Min and Max¶

"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¶

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¶

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 and Diff¶

"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¶

"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¶

"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¶

"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¶

Radius and RSquared are per "$path->n_to_radius()" and "$path->n_to_rsquared()" respectively (see "Coordinate Methods" in Math::PlanePath).

TRadius and TRSquared¶

"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 and FracXY¶

"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, BitXor¶

"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.