AbsdX,AbsdY

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

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

For example "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 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, 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, dRSquared

"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

"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

"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.