/ 
Math-PlanePath-93
River stage one • 3 direct dependents • 4 total dependents
/ Math::NumSeq::PlanePathCoord

NAME

Math::NumSeq::PlanePathCoord -- sequence of coordinate values from a PlanePath module

SYNOPSIS

use Math::NumSeq::PlanePathCoord;
my $seq = Math::NumSeq::PlanePathCoord->new
            (planepath => 'SquareSpiral',
             coordinate_type => 'X');
my ($i, $value) = $seq->next;

DESCRIPTION

This is a tie-in to present coordinates from a Math::PlanePath module as a NumSeq sequence. The NumSeq "i" index is the PlanePath "N" value.

The coordinate_type choices are

"X"            X coordinate
"Y"            Y coordinate
"Sum"          X+Y sum
"SumAbs"       abs(X)+abs(Y) sum
"Product"      X*Y product
"DiffXY"       X-Y difference
"DiffYX"       Y-X difference (negative of DiffXY)
"AbsDiff"      abs(X-Y) difference
"Radius"       sqrt(X^2+Y^2) radial distance
"RSquared"     X^2+Y^2 radius squared
"TRadius"      sqrt(X^2+3*Y^2) triangular radius
"TRSquared"    X^2+3*Y^2 triangular radius squared
"BitAnd"       X bitand Y
"BitOr"        X bitor Y
"BitXor"       X bitxor Y
"Min"          min(X,Y)
"Max"          max(X,Y)
"GCD"          greatest common divisor X,Y
"Depth"        tree_n_to_depth()
"NumChildren"  tree_n_num_children()

"Sum"=X+Y and "DiffXY=X-Y can be interpreted geometrically as coordinates on 45-degree diagonals. Sum is a measure up along the leading diagonal and DiffXY down along an anti-diagonal,

             /
\           /
 \   s=X+Y /
  \       ^\
   \     /  \
    \ | /    v
     \|/      * d=X-Y
   ---o----
     /|\
    / | \
   /  |  \
  /       \
 /         \
/           \

Or "Sum" can be thought of as a count of which anti-diagonal stripe contains X,Y or equivalently a projection onto the X=Y leading diagonal.

       Sum
\     anti-diag
 2    numbering          / / / /   DiffXY
\ \     X+Y            -1 0 1 2   diagonal
 1 2                   / / / /    numbering
\ \ \                -1 0 1 2       X-Y
 0 1 2                 / / /
  \ \ \               0 1 2

"SumAbs"=abs(X)+abs(Y) is similar, but a projection onto the cross-diagonal of whichever quadrant contains the X,Y. It's also thought of as a "taxi-cab" or Manhatten distance, being how far to travel through a square-grid city to get to X,Y. If a path uses only the first quadrant, so X>=0,Y>=0, then of course Sum and SumAbs are identical.

SumAbs = taxi-cab distance, by any square-grid travel

+-----o       +--o          o
|             |             |
|          +--+       +-----+
|          |          |
*          *          *

"DiffYX"=Y-X is simply the negative of DiffXY. It's included to give positive values on paths which are either above or below the X=Y leading diagonal. For example DiffXY is positive in CoprimeColumns which is below X=Y, whereas DiffYX is positive in CellularRule which is above X=Y.

"TRadius" and "TRSquared" are designed for use with points on a triangular lattice such as HexSpiral. On the X axis TRSquared is the same as RSquared, but any Y is scaled up by factor sqrt(3). Most triangular paths use every second X,Y point which makes TRSquared even, but some such as KochPeaks have an offset 1 from the origin making it odd instead.

"BitAnd", "BitOr" and "BitXor" treat negative X or negative Y as infinite twos-complement 1-bits, which means for example X=-1,Y=-2 has X bitand Y = -2.

...11111111    X=-1
...11111110    Y=-2
-----------
...11111110    X bitand Y = -2

This twos-complement is per Math::BigInt (it has bitwise operations in Perl 5.6 and up) and is arranged for ordinary scalars too. If X or Y are not integers then the fractional parts are treated bitwise too, though currently only to limited precision.

FUNCTIONS

See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.

$seq = Math::NumSeq::PlanePathCoord->new (planepath => $name, coordinate_type => 'X')

Create and return a new sequence object. The options are

planepath          string, name of a PlanePath module
planepath_object   PlanePath object
coordinate_type    string, as described above

planepath can be either the module part such as "SquareSpiral" or a full class name "Math::PlanePath::SquareSpiral".

$value = $seq->ith($i)

Return the coordinate at N=$i in the PlanePath.

$i = $seq->i_start()

Return the first index $i in the sequence. This is the position rewind() returns to.

This is $path->n_start() from the PlanePath, since the i numbering is the N numbering of the underlying path. For some of the Math::NumSeq::OEIS generated sequences there may be a higher i_start() corresponding to a higher starting point in the OEIS, though this is slightly experimental.

$str = $seq->oeis_anum()

Return the A-number (a string) for $seq in Sloane's Online Encyclopedia of Integer Sequences, or return undef if not in the OEIS or not known.

Known A-numbers are presented through Math::NumSeq::OEIS::Catalogue so PlanePath related sequences can be created with Math::NumSeq::OEIS by their A-number in the usual way.

SEE ALSO

Math::NumSeq, Math::NumSeq::PlanePathDelta, Math::NumSeq::PlanePathTurn, Math::NumSeq::PlanePathN, Math::NumSeq::OEIS

Math::PlanePath

HOME PAGE

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

LICENSE

Copyright 2011, 2012 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/>.