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Math::NumSeq::PlanePathDelta -- sequence of changes and directions of PlanePath coordinates
use Math::NumSeq::PlanePathDelta; my $seq = Math::NumSeq::PlanePathDelta->new (planepath => 'SquareSpiral', delta_type => 'dX'); my ($i, $value) = $seq->next;
This is a tie-in to present coordinate changes and directions from a
Math::PlanePath module in the form of a NumSeq sequence.
delta_type choices are
"dX" change in X coordinate "dY" change in Y coordinate "AbsdX" abs(dX) "AbsdY" abs(dY) "dSum" change in X+Y, equals dX+dY "dSumAbs" change in abs(X)+abs(Y) "dDiffXY" change in X-Y, equals dX-dY "dDiffYX" change in Y-X, equals dY-dX "dAbsDiff" change in abs(X-Y) "dRadius" change in Radius sqrt(X^2+Y^2) "dRSquared" change in RSquared X^2+Y^2 "Dir4" direction 0=East, 1=North, 2=West, 3=South "TDir6" triangular 0=E, 1=NE, 2=NW, 3=W, 4=SW, 5=SE
In each case the value at i is per
$path->n_to_dxdy($i), being the change from N=i to N=i+1, or from N=i to N=i+arms for paths with multiple "arms" (thus following the arm). i values start from the usual
If a path always steps NSEW by 1 then AbsdX and AbsdY behave as a boolean indicating horizontal or vertical step,
NSEW steps by 1 gives AbsdX = 0 vertical AbsdY = 0 horizontal 1 horizontal 1 vertical
If a path includes diagonal steps by 1 then those diagonals are a non-zero delta, so the indication is then
NSEW and diagonals steps by 1 gives AbsdX = 0 vertical AbsdY = 0 horizontal 1 non-vertical 1 non-horizontal ie. horiz or diag ie. vert or diag
"dSum" is the change in X+Y and is also simply dX+dY since
dSum = (Xnext+Ynext) - (X+Y) = (Xnext-X) + (Ynext-Y) = dX + dY
The sum X+Y counts anti-diagonals, as described in Math::NumSeq::PlanePathCoord. dSum is therefore a move between diagonals, or 0 if a step stays within the same diagonal.
\ \ ^ dSum > 0 dSum = step dist to North-East \/ /\ dSum < 0 v \ \
"dSumAbs" is the change in the abs(X)+abs(Y) sum,
dSumAbs = (abs(Xnext)+abs(Ynext)) - (abs(X)+abs(Y))
As described in "SumAbs" in Math::NumSeq::PlanePathCoord, SumAbs is a "Manhattan" or "taxi-cab" distance from the origin, or equivalently a move between diamond-shaped rings.
DiamondSpiral follows a diamond shape ring around and so has dSumAbs=0 until stepping out to each next diamond with dSumAbs=1.
A path might make a big X,Y jump which is only a small change in SumAbs. For example
PyramidRows in its default step=2 from the end of one row to the start of the next has dSumAbs=2.
"dDiffXY" is the change in DiffXY = X-Y, which is also simply dX-dY since
dDiffXY = (Xnext-Ynext) - (X-Y) = (Xnext-X) - (Ynext-Y) = dX - dY
The difference X-Y counts diagonals downwards to the south-east as described in "Sum and Diff" in Math::NumSeq::PlanePathCoord. dDiffXY is therefore movement between those diagonals, or 0 if a step stays within the same diagonal.
dDiffXY < 0 / \ / dDiffXY = step dist to South-East \/ /\ / v / dDiffXY > 0
"dDiffYX" is the negative of dDiffXY. Whether X-Y or Y-X is desired depends on which way you want to measure diagonals, or which way around to have the sign for the changes. dDiffYX is based on Y-X and so counts diagonals upwards to the North-West.
"dAbsDiff" is the change in AbsDiff = abs(X-Y). AbsDiff can be interpreted geometrically as distance from the leading diagonal, as described in "AbsDiff" in Math::NumSeq::PlanePathCoord. dAbsDiff is therefore movement closer to or further away from that leading diagonal, measuring perpendicular to it.
