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.\" ========================================================================
.\"
.IX Title "Math::PlanePath::Corner 3pm"
.TH Math::PlanePath::Corner 3pm "2014-06-14" "perl v5.20.1" "User Contributed Perl Documentation"
.\" For nroff, turn off justification.  Always turn off hyphenation; it makes
.\" way too many mistakes in technical documents.
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.nh
.SH "NAME"
Math::PlanePath::Corner \-\- points shaped around in a corner
.SH "SYNOPSIS"
.IX Header "SYNOPSIS"
.Vb 3
\& use Math::PlanePath::Corner;
\& my $path = Math::PlanePath::Corner\->new;
\& my ($x, $y) = $path\->n_to_xy (123);
.Ve
.SH "DESCRIPTION"
.IX Header "DESCRIPTION"
This path puts points in layers working outwards from the corner of the
first quadrant.
.PP
.Vb 10
\&      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
.Ve
.PP
The horizontal 1,4,9,16,etc along Y=0 is the
perfect squares.  This is since each further row/column \*(L"gnomon\*(R" added to a
square makes a one-bigger square,
.IX Xref "Gnomon Square numbers"
.PP
.Vb 4
\&                            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
.Ve
.PP
N=2,6,12,20,etc on the diagonal X=Y\-1 up from X=0,Y=1 is the
pronic numbers k*(k+1) which are half way between the
squares.
.IX Xref "Pronic numbers"
.PP
Each gnomon is 2 longer than the previous.  This is similar to the
\&\f(CW\*(C`PyramidRows\*(C'\fR, \f(CW\*(C`PyramidSides\*(C'\fR and \f(CW\*(C`SacksSpiral\*(C'\fR paths.  The \f(CW\*(C`Corner\*(C'\fR and
the \f(CW\*(C`PyramidSides\*(C'\fR are the same but \f(CW\*(C`PyramidSides\*(C'\fR is stretched to two
quadrants instead of one for the \f(CW\*(C`Corner\*(C'\fR here.
.SS "Wider"
.IX Subsection "Wider"
An optional \f(CW\*(C`wider => $integer\*(C'\fR makes the path wider horizontally,
becoming a rectangle.  For example
.PP
.Vb 1
\&    $path = Math::PlanePath::Corner\->new (wider => 3);
.Ve
.PP
gives
.PP
.Vb 10
\&     4  |  29\-\-30\-\-31\-\-...
\&        |
\&     3  |  19\-\-20\-\-21\-\-22\-\-23\-\-24\-\-25
\&        |                           |
\&     2  |  11\-\-12\-\-13\-\-14\-\-15\-\-16  26
\&        |                       |   |
\&     1  |   5\-\-\-6\-\-\-7\-\-\-8\-\-\-9  17  27
\&        |                   |   |   |
\&    Y=0 |   1\-\-\-2\-\-\-3\-\-\-4  10  18  28
\&        |
\&         \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-
\&            ^
\&           X=0  1   2   3   4   5   6
.Ve
.PP
Each gnomon has the horizontal part \f(CW\*(C`wider\*(C'\fR many steps longer.  Each gnomon
is still 2 longer than the previous since this widening is a constant amount
in each.
.SS "N Start"
.IX Subsection "N Start"
The default is to number points starting N=1 as shown above.  An optional
\&\f(CW\*(C`n_start\*(C'\fR can give a different start with the same shape etc.  For example
to start at 0,
.PP
.Vb 1
\&    n_start => 0
\&
\&      5  |  25 ...
\&      4  |  16 17 18 19 20
\&      3  |   9 10 11 12 21
\&      2  |   4  5  6 13 22
\&      1  |   1  2  7 14 23
\&    Y=0  |   0  3  8 15 24
\&          \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-
\&           X=0   1   2   3
.Ve
.PP
In Nstart=0 the squares are on the Y axis and the pronic numbers are on the
X=Y leading diagonal.
.SH "FUNCTIONS"
.IX Header "FUNCTIONS"
See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes.
.ie n .IP """$path = Math::PlanePath::Corner\->new ()""" 4
.el .IP "\f(CW$path = Math::PlanePath::Corner\->new ()\fR" 4
.IX Item "$path = Math::PlanePath::Corner->new ()"
.PD 0
.ie n .IP """$path = Math::PlanePath::Corner\->new (wider => $w, n_start => $n)""" 4
.el .IP "\f(CW$path = Math::PlanePath::Corner\->new (wider => $w, n_start => $n)\fR" 4
.IX Item "$path = Math::PlanePath::Corner->new (wider => $w, n_start => $n)"
.PD
Create and return a new path object.
.ie n .IP """($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 < n_start()\-0.5\*(C'\fR the return is an empty list.  There's an extra 0.5
before Nstart, but nothing further before there.
