $x1 = round_nearest ($x1);
  $y1 = round_nearest ($y1);
  $x2 = round_nearest ($x2);
  $y2 = round_nearest ($y2);
  if ($x1 > $x2) { ($x1,$x2) = ($x2,$x1); }
  if ($y1 > $y2) { ($y1,$y2) = ($y2,$y1); }

  if ($y2 < 0 || $x2 < 0) {
    return (1, 0); # rect all negative, no N
  }

  if ($x1 < 0) { $x1 *= 0; }   # "*=" to preserve bigint x1 or y1
  if ($y1 < 0) { $y1 *= 0; }

  my $wider = $self->{'wider'};
  my $ylo = $y1;
  my $xw = $x1 - $wider;

  if ($y1 <= $xw) {
    # left column is partly or wholly below X=Y diagonal
    #
    # make x1,y1 the min pos
    $y1 = ($y2 < $xw

           # wholly below diag, min "*" is at top y2 of the x1 column
           #
           # |        /
           # |       /
           # |      / *------+  y2
           # |     /  |      |
           # |    /   +------+  y1
           # |   /   x1     x2
           # +------------------
           #    ^.....^
           #    wider  xw
           #
           ? $y2

           # only partly below diag, min "*" is the X=Y+wider diagonal at x1
           #
           #            /
           # |      +------+  y2
           # |      | /    |
           # |      |/     |
           # |      *      |
           # |     /|      |
           # |    / +------+  y1
           # |   /  x1     x2
           # +------------------
           #    ^...^xw
           #    wider
           #
           : $xw);
  }

  if ($y2 <= $x2 - $wider) {
    # right column entirely at or below X=Y+wider diagonal so max is at the
    # ylo bottom end of the column
    #
    # |          /
    # |       --/---+  y2
    # |      | /    |
    # |      |/     |
    # |      /      |
    # |     /|      |
    # |    / +------+  ylo
    # |   /         x2
    # +------------------
    #    ^
    #    wider
    #
    $y2 = $ylo;   # x2,y2 now the max pos
  }

  ### min xy: "$x1,$y1"
  ### max xy: "$x2,$y2"
  return ($self->xy_to_n ($x1,$y1),
          $self->xy_to_n ($x2,$y2));
}

1;
__END__

=for stopwords pronic SacksSpiral PyramidSides PyramidRows PlanePath Ryde Math-PlanePath ie OEIS gnomon

=head1 NAME

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

=head1 SYNOPSIS

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

=head1 DESCRIPTION

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

=cut

# math-image --path=Corner --output=numbers_dash --all --size=30x14

=pod

      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

X<Gnomon>The horizontal 1,4,9,16,etc along Y=0 is the perfect squares.  This
is since each further row/column "gnomon" added to a square 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

     2x2        3x3           4x4

N=2,6,12,20,etc on the diagonal X=Y-1 up from X=0,Y=1 is the
X<Pronic numbers>pronic numbers k*(k+1) which are half way between the
squares.

Each gnomon is 2 longer than the previous.  This is similar to the
PyramidRows, PyramidSides 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 here.

=head2 Wider

An optional C<wider =E<gt> $integer> makes the path wider horizontally,
becoming a rectangle.  For example

    $path = Math::PlanePath::Corner->new (wider => 3);

gives

=cut

# math-image --path=Corner,wider=3 --all --output=numbers_dash --size=38x12

=pod

     4  |  29--30--31--...
        |
     3  |  19--20--21--22--23--24--25
        |                           |
     2  |  11--12--13--14--15--16  26
        |                       |   |
     1  |   5---6---7---8---9  17  27
        |                   |   |   |
    Y=0 |   1---2---3---4  10  18  28
        |
         -----------------------------
            ^
           X=0  1   2   3   4   5   6

Each gnomon has the horizontal part C<wider> many steps longer.  Each
gnomon is still 2 longer than the previous, since this widening is a
constant amount in each.

=head2 N Start

The default is to number points starting N=1 as shown above.  An optional
C<n_start> can give a different start with the same shape etc.  For example
to start at 0,

=cut

# math-image --path=Corner,n_start=0 --all --output=numbers --size=35x5

=pod

    n_start => 0

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

In this numbering the squares are on the Y axis and the pronic numbers are
on the X=Y leading diagonal.

=head1 FUNCTIONS

See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.

=over 4

=item C<$path = Math::PlanePath::Corner-E<gt>new ()>

=item C<$path = Math::PlanePath::Corner-E<gt>new (wider =E<gt> $w, n_start =E<gt> $n)>

Create and return a new path object.

=item C<($x,$y) = $path-E<gt>n_to_xy ($n)>

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

For C<$n < 0.5> the return is an empty list, it being considered there are
no points before 1 in the corner.

=item C<$n = $path-E<gt>xy_to_n ($x,$y)>

Return the point number for coordinates C<$x,$y>.  C<$x> and C<$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.