/ X=Y line / / ^ / \ / * dAbsDiff move towards or away from X=Y line |/ \ --o-- v /| /
When an X,Y jumps from one side of the diagonal to the other dAbsDiff is still the change in distance from the diagonal. So for example if X,Y is followed by the mirror point Y,X then dAbsDiff=0. That sort of thing happens for example in the
Diagonals path when jumping from the end of one run to the start of the next. In the
Diagonals case it's a move just 1 further away from the X=Y centre line even though it's a big jump in overall distance.
"dRadius" and "dRSquared" are the change in the Radius and RSquared as described in "Radius and RSquared" in Math::NumSeq::PlanePathCoord.
dRadius = next_Radius - Radius dRSquared = next_RSquared - RSquared
dRadius can be interpreted geometrically as movement towards (negative values) or away from (positive values) the origin, ignoring direction.
Notice that dRadius is not sqrt(dRSquared), since sqrt(n^2-t^2) != n-t unless n or t is zero. Here would mean a step either going to or coming from the origin 0,0.
"Dir4" is the curve step direction as an angle in the range 0 <= Dir4 < 4. The cardinal directions E,N,W,S are 0,1,2,3. Angles in between are a fraction.
Dir4 = atan2(dY,dX) scaled as range 0 <= Dir4 < 4 1.5 1 0.5 \ | / \|/ 2 ----o---- 0 /|\ / | \ 2.5 3 3.5
If a row such as Y=-1,X>0 just below the X axis is visited then the Dir4 approaches 4, without ever reaching it. The
$seq->value_maximum() is 4 in this case, as a supremum.
"TDir6" is the curve step direction 0 <= TDir6 < 6 taken in the triangular style of "Triangular Lattice" in Math::PlanePath. So dX=1,dY=1 is taken to be 60 degrees which is TDir6=1.
2 1.5 1 TDir6 \ | / \|/ 3 ---o--- 0 /|\ / | \ 4 4.5 5
Angles in between the six cardinal directions are fractions. North is 1.5 and South is 4.5.
The direction angle is calculated as if dY was scaled by a factor sqrt(3) to make the lattice into equilateral triangles, or equivalently as a circle stretched vertically by sqrt(3) to become an ellipse.
TDir6 = atan2(dY*sqrt(3), dX) in range 0 <= TDir6 < 6
Notice that angles on the axes dX=0 or dY=0 are not changed by the sqrt(3) factor. So TDir6 has ENWS 0, 1.5, 3, 4.5 which is steps of 1.5. Verticals North and South normally don't occur in the triangular lattice paths which go by unit steps, but TDir6 can be applied on any path.
The sqrt(3) factor increases angles in the middle of the quadrants. For example dX=1,dY=1 becomes TDir6=1 whereas a plain angle would be only 45/360*6=0.75 in the same 0 to 6 scale. The sqrt(3) is a continuous scaling, so a plain angle and a TDir6 are a one-to-one mapping. As the direction progresses through the quadrant TDir6 grows first faster and then slower than the plain angle.
See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.
$seq = Math::NumSeq::PlanePathDelta->new (key=>value,...)
Create and return a new sequence object. The options are
planepath string, name of a PlanePath module planepath_object PlanePath object delta_type string, as described above
planepathcan be either the module part such as "SquareSpiral" or a full class name "Math::PlanePath::SquareSpiral".
$value = $seq->ith($i)
Return the change at N=$i in the PlanePath.
$i = $seq->i_start()
Return the first index
$iin the sequence. This is the position
$path->n_start()from the PlanePath.
Some path sequences don't have
oeis_anum() and are not available through Math::NumSeq::OEIS entry due to the path
n_start() not matching the OEIS "offset". Paths with an
n_start parameter have suitable adjustments applied, but those without are omitted from the Math::NumSeq::OEIS mechanism presently.
Copyright 2011, 2012, 2013, 2014, 2015, 2016 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.
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