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.\" ========================================================================
.\"
.IX Title "Math::PlanePath::CornerAlternating 3pm"
.TH Math::PlanePath::CornerAlternating 3pm "2021-01-23" "perl v5.32.0" "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::CornerAlternating \-\- points shaped around a corner alternately
.SH "SYNOPSIS"
.IX Header "SYNOPSIS"
.Vb 3
\& use Math::PlanePath::CornerAlternating;
\& my $path = Math::PlanePath::CornerAlternating\->new;
\& my ($x, $y) = $path\->n_to_xy (123);
.Ve
.SH "DESCRIPTION"
.IX Header "DESCRIPTION"
This path is points in layers around a square outwards from a corner in the
first quadrant, alternately upward or downward.  Each row/column
\&\*(L"gnomon\*(R" added to a square makes a one-bigger square.
.IX Xref "Gnomon"
.PP
.Vb 11
\&      4 | 17\-\-18\-\-19\-\-20\-\-21 ...
\&        |  |               |   |
\&      3 | 16\-\-15\-\-14\-\-13  22  29
\&        |              |   |   |
\&      2 |  5\-\-\-6\-\-\-7  12  23  28
\&        |  |       |   |   |   |
\&      1 |  4\-\-\-3   8  11  24  27
\&        |      |   |   |   |   |
\&    Y=0 |  1\-\-\-2   9\-\-10  25\-\-26
\&        +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-
\&         X=0   1   2   3   4   5
.Ve
.PP
This is like the Corner path, but here gnomons go back and forward and in
particular so points are always a unit step apart.
.SS "Wider"
.IX Subsection "Wider"
An optional \f(CW\*(C`wider => $integer\*(C'\fR makes the path wider horizontally,
becoming a rectangle.  For example
.PP
.Vb 11
\&     4  |  29\-\-30\-\-31\-\-32\-\-33\-\-34\-\-35\-\-36  ...
\&        |   |                           |   |
\&     3  |  28\-\-27\-\-26\-\-25\-\-24\-\-23\-\-22  37  44      wider => 3
\&        |                           |   |   |
\&     2  |  11\-\-12\-\-13\-\-14\-\-15\-\-16  21  38  43
\&        |   |                   |   |   |   |
\&     1  |  10\-\-\-9\-\-\-8\-\-\-7\-\-\-6  17  20  39  42
\&        |                   |   |   |   |   |
\&    Y=0 |   1\-\-\-2\-\-\-3\-\-\-4\-\-\-5  18\-\-19  40\-\-41
\&        +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-
\&          X=0   1   2   3   4   5   6   7   8
.Ve
.PP
Each gnomon has the horizontal part \f(CW\*(C`wider\*(C'\fR many steps longer.  For wider=3
shown, the additional points are 2,3,4 in the first row, then 5..10 are the
next gnomon.  Each gnomon is still 2 longer than the previous since this
widening is a constant amount in each.
.SS "N Start"
.IX Subsection "N Start"
The default is to number points starting N=1 as shown above.  An optional
\&\f(CW\*(C`n_start\*(C'\fR can give a different start with the same shape etc.  For example
to start at 0,
.PP
.Vb 7
\&      4  |  16  17  18  19  20
\&      3  |  15  14  13  12  21      n_start => 0
\&      2  |   4   5   6  11  22
\&      1  |   3   2   7  10  23
\&    Y=0  |   0   1   8   9  24
\&          \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-
\&           X=0   1   2   3   4
.Ve
.PP
With Nstart=0, the pronic numbers are on the X=Y leading diagonal.
.SH "FUNCTIONS"
.IX Header "FUNCTIONS"
See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes.
.ie n .IP """$path = Math::PlanePath::CornerAlternating\->new ()""" 4
.el .IP "\f(CW$path = Math::PlanePath::CornerAlternating\->new ()\fR" 4
.IX Item "$path = Math::PlanePath::CornerAlternating->new ()"
.PD 0
.ie n .IP """$path = Math::PlanePath::CornerAlternating\->new (wider => $w, n_start => $n)""" 4
.el .IP "\f(CW$path = Math::PlanePath::CornerAlternating\->new (wider => $w, n_start => $n)\fR" 4
.IX Item "$path = Math::PlanePath::CornerAlternating->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()\*(C'\fR the return is an empty list.  Fractional \f(CW$n\fR gives
an X,Y position along a straight line between the integer positions.
