Provided by: libmath-planepath-perl_125-1_all 
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 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" 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 (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 non-leaf 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"=X-Y can be interpreted geometrically as coordinates on 45-degree
diagonals. Sum is a measure up along the leading diagonal and DiffXY down 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 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
DiffYX
"DiffYX" = Y-X 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 anti-diagonals 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 "taxi-cab" or "Manhattan" distance, being how far to travel through a
square-grid city to get to X,Y.
SumAbs = taxi-cab distance, by any square-grid 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(X-Y) can be interpreted geometrically as the distance away from the X=Y
diagonal, measured at right-angles to that line.
d=abs(X-Y)
^ / X=Y line
\ /
\/
/\
/ \
|/ \
--o-- \
/| v
/ d=abs(X-Y)
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 -2-1/4
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 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" (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 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 also presented through "Math::NumSeq::OEIS::Catalogue". This
means PlanePath related OEIS sequences can be created with "Math::NumSeq::OEIS" by
giving their A-number in the usual way for that module.
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, 2013, 2014 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/>.