Level Ranges 4^k¶

The X,Y extents of the path through to Nlevel=2^k can be expressed as a width and height measured relative to the endpoints.

       *------------------*       <-+
       |                  |         |
    *--*                  *--*      | height h[k]
    |                        |      |
    *   N=4^k         N=0    *    <-+
    |     |            |     |      | below l[k]
    *--*--*            *--*--*    <-+
    ^-----^            ^-----^
     width     2^k      width
      w[k]               w[k]           Extents to N=4^k
    <------------------------>
    total width = 2^k + 2*w[k]

N=4^k is on either the X or Y axis and for the extents here it's taken rotated as necessary to be horizontal. k=2 N=4^2=16 shown above is already horizontal. The next level k=3 N=64=4^3 would be rotated -90 degrees to be horizontal.

The width w[k] is measured from the N=0 and N=4^k endpoints. It doesn't include the 2^k length between those endpoints. The two ends are symmetric so the extent is the same at each end.

    h[k] = 2^k - 1                     0,1,3,7,15,31,etc
    w[k] = /  0            for k=0
           \  2^(k-1) - 1  for k>=1    0,0,1,3,7,15,etc
    l[k] = /  0            for k<=1
           \  2^(k-2) - 1  for k>=2    0,0,0,1,3,7,etc

The initial N=0 to N=64 shown above is k=3. h[3]=7 is the X=-7 horizontal. l[3]=1 is the X=1 horizontal. w[3]=3 is the vertical Y=3, and also Y=-11 which is 3 below the endpoint N=64 at Y=8.

Expressed as a fraction of the 2^k distance between the endpoints the extents approach total 2 wide by 1.25 high,

       *------------------*       <-+
       |                  |         |  1
    *--*                  *--*      |         total
    |                        |      |         height
    *   N=4^k         N=0    *    <-+         -> 1+1/4
    |     |            |     |      |  1/4
    *--*--*            *--*--*    <-+
    ^-----^            ^-----^
      1/2        1       1/2     total width -> 2

The extent formulas can be found by considering the self-similar blocks. The initial k=0 is a single line segment and all its extents are 0.

                          h[0] = 0
          N=1 ----- N=0
                          l[0] = 0
                    w[0] = 0

Thereafter the replication overlap as

       +-------+---+-------+
       |       |   |       |
    +------+   |   |   +------+
    |  | D |   | C |   | B |  |        <-+
    |  +-------+---+-------+  |          | 2^(k-1)
    |      |           |      |          | previous
    |      |           |      |          | level ends
    |    E |           | A    |        <-+
    +------+           +------+
         ^---------------^
        2^k this level ends
    w[k] =           max (h[k-1], w[k-1])  # right of A,B
    h[k] = 2^(k-1) + max (h[k-1], w[k-1])  # above B,C,D
    l[k] = max w[k-1], l[k-1]-2^(k-1)      # below A,E

Since h[k]=2^(k-1)+w[k] have h[k] > w[k] for k>=1 and with the initial h[0]=w[k]=0 have h[k]>=w[k] always. So the max of those two is h.

    h[k] = 2^(k-1) + h[k-1]  giving h[k] = 2^k-1     for k>=1
    w[k] = h[k-1]            giving w[k] = 2^(k-1)-1 for k>=1

The max for l[k] is always w[k-1] as l[k] is never big enough that the parts B-C and C-D can extend down past their 2^(k-1) vertical position. (l[0]=w[0]=0 and thereafter by induction l[k]<=w[k].)

    l[k] = w[k-1]   giving l[k] = 2^(k-2)-1 for k>=2

Repeated Points¶

The curve crosses itself and can repeat X,Y positions up to 4 times. The first double, triple and quadruple points are at

    visits      X,Y         N
    ------    -------    ----------------------
       2       -2,  3       7,    9
       3       18, -7     189,  279,  281
       4      -32, 55    1727, 1813, 2283, 2369

Each line segment between integer points is traversed at most 2 times, once forward and once backward. There's 4 lines reaching each integer point and this line traversal means the points are visited at most 4 times.

