mesh_surface_helpers: Weighted mesh surface helpers

Description

Convenience helpers that lift a rectangular mesh parameter grid into \(\mathbb{R}^3\) and use the induced Euclidean edge lengths as positive graph weights. These helpers are intended for benchmark families where the topology is a plain mesh but the intended metric comes from a curved ambient geometry.

Usage

mesh.surface.embedding(
  h,
  w = h,
  surface = c("saddle", "paraboloid", "ripple"),
  amplitude = 0.75,
  freq_u = 1,
  freq_v = 1,
  x_scale = 1,
  y_scale = 1
)
mesh.surface.graph(
  h,
  w = h,
  surface = c("saddle", "paraboloid", "ripple"),
  amplitude = 0.75,
  freq_u = 1,
  freq_v = 1,
  x_scale = 1,
  y_scale = 1,
  connectivity = c("orthogonal", "diagonal"),
  normalize = c("median", "mean", "none")
)

Value

mesh.surface.embedding() returns an n x 3 numeric matrix with columns x, y, and z.

mesh.surface.graph() returns a list with components:

edges: the undirected mesh edges,
n: number of vertices,
edge_weights: induced positive edge lengths,
coords_surface: the 3D surface embedding,
coords_param: the 2D parameter-grid coordinates,
weight_scale: the normalization constant applied to the raw edge lengths,
family: always "mesh",
surface: the chosen surface name,
connectivity: the chosen mesh connectivity rule,
label: a human-readable family label.

Arguments

h: Number of rows.
w: Number of columns. Defaults to h.
surface: Surface family used for the lift. One of "saddle", "paraboloid", or "ripple".
amplitude: Finite numeric amplitude controlling the non-flat displacement.
freq_u: Positive ripple frequency in the horizontal parameter direction. Used only when surface = "ripple".
freq_v: Positive ripple frequency in the vertical parameter direction. Used only when surface = "ripple".
x_scale: Positive horizontal scaling of the parameter domain.
y_scale: Positive vertical scaling of the parameter domain.
connectivity: Occupied-mesh neighborhood rule passed to edges.occupied.mesh().
normalize: Normalization applied to the induced edge lengths. One of "median", "mean", or "none".

Details

`mesh.surface.embedding()` returns the 3D coordinates of the lifted grid. `mesh.surface.graph()` returns a reusable weighted-graph bundle containing the mesh edges, induced edge weights, the 3D surface coordinates, and the 2D parameter coordinates.