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 tiein to make a "NumSeq" sequence giving coordinate values from a "Math::PlanePath". The NumSeq ``i'' index is the PlanePath ``N'' value.The "coordinate_type" choices are as follows. Generally they have some sort of geometric interpretation or are related to fractions X/Y.
"X" X coordinate "Y" Y coordinate "Min" min(X,Y) "Max" max(X,Y) "MinAbs" min(abs(X),abs(Y)) "MaxAbs" max(abs(X),abs(Y)) "Sum" X+Y sum "SumAbs" abs(X)+abs(Y) sum "Product" X*Y product "DiffXY" XY difference "DiffYX" YX difference (negative of DiffXY) "AbsDiff" abs(XY) 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 "IntXY" int(X/Y) division rounded towards zero "FracXY" frac(X/Y) division rounded towards zero "BitAnd" X bitand Y "BitOr" X bitor Y "BitXor" X bitxor Y "GCD" greatest common divisor X,Y "Depth" tree_n_to_depth() "SubHeight" tree_n_to_subheight() "NumChildren" tree_n_num_children() "NumSiblings" not including self "RootN" the N which is the tree root "IsLeaf" 0 or 1 whether a leaf node (no children) "IsNonLeaf" 0 or 1 whether a nonleaf node (has children) also called an "internal" node
Min and Max
``Min'' and ``Max'' are the minimum or maximum of X and Y. The geometric interpretation of ``Min'' is to select X at any point above the X=Y diagonal or Y for any point below. Conversely ``Max'' is Y above and X below. On the X=Y diagonal itself X=Y=Min=Max.
Max=Y / X=Y diagonal Min=X  / / o / /  Max=X / Min=Y
Min and Max can also be interpreted as counting which gnomon shaped line the X,Y falls on.
    Min=gnomon 2 . Max=gnomon     1 .      ... 0 o       1 1 .      o 0 ...       1      2    
MinAbs
MinAbs = min(abs(X),abs(Y)) can be interpreted geometrically as counting gnomons successively away from the origin. This is like Min above, but within the quadrant containing X,Y.
     MinAbs=gnomon counted away from the origin      2      2 1    1 0 o 0 1    1 2      2          
MaxAbs
MaxAbs = max(abs(X),abs(Y)) can be interpreted geometrically as counting successive squares around the origin.
++ MaxAbs=which square  ++    ++      o      ++    ++  ++
For example Math::PlanePath::SquareSpiral loops around in squares and so its MaxAbs is unchanged until it steps out to the next bigger square.
Sum and Diff
``Sum''=X+Y and ``DiffXY''=XY can be interpreted geometrically as coordinates on 45degree diagonals. Sum is a measure up along the leading diagonal and DiffXY down an antidiagonal,
\ / \ s=X+Y / \ ^\ \ / \ \  / v \/ * d=XY o /\ /  \ /  \ / \ / \ / \
Or ``Sum'' can be thought of as a count of which antidiagonal stripe contains X,Y, or a projection onto the X=Y leading diagonal.
Sum \ = antidiag 2 numbering / / / / DiffXY \ \ X+Y 1 0 1 2 = diagonal 1 2 / / / / numbering \ \ \ 1 0 1 2 XY 0 1 2 / / / \ \ \ 0 1 2
DiffYX
``DiffYX'' = YX is simply the negative of DiffXY. It's included to give positive values on paths which are above 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.SumAbs
``SumAbs'' = abs(X)+abs(Y) is similar to the projection described above for Sum or Diff, but SumAbs projects onto the central diagonal of whichever quadrant contains the X,Y. Or equivalently it's a numbering of antidiagonals within that quadrant, so numbering which diamond shape the X,Y falls on.
 /\ SumAbs = which diamond X,Y falls on /  \ /  \ o \  / \  / \/ 
As an example, the "DiamondSpiral" path loops around on such diamonds, so its SumAbs is unchanged until completing a loop and stepping out to the next bigger.
SumAbs is also a ``taxicab'' or ``Manhattan'' distance, being how far to travel through a squaregrid city to get to X,Y.
SumAbs = taxicab distance, by any squaregrid travel +o +o o     ++ ++    * * *
If a path is entirely X>=0,Y>=0 in the first quadrant then Sum and SumAbs are identical.
AbsDiff
``AbsDiff'' = abs(XY) can be interpreted geometrically as the distance away from the X=Y diagonal, measured at rightangles to that line.
d=abs(XY) ^ / X=Y line \ / \/ /\ / \ / \ o \ / v / d=abs(XY)
If a path is entirely below the X=Y line, so X>=Y, then AbsDiff is the same as DiffXY. Or if a path is entirely above the X=Y line, so Y>=X, then AbsDiff is the same as DiffYX.
Radius and RSquared
Radius and RSquared are per "$path>n_to_radius()" and "$path>n_to_rsquared()" respectively (see ``Coordinate Methods'' in Math::PlanePath).TRadius and TRSquared
``TRadius'' and ``TRSquared'' are designed for use with points on a triangular lattice as per ``Triangular Lattice'' in Math::PlanePath. For points on the X axis TRSquared is the same as RSquared but off the axis Y is scaled up by factor sqrt(3).Most triangular paths use ``even'' points X==Y mod 2 and for them TRSquared is always even. Some triangular paths such as "KochPeaks" have an offset from the origin and use ``odd'' points X!=Y mod 2 and for them TRSquared is odd.
IntXY and FracXY
``IntXY'' = int(X/Y) is the quotient from X divide Y rounded to an integer towards zero. This is like the integer part of a fraction, for example X=9,Y=4 is 9/4 = 2+1/4 so IntXY=2. Negatives are reckoned with the fraction part negated too, so 2 1/4 is 21/4 and thus IntXY=2.Geometrically IntXY gives which wedge of slope 1, 2, 3, etc the point X,Y falls in. For example IntXY is 3 for all points in the wedge 3Y<=X<4Y.
X=Y X=2Y X=3Y X=4Y * 2 * 1 * 0  0 * 1 * 2 * 3 * * * *  * * * * * * *  * * * * * * *  * * * * * * *  * * * * * * *  * * * * ******* + **** * *  * * * *  * * * *  * * * *  * * 2 * 1 * 0  0 * 1 * 2
``FracXY'' is the fraction part which goes with IntXY. In all cases
X/Y = IntXY + FracXY
IntXY rounds towards zero so the remaining FracXY has the same sign as IntXY.
BitAnd, BitOr, BitXor
``BitAnd'', ``BitOr'' and ``BitXor'' treat negative X or negative Y as infinite twoscomplement 1bits, which means for example X=1,Y=2 has X bitand Y = 2.
...11111111 X=1 ...11111110 Y=2  ...11111110 X bitand Y = 2
This twoscomplement is per "Math::BigInt" (which has bitwise operations in Perl 5.6 and up). The code here arranges the same on ordinary scalars.
If X or Y are not integers then the fractional parts are treated bitwise too, but currently only to limited precision.
FUNCTIONS
See ``FUNCTIONS'' in Math::NumSeq for behaviour common to all sequence classes. "$seq = Math::NumSeq::PlanePathCoord>new (planepath => $name, coordinate_type => $str)"

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 Anumber (a string) for $seq in Sloane's Online Encyclopedia
of Integer Sequences, or return "undef" if not in the OEIS or not known.
Known Anumbers are also presented through "Math::NumSeq::OEIS::Catalogue". This means PlanePath related OEIS sequences can be created with "Math::NumSeq::OEIS" by giving their Anumber in the usual way for that module.
LICENSE
Copyright 2011, 2012, 2013, 2014 Kevin RydeThis file is part of MathPlanePath.
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.