As per "Direction" below the direction of the curve is given by the count of 1-bits in N. Since no line is repeated each of the N values at a given X,Y have a different count-1-bits mod 4. For example N=7 is 3 1-bits and N=9 is 2 1-bits. The full counts need not be consecutive, as for example N=1727 is 9 1-bits and N=2369 is 4 1-bits.

The maximum of 2 line segment traversals can be seen from the way the curve replicates. Suppose the entire plane had all line segments traversed forward and backward.

      v |         v |
    --   <--------   <-
     [0,1]       [1,1]           [X,Y] = integer points
    ->   -------->   --          each edge traversed
      | ^         | ^            forward and backward
      | |         | |
      | |         | |
      v |         v |
    --   <--------   <--
     [0,0]       [1,0]
    ->   -------->   --
      | ^         | ^

Then when each line segment expands on the right the result is the same pattern of traversals -- viewed rotated by 45-degrees and scaled by factor sqrt(2).

     \ v / v        \ v  / v
      [0,1]           [1,1]
     / / ^ \         ^ / ^ \
    / /   \ \       / /   \ \
           \ \     / /
            \ v   / v
             [1/2,1/2]
            ^ /   ^ \
           / /     \ \
    \ \   / /       \ \   / /
     \ v / v         \ v / v
      [0,0]            1,0
     ^ / ^ \         ^ / ^ \

The curve is a subset of this pattern. It begins as a single line segment which has this pattern and thereafter the pattern preserves itself. Hence at most 2 segment traversals in the curve.

Tiling¶

The segment traversal argument above can also be made by taking the line segments as triangles which are a quarter of a unit square with peak pointing to the right of the traversal direction.

       to  *
           ^\
           | \
           |  \ triangle peak
           |  /
           | /
           |/       quarter of a unit square
      from *

These triangles in the two directions tile the plane. On expansion each splits into 2 halves in new positions. Those parts don't overlap and the plane is still tiled. See for example

For the integer version of the curve this kind of tiling can be used to combine copies of the curve so that each every point is visited precisely 4 times. The h[k], w[k] and l[k] extents above are less than the 2^k endpoint length, so a square of side 2^k can be fully tiled with copies of the curve at each corner,

             | ^         | ^
             | |         | |               24 copies of the curve
             | |         | |               to visit all points of the
             v |         v |               inside square ABCD
    <-------    <--------   <--------      precisely 4 times each
              A           B
    -------->   -------->   -------->      each part points
             | ^         | ^               N=0 to N=4^k-1
             | |         | |               rotated and shifted
             | |         | |               suitably
             v |         v |
    <--------   <--------   <--------
              C           D
    --------    -------->   -------->
             | ^         | ^
             | |         | |
             | |         | |
             v |         v |

The four innermost copies of the curve cover most of the inside square, but the other copies surrounding them loop into the square and fill in the remainder to make 4 visits at every point.

It's interesting to note that a set of 8 curves at the origin only covers the axes with 4-fold visits,

             | ^              8 arms at the origin
             | |              cover only X,Y axes
             v |              with 4-visits
    <--------   <--------
             0,0              away from the axes
    --------    -------->     some points < 4 visits
             | ^
             | |
             v |

This means that if the path had some sort of "arms" of multiple curves extending from the origin then it would visit all points on the axes X=0 Y=0 a full 4 times, but off the axes there would be points without full 4 visits.

See examples/c-curve-wx.pl for a wxWidgets program drawing various forms and tilings of the curve.

Level Methods¶

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

Return "(0, 2**$level)".

Direction¶

The direction or net turn of the curve is the count of 1 bits in N,

    direction = count_1_bits(N) * 90degrees

For example N=11 is binary 1011 has three 1 bits, so direction 3*90=270 degrees, ie. to the south.

This bit count is because at each power-of-2 position the curve is a copy of the lower bits but turned +90 degrees, so +90 for each 1-bit.

For powers-of-2 N=2,4,8,16, etc, there's only a single 1-bit so the direction is always +90 degrees there, ie. always upwards.

