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 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 (norm)
"TRadius" sqrt(X^2+3*Y^2) triangular radius
"TRSquared" X^2+3*Y^2 triangular radius squared (norm)
"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 positionrewind()
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 theMath::NumSeq::OEIS
generated sequences there may be a higheri_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 returnundef
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 withMath::NumSeq::OEIS
by giving their Anumber in the usual way for that module.
SEE ALSO
Math::NumSeq, Math::NumSeq::PlanePathDelta, Math::NumSeq::PlanePathTurn, Math::NumSeq::PlanePathN, Math::NumSeq::OEIS
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2011, 2012, 2013, 2014 Kevin Ryde
This 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/>.