.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,$y\*(C'\fR.
.Sp
\&\f(CW$x\fR and \f(CW$y\fR are each rounded to the nearest integer, which has the
effect of treating each point as a square of side 1, so the quadrant x>=\-0.5
and y>=\-0.5 is entirely covered.
.ie n .IP """($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 "N to X,Y"
.IX Subsection "N to X,Y"
Counting d=0 for the first L\-shaped gnomon at Y=0, then the start of the
gnomon is
.PP
.Vb 1
\&    StartN(d) = d^2 + 1 = 1,2,5,10,17,etc
.Ve
.PP
The current \f(CW\*(C`n_to_xy()\*(C'\fR 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
.PP
.Vb 1
\&    StartNfrac(d) = d^2 + 0.5
.Ve
.PP
Inverting gives the gnomon d number for an N,
.PP
.Vb 1
\&    d = floor(sqrt(N \- 0.5))
.Ve
.PP
Subtracting the gnomon start gives an offset into that gnomon
.PP
.Vb 1
\&    OffStart = N \- StartNfrac(d)
.Ve
.PP
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,
.PP
.Vb 2
\&    Off = OffStart \- (d+0.5)
\&        = N \- (d*(d+1) + 1)
.Ve
.PP
Then the X,Y coordinates are
.PP
.Vb 2
\&    if (Off < 0)  then  X=d+Off, Y=d
\&    if (Off >= 0) then  X=d,     Y=d\-Off
.Ve
.SS "X,Y to N"
.IX Subsection "X,Y to N"
For a given X,Y the bigger of X or Y determines the d gnomon.
.PP
If Y>=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.
.PP
.Vb 4
\&    if Y >= X then
\&      d=Y
\&      N = StartN(d) + X
\&        = Y^2 + 1 + X
.Ve
.PP
Otherwise if Y<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,
.PP
.Vb 6
\&    if Y <= X then
\&      d=X
\&      LastN(d) = StartN(d+1) \- 1
\&               = (d+1)^2
\&      N = LastN(d) \- Y
\&        = (X+1)^2 \- Y
.Ve
.SS "Rectangle N Range"
.IX Subsection "Rectangle N Range"
For \f(CW\*(C`rect_to_n_range()\*(C'\fR, 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.
.PP
.Vb 4
\&    |
\&    |  \-\-\-\-\-\->  N increasing
\&    |
\&     \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-
.Ve
.PP
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.
.PP
.Vb 9
\&    |    ^  N increasing up from X=Y diagonal
\&    |    |
\&    |    |/
\&    |    /
\&    |   /|
\&    |  / |  N increasing down from X=Y diagonal
\&    | /  v
\&    |/
\&     \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-
.Ve
.PP
This means the biggest N is the top right corner if that corner is Y>=X,
otherwise the bottom right corner.
.PP
.Vb 9
\&                                           max N at top right
\&    |      /                          | \-\-+     if corner Y>=X
\&    |     / \-\-+                       |   | /
\&    |    /    |                       |   |/
\&    |   /     |                       |   |
\&    |  /  \-\-\-\-v                       |  /|
\&    | /     max N at bottom right     | \-\-+
\&    |/        if corner Y<=X          |/
\&     \-\-\-\-\-\-\-\-\-\-                        \-\-\-\-\-\-\-
.Ve
.PP
For the smallest N, if the bottom left corner has Y>X then it's in the
\&\*(L"increasing\*(R" part and that bottom left corner is the smallest N.  Otherwise
Y<=X means some of the \*(L"decreasing\*(R" 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.
.PP
.Vb 8
\&    |      /
\&    | |   /
\&    | |  /  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
.Ve
.SH "OEIS"
.IX Header "OEIS"
This path is in Sloane's Online Encyclopedia of Integer Sequences as,
.Sp
.RS 4
<http://oeis.org/A196199> (etc)
.RE
.PP
.Vb 4
\&    wider=0, n_start=1 (the defaults)
\&      A213088    X+Y sum
\&      A196199    X\-Y diff, being runs \-n to +n
\&      A053615    abs(X\-Y), runs n to 0 to n, 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
\&
\&      A027709    boundary length of N unit squares
\&      A078633    grid sticks of N points
\&
\&    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)
.Ve
.SH "SEE ALSO"
.IX Header "SEE ALSO"
Math::PlanePath,
Math::PlanePath::PyramidSides,
Math::PlanePath::PyramidRows,
Math::PlanePath::SacksSpiral,
Math::PlanePath::Diagonals
.SH "HOME PAGE"
.IX Header "HOME PAGE"
<http://user42.tuxfamily.org/math\-planepath/index.html>
.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 <http://www.gnu.org/licenses/>.