Turn¶

At each point N the curve can turn in any direction: left, right, straight, or 180 degrees back. The turn is given by the number of low 0-bits of N,

    turn right = (count_low_0_bits(N) - 1) * 90degrees

For example N=8 is binary 0b100 which is 2 low 0-bits for turn=(2-1)*90=90 degrees to the right.

When N is odd there's no low zero bits and the turn is always (0-1)*90=-90 to the right, so every second turn is 90 degrees to the left.

Next Turn¶

The turn at the point following N, ie. at N+1, can be calculated by counting the low 1-bits of N,

    next turn right = (count_low_1_bits(N) - 1) * 90degrees

For example N=11 is binary 0b1011 which is 2 low one bits for nextturn=(2-1)*90=90 degrees to the right at the following point, ie. at N=12.

This works simply because low 1-bits like ..0111 increment to low 0-bits ..1000 to become N+1. The low 1-bits at N are thus the low 0-bits at N+1.

N to dX,dY¶

"n_to_dxdy()" is implemented using the direction described above. For integer N the count mod 4 gives the direction for dX,dY.

    dir = count_1_bits(N) mod 4
    dx = dir_to_dx[dir]    # table 0 to 3
    dy = dir_to_dy[dir]

For fractional N the direction at int(N)+1 can be obtained from the direction at int(N) and the turn at int(N)+1, which is the low 1-bits of N per "Next Turn" above. Those two directions can then be combined as described in "N to dX,dY -- Fractional" in Math::PlanePath.

    # apply turn to make direction at Nint+1
    turn = count_low_1_bits(N) - 1      # N integer part
    dir = (dir - turn) mod 4            # direction at N+1
    # adjust dx,dy by fractional amount in this direction
    dx += Nfrac * (dir_to_dx[dir] - dx)
    dy += Nfrac * (dir_to_dy[dir] - dy)

A small optimization can be made by working the "-1" of the turn formula into a +90 degree rotation of the "dir_to_dx[]" and "dir_to_dy[]" parts by swap and sign change,

    turn_plus_1 = count_low_1_bits(N)     # on N integer part
    dir = (dir - turn_plus_1) mod 4       # direction-1 at N+1
    # adjustment including extra +90 degrees on dir
    dx -= $n*(dir_to_dy[dir] + dx)
    dy += $n*(dir_to_dx[dir] - dy)

X,Y to N¶

The N values at a given X,Y can be found by taking terms low to high from the complex number formula (the same as given above)

    X+iY = b^k            N = 2^k + 2^(k1) + 2^(k2) + ... in binary
         + b^k1 * i       base b=1+i
         + b^k2 * i^2
         + ...

If the lowest term is b^0 then X+iY has X+Y odd. If the lowest term is not b^0 but instead some power b^n then X+iY has X+Y even. This is because a multiple of b=1+i,

    X+iY = (x+iy)*(1+i)
         = (x-y) + (x+y)i
    so X=x-y Y=x+y
    sum X+Y = 2x is even if X+iY a multiple of 1+i

So the lowest bit of N is found by

    bit = (X+Y) mod 2

If bit=1 then a power i^p is to be subtracted from X+iY. p is how many 1-bits are above that point, and this is not yet known. It represents a direction to move X,Y to put it on an even position. It's also the direction of the step N-2^l to N, where 2^l is the lowest 1-bit of N.

The reduction should be attempted with p commencing as each of the four possible directions N,S,E,W. Some or all will lead to an N. For quadrupled points (such as X=-32, Y=55 described above) all four will lead to an N.

    for p 0 to 3
      dX,dY = i^p   # directions [1,0]  [0,1]  [-1,0]  [0,-1]
      loop until X,Y = [0,0] or [1,0] or [-1,0] or [0,1] or [0,-1]
      {
        bit = X+Y mod 2       # bits of N from low to high
        if bit == 1 {
          X -= dX             # move to "even" X+Y == 0 mod 2
          Y -= dY
          (dX,dY) = (dY,-dX)       # rotate -90 as for p-1
        }
        X,Y = (X+Y)/2, (Y-X)/2   # divide (X+iY)/(1+i)
      }
      if not (dX=1 and dY=0)
        wrong final direction, try next p
      if X=dX and Y=dY
        further high 1-bit for N
        found an N
      if X=0 and Y=0
        found an N

