Level Ranges¶

Within a power-of-2 square 2x2, 4x4, 8x8, 16x16 etc (2^k)x(2^k) at the origin, all the N values 0 to 2^(2*k)-1 are within the square. The maximum 3, 15, 63, 255 etc 2^(2*k)-1 is alternately at the top left or bottom right corner.

Because each step is by 1, the distance along the curve between two X,Y points is the difference in their N values (as from "xy_to_n()").

On the X=Y diagonal N=0,2,8,10,32,etc is the integers using only digits 0 and 2 in base 4, or equivalently have even-numbered bits 0, like x0y0...z0.

Locality¶

The Hilbert curve is fairly well localized in the sense that a small rectangle (or other shape) is usually a small range of N. This property is used in some database systems to store X,Y coordinates using the resulting Hilbert curve N as an index. A search through a 2-D region is then usually a fairly modest linear search through N values. "rect_to_n_range()" gives exact N range for a rectangle, or see notes "Rectangle to N Range" below for calculating on any shape.

The N range can be large when crossing sub-parts. In the sample above it can be seen for instance adjacent points X=0,Y=3 and X=0,Y=4 have rather widely spaced N values 5 and 58.

Fractional X,Y values can be indexed by extending the N calculation down into X,Y binary fractions. The code here doesn't do that, but could be pressed into service by moving the binary point in X and Y an even number of places, the same in each. (An odd number of bits would require swapping X,Y to compensate for the alternating transpose in part 0.) The resulting integer N is then divided down by a corresponding multiple-of-4 binary places.

Level Methods¶

"($n_lo, $n_hi) = $path->level_to_n_range($level)"

Return "(0, 4**$level - 1)".

N to X,Y¶

Converting N to X,Y coordinates is reasonably straightforward. The top two bits of N is a configuration

    3--2                    1--2
       |    or transpose    |  |
    0--1                    0  3

according to whether it's an odd or even bit-pair position. Then within each of the "3" sub-parts there's also inverted forms

    1--0        3  0
    |           |  |
    2--3        2--1

Working N from high to low with a state variable can record whether there's a transpose, an invert, or both, being four states altogether. A bit pair 0,1,2,3 from N then gives a bit each of X,Y according to the configuration and a new state which is the orientation of that sub-part. Bill Gosper's HAKMEM item 115 has this with either bit operations or a table for the state and X,Y bits,

And C++ code based on that in Jorg Arndt's book,

It also works to process N from low to high, at each stage applying any transpose (swap X,Y) and/or invert (bitwise NOT) to the low X,Y bits generated so far. This works because there's no "reverse" sections, or equivalently the curve is the same forward and reverse. Low to high saves locating the top bits of N, but if using bignums then the bitwise inverts of the full X,Y values will be much more work.

X,Y to N¶

X,Y to N can follow the table approach from high to low taking one bit from X and Y each time. The state table of N-pair -> X-bit,Y-bit is reversible, and a new state is based on the N-pair thus obtained (or could be based on the X,Y bits if that mapping is combined into the state transition table).

Rectangle to N Range¶

An easy over-estimate of the maximum N in a region can be had by finding the next bigger (2^k)x(2^k) square enclosing the region. This means the biggest X or Y rounded up to the next power of 2, so

    find lowest k with 2^k > max(X,Y)
    N_max = 2^(2k) - 1

Or equivalently rounding down to the next lower power of 2,

    find highest k with 2^k <= max(X,Y)
    N_max = 2^(2*(k+1)) - 1

An exact N range can be found by following the high to low N to X,Y procedure above. Start at the 2^(2k) bit pair position in an N bigger than the desired region and choose 2 bits for N to give a bit each of X and Y. The X,Y bits are based on the state table as above and the bits chosen for N are those for which the resulting X,Y sub-square overlaps some of the target region. The smallest N similarly, choosing the smallest bit pair for N which overlaps.

The biggest and smallest N digit for a sub-part can be found with a lookup table. The X range might cover one or both sub-parts, and the Y range similarly, for a total 4 possible configurations. Then a table of state+coverage -> digit gives the minimum and maximum N bit-pair, and state+digit gives a new state the same as X,Y to N.

Biggest and smallest N must be calculated with separate state and X,Y values since they track down different N bits and thus different states. But they take the same number of steps from an enclosing level down to level 0 and can thus be done in a single loop.

The N range for any shape can be found this way, not just a rectangle like "rect_to_n_range()". At each level the procedure only depends on asking which of the four sub-parts overlaps some of the target area.

Direction¶

Each step between successive N values is always 1 up, down, left or right. The next direction can be calculated from N in the high-to-low procedure above by watching for the lowest non-3 digit and noting the direction from that digit towards digit+1. That can be had from the state+digit -> X,Y table looking up digit and digit+1, or alternatively a further table encoding state+digit -> direction.

The reason for taking only the lowest non-3 digit is that in a 3 sub-part the direction it goes is determined by the next higher level. For example at N=11 the direction is down for the inverted-U of the next higher level N=0,4,8,12.

This non-3 (or non whatever highest digit) is a general procedure and can be used on any state-based high-to-low procedure of self-similar curves. In the current code, it's used to apply a fractional part of N in the correct direction but is not otherwise made directly available.

