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 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" 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 = 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 = 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"=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 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" = 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" = 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" 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.
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::NumSeq, Math::NumSeq::PlanePathDelta, Math::NumSeq::PlanePathTurn, Math::NumSeq::PlanePathN, Math::NumSeq::OEIS
http://user42.tuxfamily.org/math-planepath/index.html
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/>.