=item C<($n_lo, $n_hi) = $path-E<gt>rect_to_n_range ($x1,$y1, $x2,$y2)>

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

=back

=head1 FORMULAS

=head2 N to X,Y

Counting d=0 for the first L-shaped gnomon at Y=0, then the start of the
gnomon is

    StartN(d) = d^2 + 1 = 1,2,5,10,17,etc

The current C<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 gnomon d is

    StartNfrac(d) = d^2 + 0.5

Inverting gives the gnomon d number for an N,

    d = floor(sqrt(N - 0.5))

Subtracting the gnomon start gives an offset into that gnomon

    OffStart = N - StartNfrac(d)

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

    Off = OffStart - (d+0.5)
        = N - (d*(d+1) + 1)

Then the X,Y coordinates are

    if (Off < 0)  then  X=d+Off, Y=d
    if (Off >= 0) then  X=d,     Y=d-Off

=head2 X,Y to N

For a given X,Y the bigger of X or Y determines the d gnomon.

If YE<gt>=X then X,Y is on the horizontal part.  At X=0 have N=StartN(d) per
the Start(N) formula above, and any further X is an offset from there.

    if Y >= X then
      d=Y
      N = StartN(d) + X
        = Y^2 + 1 + X

Otherwise if YE<lt>X then X,Y is on the vertical part.  At Y=0 N is the last
point on the gnomon, and one back from the start of the following gnomon,

    if Y <= X then
      d=X
      LastN(d) = StartN(d+1) - 1
               = (d+1)^2
      N = LastN(d) - Y
        = (X+1)^2 - Y

=head2 Rectangle N Range

For C<rect_to_n_range()>, in each row increasing X is increasing N so the
smallest N is in the leftmost column and the biggest N in the rightmost
column.

    |
    |  ------>  N increasing
    |
     -----------------------

Going up a column, N values are increasing away from the X=Y diagonal up or
down, and all N values above X=Y are bigger than the ones below.

    |    ^  N increasing up from X=Y diagonal
    |    |
    |    |/
    |    /
    |   /|
    |  / |  N increasing down from X=Y diagonal
    | /  v
    |/
     -----------------------

This means the biggest N is the top right corner if that corner is YE<gt>=X,
otherwise the bottom right corner.

                                           max N at top right
    |      /                          | --+     if corner Y>=X
    |     / --+                       |   | /
    |    /    |                       |   |/
    |   /     |                       |   |
    |  /  ----v                       |  /|
    | /     max N at bottom right     | --+
    |/        if corner Y<=X          |/
     ----------                        -------

For the smallest N, if the bottom left corner has YE<gt>X then it's in the
"increasing" part and that bottom left corner is the smallest N.  Otherwise
YE<lt>=X means some of the "decreasing" part is covered and the smallest N
is at Y=min(X,Ymax), ie. either the Y=X diagonal if it's in the rectangle or
the top right corner otherwise.

    |      /
    | |   /
    | |  /  min N at bottom left
    | +----  if corner Y>X
    |  /
    | /
    |/
     ----------

    |      /                           |      /
    |   | /                            |     /
    |   |/  min N at X=Y               |    /
    |   *    if diagonal crossed       |   / +-- min N at top left
    |  /|                              |  /  |    if corner Y<X
    | / +-----                         | /   |
    |/                                 |/    
     ----------                         ----------

    min N at Xmin,Ymin            if Ymin >= Xmin
             Xmin,min(Xmin,Ymax)  if Ymin <= Xmin


=head1 OEIS

This path is in Sloane's Online Encyclopedia of Integer Sequences as,

    http://oeis.org/A196199  (etc)

    wider=0, n_start=1 (the defaults)
      A213088    X+Y sum
      A196199    X-Y diff, being runs -n to +n
      A053615    abs(X-Y), distance to next pronic

      A000290    N on X axis, perfect squares starting from 1
      A002522    N on Y axis, Y^2+1
      A002061    N on X=Y diagonal, extra initial 1
      A004201    N on and below X=Y diagonal, so X>=Y

      A020703    permutation N at transpose Y,X
      A060734    permutation N by diagonals up from X axis
      A064790     inverse
      A060736    permutation N by diagonals down from Y axis
      A064788     inverse

    n_start=0
      A000196    max(X,Y), being floor(sqrt(N))

      A005563    N on X axis, n*(n+2)
      A000290    N on Y axis, perfect squares
      A002378    N on X=Y diagonal, pronic numbers

    n_start=2
      A059100    N on Y axis, Y^2+2
      A014206    N on X=Y diagonal, pronic+2

    wider=1
      A053188    abs(X-Y), dist to nearest square, extra initial 0
    wider=1, n_start=0
      A002378    N on Y axis, pronic numbers
      A005563    N on X=Y diagonal, n*(n+2)
    wider=1, n_start=2
      A014206    N on Y axis, pronic+2

    wider=2, n_start=0
      A005563    N on Y axis, (Y+1)^2-1
      A028552    N on X=Y diagonal, k*(k+3)

    wider=3, n_start=0
      A028552    N on Y axis, k*(k+3)

=head1 SEE ALSO

L<Math::PlanePath>,
L<Math::PlanePath::PyramidRows>,
L<Math::PlanePath::PyramidSides>,
L<Math::PlanePath::SacksSpiral>,
L<Math::PlanePath::Diagonals>

=head1 HOME PAGE

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

=head1 LICENSE

Copyright 2010, 2011, 2012, 2013 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/>.

=cut
