Math::NumSeq::PlanePathCoord -- sequence of coordinate values from a PlanePath module
use Math::NumSeq::PlanePathCoord; my $seq = Math::NumSeq::PlanePathCoord->new (planepath => 'SquareSpiral', coordinate_type => 'X'); my ($i, $value) = $seq->next;
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.
Math::PlanePath
The coordinate_type choices are
coordinate_type
"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.
Math::BigInt
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".
planepath
$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.
$i
rewind()
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.
$path->n_start()
Math::NumSeq::OEIS
i_start()
$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.
$seq
undef
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.
Math::NumSeq::OEIS::Catalogue
Math::NumSeq, Math::NumSeq::PlanePathDelta, Math::NumSeq::PlanePathTurn, Math::NumSeq::PlanePathN, Math::NumSeq::OEIS
http://user42.tuxfamily.org/math-planepath/index.html
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/>.
To install Math::PlanePath, copy and paste the appropriate command in to your terminal.
cpanm
cpanm Math::PlanePath
CPAN shell
perl -MCPAN -e shell install Math::PlanePath
For more information on module installation, please visit the detailed CPAN module installation guide.