The "loop until" ends at one of the five points

            0,1
             |
    -1,0 -- 0,0 -- 1,0
             |
            0,-1

It's not possible to wait for X=0,Y=0 to be reached because some dX,dY directions will step infinitely among the four non-zeros. Only the case X=dX,Y=dY is sure to reach 0,0.

The successive p decrements which rotate dX,dY by -90 degrees must end at p == 0 mod 4 for highest term in the X+iY formula having i^0=1. This means must end dX=1,dY=0 East. If this doesn't happen then there is no N for that p direction.

The number of 1-bits in N is == p mod 4. So the order the N values are obtained follows the order the p directions are attempted. In general the N values will not be smallest to biggest N so a little sort is necessary if that's desired.

It can be seen that sum X+Y is used for the bit calculation and then again in the divide by 1+i. It's convenient to write the whole loop in terms of sum S=X+Y and difference D=Y-X.

    for dS = +1 or -1      # four directions
      for dD = +1 or -1    #
        S = X+Y
        D = Y-X
        loop until -1 <= S <= 1 and -1 <= D <= 1 {
          bit = S mod 2       # bits of N from low to high
          if bit == 1 {
            S -= dS              # move to "even" S+D == 0 mod 2
            D -= dD
            (dS,dD) = (dD,-dS)   # rotate -90
          }
          (S,D) = (S+D)/2, (D-S)/2   # divide (S+iD)/(1+i)
        }
        if not (dS=1 and dD=-1)
          wrong final direction, try next dS,dD direction
        if S=dS and D=dD
          further high 1-bit for N
          found an N
        if S=0 and D=0
          found an N

The effect of S=X+Y, D=Y-D is to rotate by -45 degrees and use every second point of the plane.

    D= 2                      X=0,Y=2       .              rotate -45
    D= 1            X=0,Y=1      .       X=1,Y=2       .
    D= 0  X=0,Y=0      .      X=1,Y=1       .       X=2,Y=2
    D=-1            X=1,Y=0      .       X=2,Y=1       .
    D=-2                      X=2,Y=0       .
           S=0        S=1       S=2        S=3        S=4

The final five points described above are then in a 3x3 block at the origin. The four in-between points S=0,D=1 etc don't occur so range tests -1<=S<=1 and -1<=D<=1 can be used.

     S=-1,D=1      .      S=1,D=1
        .       S=0,D=0      .
     S=-1,D=-1     .      S=1,D=-1

Segments by Direction¶

In a level N=0 to N=2^k-1 inclusive, the number of segments in each direction 0=East, 1=North, 2=West, 3=South are given by

           k=0        for k >= 1
           ---        ----------
    M0[k] = 1,    2^(k-2) + d(k+2)*2^(h-1)
    M1[k] = 0,    2^(k-2) + d(k+0)*2^(h-1)
    M2[k] = 0,    2^(k-2) + d(k-2)*2^(h-1)
    M3[k] = 0,    2^(k-2) + d(k-4)*2^(h-1)
    where h = floor(k/2)
    and   d(m) = 0  1  1  1  0 -1 -1 -1
                 for m == 0 to 7 mod 8
    M0[k] = 1, 1, 1, 1, 2,  6, 16, 36, 72, 136, 256, ...
    M1[k] = 0, 1, 2, 3, 4,  6, 12, 28, 64, 136, 272, ...
    M2[k] = 0, 0, 1, 3, 6, 10, 16, 28, 56, 120, 256, ...
    M3[k] = 0, 0, 0, 1, 4, 10, 20, 36, 64, 120, 240, ...

d(n) is a factor +1, -1 or 0 according to n mod 8. Each M goes as a power 2^(k-2), so roughly 1/4 each, but a half power 2^(h-1) possibly added or subtracted in a k mod 8 pattern. In binary this is a 2^(k-2) high 1-bit with another 1-bit in the middle added or subtracted.

