irregular_double_torus_surface_helpers: Weighted irregular double-torus surface helpers

Description

Convenience helpers that build a closed genus-2 surface from a deterministically irregular slice family whose cyclic cross-sections follow a `1 -> 3 -> 1` loop transition between two poles. This gives a non-lattice, point-sampled double-torus graph together with either the canonical 3D embedding or simple bulged, twisted, and wavy deformations of that geometry.

Usage

irregular.double.torus.surface.embedding(
  slices = 11,
  tube_count = 14,
  branch_length = 0.85,
  branch_offset = 0.72,
  tube_radius = 0.28,
  transition_width = 0.42,
  count_irregularity = 0.2,
  axial_irregularity = 0.3,
  phase_twist = 0.35,
  surface = c("standard", "bulged", "twisted", "wavy"),
  amplitude = 0.25,
  freq_x = 2,
  freq_theta = 2,
  twist = 0.6
)
irregular.double.torus.surface.graph(
  slices = 11,
  tube_count = 14,
  branch_length = 0.85,
  branch_offset = 0.72,
  tube_radius = 0.28,
  transition_width = 0.42,
  count_irregularity = 0.2,
  axial_irregularity = 0.3,
  phase_twist = 0.35,
  surface = c("standard", "bulged", "twisted", "wavy"),
  amplitude = 0.25,
  freq_x = 2,
  freq_theta = 2,
  twist = 0.6,
  normalize = c("median", "mean", "none")
)

Value

irregular.double.torus.surface.embedding() returns an n x 3

numeric matrix with columns x, y, and z.

irregular.double.torus.surface.graph() returns a list with components:

edges: the irregular double-torus edges,
n: number of vertices,
edge_weights: induced positive edge lengths,
coords_surface: the 3D embedding,
coords_param: the canonical 3D double-torus coordinates,
weight_scale: the normalization constant applied to the raw edge lengths,
family: always "irregular.double.torus",
surface: the chosen surface name,
slices: number of non-pole slices,
slice_sizes: vertex counts in each slice,
slice_components: number of cyclic components in each slice,
label: a human-readable family label.

Arguments

slices: Number of non-pole slices through the double torus.
tube_count: Approximate number of vertices around each tube-like loop.
branch_length: Half-length of the three-loop central region.
branch_offset: Offset of the outer loop centers from the central loop.
tube_radius: Baseline radius of each tube-like loop.
transition_width: Width of the left and right transition regions between the single-loop and three-loop slices.
count_irregularity: Irregularity level for the per-component sample counts. Must lie in [0, 1).
axial_irregularity: Irregularity level for the slice spacing. Must lie in [0, 1].
phase_twist: Finite phase offset used to desynchronize neighboring cyclic slices.
surface: Double-torus geometry family. One of "standard", "bulged", "twisted", or "wavy".
amplitude: Finite deformation amplitude.
freq_x: Positive modulation frequency along the axial direction. Used only when surface = "wavy".
freq_theta: Positive angular modulation frequency around the local tube direction. Used only when surface = "wavy".
twist: Finite twist strength used only when surface = "twisted".
normalize: Normalization applied to the induced edge lengths. One of "median", "mean", or "none".