.ie n .IP """$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"
Most calculations are similar to the Corner path (without the 0.5 fractional
part), and a reversal applied when the d gnomon number is odd.  When
wider>0, that reversal must allow for the horizontals and verticals
different lengths.
.SS "Rectangle N Range"
.IX Subsection "Rectangle N Range"
For \f(CW\*(C`rect_to_n_range()\*(C'\fR, the largest gnomon is either the top or right of
the rectangle, depending where the top right corner x2,y2 falls relative to
the leading diagonal,
.PP
.Vb 7
\&    |  A\-\-\-B /    x2<y2         |       /       x2>y2
\&    |  |   |/     top           |  +\-\-\-\-\-\-B     right
\&    |  |   |      row           |  |  /   |     side
\&    |  |  /|     biggest        |  | /    |     biggest
\&    |  +\-\-\-+     gnomon         |  +\-\-\-\-\-\-C     gnomon
\&    |   /                       |  /
\&    +\-\-\-\-\-\-\-\-\-                  +\-\-\-\-\-\-\-\-\-\-\-
.Ve
.PP
Then the maximum is at A or B, or B or C according as which way that gnomon
goes, so odd or even.
.PP
If it happens that B is on the diagonal, so x2=y2, then it's either A or C
according as the gnomon odd or even
.PP
.Vb 7
\&    |        /
\&    |  A\-\-\-\-+     x2=y2
\&    |  |   /|
\&    |  |  / |
\&    |  +\-\-\-\-C
\&    |   /
\&    +\-\-\-\-\-\-\-\-\-\-\-
.Ve
.PP
For wider > 0, the diagonal shifts across so that x2\-wider <=>
y2 is the relevant test.
.SH "OEIS"
.IX Header "OEIS"
This path is in Sloane's Online Encyclopedia of Integer Sequences as,
.Sp
.RS 4
<http://oeis.org/A319289> (etc)
.RE
.PP
.Vb 10
\&    wider=0, n_start=1 (the defaults)
\&      A220603    X+1 coordinate
\&      A220604    Y+1 coordinate
\&      A213088    X+Y sum
\&      A081346    N on X axis
\&      A081345    N on Y axis
\&      A002061    N on X=Y diagonal, extra initial 1
\&      A081344    permutation N by diagonals
\&      A194280      inverse
\&      A020703    permutation N at transpose Y,X
\&
\&      A027709    boundary length of N unit squares
\&      A078633    grid sticks of N points
\&
\&    n_start=0
\&      A319290    X coordinate
\&      A319289    Y coordinate
\&      A319514    Y,X coordinate pairs
\&      A329116    X\-Y diff
\&      A053615    abs(X\-Y) diff
\&      A000196    max(X,Y), being floor(sqrt(N))
\&      A339265    dX\-dY increments (runs +1,\-1)
\&      A002378    N on X=Y diagonal, pronic numbers
\&      A220516    permutation N by diagonals
\&
\&    n_start=2
\&      A014206    N on X=Y diagonal, pronic+2
\&
\&    wider=1, n_start=1
\&      A081347    N on X axis
\&      A081348    N on Y axis
\&      A080335    N on X=Y diagonal
\&      A093650    permutation N by diagonals
\&
\&    wider=1, n_start=0
\&      A180714    X\-Y diff
\&
\&    wider=2, n_start=1
\&      A081350    N on X axis
\&      A081351    N on Y axis
\&      A081352    N on X=Y diagonal
\&      A081349    permutation N by diagonals
.Ve
.SH "SEE ALSO"
.IX Header "SEE ALSO"
Math::PlanePath,
Math::PlanePath::Corner,
Math::PlanePath::DiagonalsAlternating
.SH "HOME PAGE"
.IX Header "HOME PAGE"
<http://user42.tuxfamily.org/math\-planepath/index.html>
.SH "LICENSE"
.IX Header "LICENSE"
Copyright 2021 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/>.