Kevin Ryde > Math-PlanePath-116 > Math::NumSeq::PlanePathCoord

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Module Version: 116   Source   Latest Release: Math-PlanePath-117

# 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
"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 "Manhatten" 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 are per \$path->n_to_radius() and \$path->n_to_rsquared() respectively (see "Coordinate Methods" in Math::PlanePath).

"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.

Math::PlanePath

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