Kevin Ryde > Math-PlanePath-110 > Math::NumSeq::PlanePathDelta

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Module Version: 110   Source   Latest Release: Math-PlanePath-112

# NAME

Math::NumSeq::PlanePathDelta -- sequence of changes and directions of PlanePath coordinates

# SYNOPSIS

``` use Math::NumSeq::PlanePathDelta;
my \$seq = Math::NumSeq::PlanePathDelta->new
(planepath => 'SquareSpiral',
delta_type => 'dX');
my (\$i, \$value) = \$seq->next;```

# DESCRIPTION

This is a tie-in to present coordinate changes and directions from a `Math::PlanePath` module in the form of a NumSeq sequence.

The `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)
"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 a particular arm). i values start from the usual `\$path->n_start()`.

## AbsdX,AbsdY

If a path always step NSEW by 1 then AbsdX and AbsdY behave as a boolean indicating horizontal or vertical step,

```    NSEW steps by 1

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

AbsdX = 0 vertical            AbsdY = 0 horizontal
1 non-vertical                1 non-horizontal
ie. horiz or diag             ie. vert or diag```

## dSum

"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

"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 "taxi-cab" distance from the origin, or equivalently a move between diamond shaped rings.

As an example, a path such as `DiamondSpiral` follows a diamond shape ring around and so has dSumAbs=0 until stepping out to the next diamond with dSumAbs=1.

A path might make a big jump which is only a small change in SumAbs. For example `PyramidRows` in its default step=2 going from the end of one row to the start of the next has dSumAbs=2.

## dDiffXY and dDiffYX

"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

"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, measured 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.

## Dir4

"Dir4" is the curve step direction as an angle scaled to range 0 to 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```

## TDir6

"TDir6" is the curve step direction in the triangular style of "Triangular Lattice" in Math::PlanePath. So dX=1,dY=1 is 60 degrees and a full circle ranges 0 to 6.

```      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 to become an ellipse.

`    TDir6 = atan2(dY*sqrt(3), dX)      in range 0 <= TDir6 < 6`

Notice that angles for dX=0 or dY=0 which are the axes are not changed by the sqrt(3) factor. So TDir6 has ENWS 0, 1.5, 3, 4.5 which is in steps of 1.5. Verticals North and South normally doesn't occur in the triangular lattice paths but TDir6 can be applied to other paths.

The sqrt(3) factor increases angles in the middle of the quadrants away from the axes. 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.

# FUNCTIONS

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```

`planepath` can 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 `\$i` in the sequence. This is the position `\$seq->rewind()` returns to.

This is `\$path->n_start()` from the PlanePath.

Math::PlanePath

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