sfc_shape: Shape of the curve

In sfcurve: 2x2, 3x3 and Nxn Space-Filling Curves

Shape of the curve

Description

Shape of the curve

Usage

## S4 method for signature 'sfc_2x2'
sfc_shape(p)

all_2x2_shapes(level = 2)

## S4 method for signature 'sfc_3x3_peano'
sfc_shape(p)

all_3x3_peano_shapes(level = 2)

## S4 method for signature 'sfc_3x3_meander'
sfc_shape(p)

all_3x3_meander_shapes(level = 2)

Arguments

p	An sfc_2x2 object.
level	Level of the 2x2 curve.

Details

The shape of the curve is defined as a form of the curve without considering entry/exit directions, rotation, flipping (reflection) nor reversing.

2x2 curve

The process of selecting the shape segment of the curve denoted as P is:

1. The entry-point should locate in the bottom left subunit and the exit-point should locate in the bottom right subunit. We try the four rotations (0, 90, 180, 270), and the four rotations on the horizontally flipped curve. Once we find the transformed curve that satisfies this criterion, we name it as P2.

2. We also generate P3 which is a horizontally flipped version of rev(P2).

3. We compare the first point p of P2 and P3, and select the one whose p has the smaller x-coordinate (i.e. more to the left of the curve). If the x-coordinates of p are the same in P2 and P3, we select the one whose p has the smaller y-coordinate.

3x3 Peano curve

The process of selecting the shape segment of the curve denoted as P is:

1. The entry-point should locate in the bottom left subunit and the exit-point should locate in the top right subunit. We try the four rotations (0, 90, 180, 270). Once we find the transformed curve that satisfies this criterion, we name it as P2.

2. We also generate P3 which is a 180 degrees rotation on the reversed P2, P4 which is a diagonal flip with slop of 1 on P2 and P5 which is a diagonal flip with slop of 1 on P3.

3. We calculate the "UID" of P(2-5) and pick the one with the smallest UID as the final curve.

The UID of a 3x3 Peano curve is based on the hierarchical indices of the units on it. The hierarchy of the Peano curve is traversed in a depth-first manner. On each node, the orientation of the corresponding unit is calculated where vertical is 1 and horizontal is 2. The digits are concatenated into a long string.

all_3x3_peano_shapes() only calculates all shapes for Peano curve on level 2.

all_3x3_meander_shapes() only considers the Meander curve with all subunits on all levels in forward orientation.

Value

sfc_shape() returns a two-column data frame of the xy-coordinates of the shape curve.

all_2x2_shapes() returns a list of n two-column data frames where each data frame corresponds to the xy-coordnates of the corresponding shape curve.

Examples

p1 = sfc_2x2("I", 11)
p2 = sfc_2x2("R", 22)
draw_multiple_curves(
    p1, p2, 
    sfc_shape(p1), sfc_shape(p2), 
    col = "black")
sl = all_2x2_shapes(2)
draw_multiple_curves(list = sl, lwd = 2, col = "black")
sl = all_2x2_shapes(3)
draw_multiple_curves(list = sl, lwd = 2, col = "black")
p = sfc_3x3_peano("I", 11)
draw_multiple_curves(
    p, sfc_dflip(p),
    sfc_shape(p), sfc_shape(sfc_dflip(p)),
    col = "black")

sl = all_3x3_peano_shapes()
length(sl)
# the first 8 shapes
draw_multiple_curves(sl[1:8], col = "black")

sl = all_3x3_meander_shapes(2)
draw_multiple_curves(list = sl, lwd = 2, col = "black")

sfcurve documentation built on Sept. 14, 2024, 1:07 a.m.