Because the Hilbert curve has no "reverse" sections it also works to build a direction from low to high N digits. 1 and 2 digits make no change to the orientation, 0 digit is a transpose, and a 3 digit is a rotate and transpose, except that low 3s are transpose-only (no rotate) for the same reason as taking the lowest non-3 above.

Jorg Arndt in the fxtbook above notes the direction can be obtained just by counting 3s in n and -n (the twos-complement). This numbers segments starting n=1, unlike PlanePath here starting N=0, so it becomes

    N+1 count 3s  / 0 mod 2   S or E
                  \ 1 mod 2   N or W
    -(N+1) count 3s  / 0 mod 2   N or E
                     \ 1 mod 2   S or W

For the twos-complement negation, an even number of base-4 digits of N must be taken. Because -(N+1) = ~N, ie. a ones-complement, the second part is also

    N count 0s          / 0 mod 2   N or E
    in even num digits  \ 1 mod 2   S or W

Putting the two together then

    N count 0s   N+1 count 3s    direction (0=E,1=N,etc)
    in base 4    in base 4
      0 mod 2      0 mod 2          0
      1 mod 2      0 mod 2          3
      0 mod 2      1 mod 2          1
      1 mod 2      1 mod 2          2

Segments in Direction¶

The number of segments in each direction is calculated in

Their form is based on keeping the top-most U shape fixed and expanding sub-parts. This means the end segments alternate vertical and horizontal in successive expansion levels.

    direction            k=1              2       2
      1 to 4                            *---*   *---*
                           2           1|  3|   |1  |3
        1                *---*          *   *---*   *
        |               1|   |3        1| 4   2   4 |3
    4--- ---2            *   *          *---*   *---*
        |                                  1|   |3       k=2
        3                               *---*   *---*
                                          2       2
    count segments in direction, for k >= 1
    d(1,k) = 4^(k-1)                = 1,4,16,64,256,1024,4096,...
    d(2,k) = 4^(k-1) + 2^(k-1) - 1  = 1,5,19,71,271,1055,4159,...
    d(3,k) = 4^(k-1)                = 1,4,16,64,256,1024,4096,...
    d(4,k) = 4^(k-1) - 2^(k-1)      = 0,2,12,56,240, 992,4032,...
                             (A000302, A099393, A000302, A020522)
    total segments d(1,k)+d(2,k)+d(3,k)+d(4,k) = 4^k - 1

The path form here keeps the first segment direction fixed. This means a transpose 1<->2 and 3<->4 in odd levels. The result is to take the alternate d values as follows. For k=0 there is a single point N=0 so no line segments at all and so c(dir,0)=0.

    first 4^k-1 segments
    c(1,k) = / 0                        if k=0
     North   | 4^(k-1) + 2^(k-1) - 1    if k odd >= 1
             \ 4^(k-1)                  if k even >= 2
      = 0, 1, 4, 19, 64, 271, 1024, 4159, 16384, ...
    c(2,k) = / 0                        if k=0
     East    | 4^(k-1)                  if k odd >= 1
             \ 4^(k-1) + 2^(k-1) - 1    if k even >= 2
      = 0, 1, 5, 16, 71, 256, 1055, 4096, 16511, ...
    c(3,k) = / 0                        if k=0
     South   | 4^(k-1) - 2^(k-1)        if k odd >= 1
             \ 4^(k-1)                  if k even >= 2
      = 0, 0, 4, 12, 64, 240, 1024, 4032, 16384, ...
    c(4,k) = / 0                        if k=0
     West    | 4^(k-1)                  if k odd >= 1
             \ 4^(k-1) - 2^(k-1)        if k even >= 2
      = 0, 1, 2, 16, 56, 256, 992, 4096, 16256, ...

The segment N=4^k-1 to N=4^k is North (direction 1) when k odd, or East (direction 2) when k even. That could be added to the respective cases in c(1,k) and c(2,k) if desired.

Hamming Distance¶

The Hamming distance between integers X and Y is the number of bit positions where the two values differ when written in binary. On the Hilbert curve each bit-pair of N becomes a bit of X and a bit of Y,

       N      X   Y
    ------   --- ---
    0 = 00    0   0
    1 = 01    1   0     <- difference 1 bit
    2 = 10    1   1
    3 = 11    0   1     <- difference 1 bit

So the Hamming distance for N=0to3 is 1 at N=1 and N=3. At higher levels, these X,Y bits may be transposed (swapped) or rotated by 180 or both. A transpose swapping X<->Y doesn't change the bit difference. A rotate by 180 is a flip 0<->1 of the bit in both X and Y, so that doesn't change the bit difference either.

On that basis, the Hamming distance X,Y is the number of base4 digits of N which are 01 or 11. If bit positions are counted from 0 for the least significant bit then

    X,Y coordinates of N
    HammingDist(X,Y) = count 1-bits at even bit positions in N
           = 0,1,0,1, 1,2,1,2, 0,1,0,1, 1,2,1,2, ... (A139351)

See also "Hamming Distance" in Math::PlanePath::CornerReplicate which is the same formula, but arising directly from 01 or 11, no transpose or rotate.