The total is 2^k since there are a total 2^k points from N=0 to 2^k-1 inclusive.

    M0[k] + M1[k] + M2[k] + M3[k] = 2^k

It can be seen that the d(n) parts sum to 0 so the 2^(h-1) parts cancel out leaving 4*2^(k-2) = 2^k.

    d(0) + d(2) + d(4) + d(6) = 0
    d(1) + d(3) + d(5) + d(7) = 0

The counts can be calculated in two ways. Firstly they satisfy mutual recurrences. Each adds the preceding rotated M.

    M0[k+1] = M0[k] + M3[k]        initially M0[0] = 1 (N=0 to N=1)
    M1[k+1] = M1[k] + M0[k]                  M1[0] = 0
    M2[k+1] = M2[k] + M1[k]                  M2[0] = 0
    M3[k+1] = M3[k] + M2[k]                  M3[0] = 0

Geometrically this can be seen from the way each level extends by a copy of the previous level rotated +90,

    7---6---5            Easts in N=0 to 8
    |       |            =   Easts in N=0 to 4
    8       4---3          + Wests in N=0 to 4
                |             since N=4 to N=8 is
                2             the N=0 to N=4 rotated +90
                |
            0---1

For the bits in N, level k+1 introduces a new bit either 0 or 1. In M0[k+1] the a 0-bit is count M0[k] the same direction, and when a 1-bit is M3[k] since one less bit mod 4. Similarly the other counts.

Some substitutions give 3rd order recurrences

    for k >= 4
    M0[k] = 4*M0[k-1] - 6*M0[k-2] + 4*M0[k-3]    initial 1,1,1,1
    M1[k] = 4*M1[k-1] - 6*M1[k-2] + 4*M1[k-3]    initial 0,1,2,3
    M2[k] = 4*M2[k-1] - 6*M2[k-2] + 4*M2[k-3]    initial 0,0,1,3
    M3[k] = 4*M3[k-1] - 6*M3[k-2] + 4*M3[k-3]    initial 0,0,0,1

The characteristic polynomial of these recurrences is

    x^3 - 4x^2 + 6x - 4
    = (x-2) * (x - (1-i)) * (x - (1+i))

So explicit formulas can be written in powers of the roots 2, 1-i and 1+i,

    M0[k] = ( 2^k +   (1-i)^k +   (1+i)^k )/4      for k>=1
    M1[k] = ( 2^k + i*(1-i)^k - i*(1+i)^k )/4
    M2[k] = ( 2^k -   (1-i)^k -   (1+i)^k )/4
    M3[k] = ( 2^k - i*(1-i)^k + i*(1+i)^k )/4

The complex numbers 1-i and 1+i are 45 degree lines clockwise and anti-clockwise respectively. The powers turn them in opposite directions so the imaginary parts always cancel out. The remaining real parts can be had by a half power h=floor(k/2) which is the magnitude abs(1-i)=sqrt(2) projected onto the real axis. The sign selector d(n) above is whether the positive or negative part of the real axis, or zero when at the origin.

The second way to calculate is the combinatorial interpretation that per "Direction" above the direction is count_1_bits(N) mod 4 so East segments are all N values with count_1_bits(N) == 0 mod 4, ie. N with 0, 4, 8, etc many 1-bits. The number of ways to have those bit counts within total k bits is k choose 0, 4, 8 etc.

    M0[k] = /k\ + /k\ + ... + / k\      m = floor(k/4)
            \0/   \4/         \4m/
    M1[k] = /k\ + /k\ + ... + / k  \    m = floor((k-1)/4)
            \1/   \5/         \4m+1/
    M2[k] = /k\ + /k\ + ... + / k  \    m = floor((k-2)/4)
            \2/   \6/         \4m+2/
    M3[k] = /k\ + /k\ + ... + / k  \    m = floor((k-3)/4)
            \3/   \7/         \4m+3/

The power forms above are cases of the identity by Ramus for sums of binomial coefficients in arithmetic progression like this. (See Knuth volume 1 section 1.2.6 exercise 30 for a form with cosines resulting from w=i+1 as 8th roots of unity.)

