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 make a
NumSeq sequence giving coordinate values of a
Math::PlanePath. The NumSeq "i" index is the PlanePath "N" value.
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) "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 "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() "NumChildren" tree_n_num_children() "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" are the minimum or maximum of X and Y. The geometric interpretation of "Max" is to select Y at any X,Y point above the X=Y diagonal or X for any X,Y point below. Conversely "Min" is X above and Y below. On the X=Y diagonal itself X=Y=Min=Max, a single value.
Max=Y / X=Y diagonal Min=X / | / |/ ---o---- /| / | / Max=X / Min=Y
"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" = 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" = abs(X)+abs(Y) is similar to the projection described above for Sum or Diff, but SumAbs projects onto the diagonal of whichever quadrant contains the X,Y, or a numbering of anti-diagonals within that quadrant. Those anti-diagonals make a diamond shape,
| /|\ SumAbs = which size diamond X,Y falls on / | \ / | \ -----o----- \ | / \ | / \|/ |
For example the
DiamondSpiral path follows around such diamonds, thus keeping SumAbs unchanged until stepping out to the next bigger diamond.
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 within the first quadrant X>=0,Y>=0 then Sum and SumAbs are identical.
"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
"TRadius" and "TRSquared" are designed for use with points on a triangular lattice such as
HexSpiral. 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" = 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 is 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
Since IntXY rounds towards zero the remaining FracXY has the same sign as that integer part.
"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.
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.
$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.
Copyright 2011, 2012, 2013 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/>.