Unit flipping

Zuguang Gu (z.gu@dkfz.de)

2024-09-05

If the level-1 unit is not symmetric, there will be a flipped version of the expansion rules. The flip can happen on the diagonal if the corner value of the unit is 11/22 or vertical/horizontal if the corner value of the unit is 12/21.

We will use the Peano curve as an example because it is easier to identify different flipped patterns by eyes.

p = sfc_3x3_peano("I", rot = 90, level = 3)

Flip individual units

Each unit on a certain level of the curve can be independently flipped. The unit is specified by its “hierarchy index”. The units are organized in a hierarchical way on the curve. See the following examples:

par(mfrow = c(1, 3))
sfcurve:::test_sfc_index(p, "") # the complete curve
sfcurve:::test_sfc_index(p, "4")
sfcurve:::test_sfc_index(p, "4:2")

sfc_flip_unit() flips a single unit or a list of units. Whether it is a diagonal flip or a vertical/horizontal flip is automatically detected.

p2 = sfc_flip_unit(p, "7:3") # the most topright 3x3 unit 
p3 = sfc_flip_unit(p, as.character(1:9)) # every level-2 square units
draw_multiple_curves(
    p, p2, p3, 
    nrow = 1, lwd = 2)

Flip level-1 units

There is a helper function change_level1_unit_orientation() which change the orientation of all level-1 units simultaneously.

p2 = change_level1_unit_orientation(p,  "vertical")
p3 = change_level1_unit_orientation(p,  "horizontal")
draw_multiple_curves(
    p, p2, p3, 
    nrow = 1, lwd = 2)

Note this type of flipping does not affect the higher level structure.

It internally applies sfc_apply() on the depth of sfc_level(p)-1 (i.e. level = 1) which will introduced in later sections.

Set flipping when creating the curve

As has already shown in the vignette “Generalized Peano Curve”, sfc_3x3_peano() accepts its argument flip as a self-defined function.

p2 = sfc_3x3_peano("I", level = 3, flip = function(p) {
    p@rot %in% c(90, 270)
})
p3 = sfc_3x3_peano("I", level = 3, flip = function(p) {
    n = length(p)
    l = rep(FALSE, n)
    ind = seq(1, n/3)
    l[ind] = p@rot[ind] %in% c(90, 270)

    ind = seq(n/3+1, n/3*2)
    l[ind] = p@rot[ind] %in% c(0, 180)

    ind = seq(n/3*2+1, n)
    l[ind] = p@rot[ind] %in% c(90, 270)
    l
})
draw_multiple_curves(p, p2, p3, 
    nrow = 1, lwd = 2)

Being different from change_level1_unit_orientation() which is only applied on level 1, the self-defined function flip() is appied to every level of the curve.

Adjust flipping on a generated curve

We can also adjust the structure of the curve by sfc_apply() on a certain depth. To better control the flipping patterns, we use p2 where all level-1 units are all vertical.

p3 = sfc_apply(p2, 1, function(x, i) {
    if(i %% 2 == 1) {
        sfc_flip_unit(x)
    } else {
        x
    }
})
p4 = sfc_apply(p2, 2, function(x, i) {
    if(i %% 9 %in% 4:6) {
        sfc_flip_unit(x, to = "horizontal")
    } else {
        x
    }
})
draw_multiple_curves(
    p, p3, p4, 
    nrow = 1, lwd = 2)

Seed as a sequence

sfc_flip_unit(), change_level1_unit_orientation() and sfc_apply() only work on curves with a single letter as the seed because internally they use sfc_index() to get subunits and sfc_index() only works on square curves. Only the “flip” function works when the seed is a sequence.

sfc_3x3_peano("IJ", level = 3, rot = 90, flip = function(p) {
    p@rot %in% c(90, 270)
}) |> plot()