The total M0+M1+M2+M3=2^k is the total binomials across a row of Pascal's triangle.

    /k\ + /k\ + ... + /k\ = 2^k
    \0/   \1/         \k/

It's interesting to note the M counts here are the same in the dragon curve (Math::PlanePath::DragonCurve). The shapes of the curves are different since the segments are in a different order, but the total puts points N=2^k at the same X,Y position.

Right Boundary¶

The length of the right-side boundary of the curve, which is the outside of the "C", from N=0 to N=2^k is

    R[k] = /  7*2^h - 2k - 6     if k even
           \ 10*2^h - 2k - 6     if k odd
           where h = floor(k/2)
         = 1, 2, 4, 8, 14, 24, 38, 60, 90, 136, 198, 292, 418, ...
    R[k] =   (7/2 + 5/2 * sqrt(2)) * ( sqrt(2))^k
           + (7/2 - 5/2 * sqrt(2)) * (-sqrt(2))^k
           - 2*k - 6
    R[k] = 2*R[k-1] + R[k-2] - 4*R[k-3] + 2*R[k-4]

The length doubles until R[4]=14 which is points N=0 to N=2^4=16. At k=4 the points N=7,8,9 have turned inward and closed off some of the outside of the curve so the boundary less than 2x.

        11--10--9,7--6--5        right boundary
         |       |      |        around "outside"
    13--12       8      4--3     N=0 to N=2^4=16
     |                     |
    14                     2        R[4]=14
     |                     |
    15--16              0--1

The floor(k/2) and odd/even cases are eliminated by the +/-sqrt(2) powering shown. Those powers are also per the characteristic equation of the recurrence,

    x^4 - 2*X^3 - x^2 + 4*x - 2
      = (x - 1)^2 * (x + sqrt(2)) * (x - sqrt(2))
    roots 1, sqrt(2), -sqrt(2)

The right boundary comprises runs of straight lines and zig-zags. When it expands the straight lines become zig-zags and the zig-zags become straight lines. The straight lines all point "forward", which is anti-clockwise.

                                      c     *     a
                                     / ^   / ^   / ^
                            =>      v   \ v   \ v   \
    D<----C<----B<----A            D     C     B     A
    |                 ^           /                   ^
    v                 |          v                     \
       straight S=3                zig-zag Z[k+1] = 2S[k]-2 = 4

The count Z here is both sides of each "V" shape from points "a" through to "c". So Z counts the boundary length (rather than the number of "V"s). Each S becomes an upward peak. The first and last side of those peaks become part of the following "straight" section (at A and D), hence Z[k+1]=2*S[k]-2.

The zigzags all point "forward" too. When they expand they close off the V shape and become 2 straight lines for each V, which means 1 straight line for each Z side. The segment immediately before and after contribute a segment to the resulting straight run too, hence S[k+1]=Z[k]+2.

         C     B     A               *<---C<---*<---B<---*<---A<---*
        / ^   / ^   / ^              |         |         |         |
       v   \ v   \ v   \      =>     |         |         |         |
      *     *     *     *         <--*         *         *         *<--
     /                   ^
    v                     \
      zig-zag Z=4 segments             straight S[k+1] = Z[k]+2 = 6

The initial N=0 to N=1 is a single straight segment S[0]=1 and from there the runs grow. N=1 to N=3 is a straight section S[1]=2. Z[0]=0 represents an empty zigzag at N=1. Z[1] is the first non-empty at N=3 to N=5.

     h   S[h]     Z[h]       Z[h]   = 2*S[h]-2
    --   ----     ----       S[h+1] = Z[h]+2
     0     1        0
     1     2        2        S[h+1] = 2*S[h]-2+2 = 2*S[h]
     2     4        6        so
     3     8       14        S[h] = 2^h
     4    16       30        Z[h] = 2*2^h-2
     5    32       62
     5    64      126

The curve N=0 to N=2^k is symmetric at each end and is made up of runs S[0], Z[0], S[1], Z[1], etc, of straight and zigzag alternately at each end. When k is even there's a single copy of a middle S[k/2]. When k is odd there's a single middle Z[(k-1)/2] (with an S[(k-1)/2] before and after). So

                / i=h-1          \           # where h = floor(k/2)
    R[k] = 2 * | sum   S[i]+Z[i]  |
                \ i=0            /
           + S[h]
           + / S[h]+Z[h]  if k odd
             \ 0          if k even
         =  2*(  1+2+4+...+2^(h-1)           # S[0] to S[h-1]
               + 2+4+8+...+2^h  -  2*h)      # Z[0] to Z[h-1]
           + 2^h                             # S[h]
           + if k odd (2^h + 2*2^h - 2)      # possible S[h]+Z[h]
         = 2*(2^h-1 + 2*2^h-2 - 2h) + 2^h + (k odd 3*2^h - 2)
         = 7*2^h - 4h-6 + (if k odd then + 3*2^h - 2)
         = 7*2^h - 2k-6 + (if k odd then + 3*2^h)

Convex Hull Boundary¶

A convex hull is the smallest convex polygon which contains a given set of points. For the C curve the boundary length of the convex hull for points N=0 to N=2^k inclusive is

    hull boundary[k]
        / 2                                    if k=0
        | 2+sqrt(2)                            if k=1
      = | 6                                    if k=2
        | 6*2^(h-1) + (7*2^(h-1) - 4)*sqrt(2)  if k odd  >=3
        \ 7*2^(h-1) + (3*2^(h-1) - 4)*sqrt(2)  if k even >=4
    where h = floor(k/2)
      k              hull boundary
     ---      ----------------------------
      0        2 +  0 * sqrt(2)  =    2
      1        2 +  1 * sqrt(2)  =    3.41
      2        6 +  0 * sqrt(2)  =    6
      3        6 +  3 * sqrt(2)  =   10.24
      4       14 +  2 * sqrt(2)  =   16.82
      5       12 + 10 * sqrt(2)  =   26.14
      6       28 +  8 * sqrt(2)  =   39.31
      7       24 + 24 * sqrt(2)  =   57.94
      8       56 + 20 * sqrt(2)  =   84.28
      9       48 + 52 * sqrt(2)  =  121.53

The integer part is the straight sides of the hull and the sqrt(2) part is the diagonal sides of the hull.

When k is even the hull has the following shape. The sides are as per the right boundary above but after Z[h-2] the curl goes inwards and so parts beyond Z[h-2] are not part of the hull. Each Z stair-step diagonal becomes a sqrt(2) length for the hull. Z counts both vertical and horizontal of each stairstep, hence sqrt(2)*Z/2 for the hull boundary.

                  S[h]
               *--------*                  *       Z=2
      Z[h-1]  /          \  Z[h-1]         | \     diagonal
             /            \                |  \    sqrt(2)*Z/2
            *              *               *----*  = sqrt(2)
    S[h-1]  |              |  S[h-1]
            |              |
            *              *
      Z[h-2] \            / Z[h-2]
              *--      --*
                S[h] + Z[h-2]-Z[h-1]
    k even
    hull boundary[k] = S[h] + 2*S[h-1] + S[h+Z[h-2]-Z[h-1]
                       + sqrt(2)*(2*Z[h-1] + 2*Z[h-2])/2

When k is odd the shape is similar but Z[h] in the middle.

                        S[h]
                       *----*
               Z[h-1] /      \  middle
                     *        \  Z[h]
             S[h-1]  |         \
                     *          *
                      \         |  S[h]
          Z[h]                  |
    + 2*(S[h]-S[h-1])           *
                         \     /  Z[h-1]
                          *---*
                          S[h-1]
    k odd
    hull boundary[k] = 2*S[h] + 2*S[h-1]
                       + sqrt(2)*(Z[h]/2 + 2*Z[h-1]/2
                                  + Z[h]/2 + S[h]-S[h-1]

Convex Hull Area¶

The area of the convex hull for points N=0 to N=2^k inclusive is

            / 0                                if k=0
            | 1/2                              if k=1
    HA[k] = | 2                                if k=2
            | 35*2^(k-4) - 13*2^(h-1) + 2      if k odd  >=3
            \ 35*2^(k-4) - 10*2^(h-1) + 2      if k even >=4
          where h = floor(k/2)
    = 0, 1/2, 2, 13/2, 17, 46, 102, 230, 482, 1018, 2082, 4274, ...

HA[1] and HA[3] are fractions but all others are integers.

The area can be calculated from the shapes shown for the hull boundary above. For k odd it can be noted the width and height are equal, then the various corners are cut off.

Line Points¶

The number of points which fall on straight and diagonal lines from the endpoints can be calculated by considering how the previous level duplicates to make the next.

              d   d
            c  \ /  c
        b   |   +   |   b
         \  |  / \  |  /           curve endpoints
          \ | /   \ | /             "S" start
           \|/     \|/              "E" end
     a------E---e---S------a
           /|\     /|\
          / | \   / | \
         /  |  f f  |  \
        h   g       g   h

The curve is rotated to make the endpoints horizontal. Each "a" through "h" is the number of points which fall on the respective line. The curve is symmetric in left to right so the line counts are the same each side in mirror image.

"S" start and "E" end points are not included in any of the counts. "e" is the count in between S and E. The two "d" lines meet at point "+" and that point is counted in d. That point is where two previous level curves meet for k>=1. Points are visited up to 4 times (per "Repeated Points" above) and all those multiple visits are counted.

The following diagram shows how curve level k+1 is made from two level k curves. One is from S to M and another M to E.

            |\       /|            curve level k copies
            | \     / |            S to M and M to E
            | c+a c+a |            making curve k+1 S to E
            |   \|/   |
       \    |  --M--  |    /
        \   |   /|\   |   c        a[k+1] = b[k]
         c  d e+g e+g d  /         b[k+1] = c[k]
          \ | /     \ | /          c[k+1] = d[k]
           \|/       \|/           d[k+1] = a[k]+c[k] + e[k]+g[k] + 1
    b-------E--f---f--S-------b    e[k+1] = 2*f[k]
           /|\       /|\           f[k+1] = g[k]
          a | g     g | a          g[k+1] = h[k]
         /  h  \   /  h  \         h[k+1] = a[k]
        /   |   \ /   |   \
       /    |         |    \

For example the line S to M is an e[k], but also the M to E contributes a g[k] on that same line so e+g. Similarly c[k] and a[k] on the outer sides of M. Point M itself is visited too so the grand total for d[k+1] is a+c+e+g+1. The other lines are simpler, being just rotations except for the middle line e[k+1] which is made of two f[k].

The successive g[k+1]=h[k]=a[k-1]=b[k-2]=c[k-3]=d[k-4] can be substituted into the d to give a recurrence

    d[k+1] = d[k-1] + d[k-3] + d[k-5] + 2*d[k-7] + 1
           = 0,1,1,2,2,4,4,8,8,17,17,34,34,68,68,136,136,273,273,...
                               x + x^2          (common factor 1+x
    generating function  -------------------     in numerator and
                         (1-2*x^2) * (1-x^8)     denominator)
    d[2h-1] = d[2h] = floor( 8/15 * 2^h )

The recurrence begins with the single segment N=0 to N=1 and the two endpoints are not included so initial all zeros a[0]=...=h[0]=0.

As an example, the N=0 to N=64 picture above is level k=6 and its "d" line relative to those endpoints is the South-West diagonal down from N=0. The points on that line are N=32,30,40,42 giving d[6]=4.

All the measures are relative to the endpoint direction. The points on the fixed X or Y axis or diagonal can be found by taking the appropriate a through h, or sum of two of them for both positive and negative of a direction.