vectorialcalculus (version 1.0.5)

osculating_ribbon3d: Osculating ribbon along a 3D parametric curve

Description

Constructs a narrow ribbon that follows a three-dimensional parametric curve. The ribbon is based on the Frenet-Serret frame of the curve, computed numerically along a set of sample points. The ribbon extends a small distance in the normal direction of the curve, producing a thin band that helps visualize how the curve bends and twists in space.

Usage

osculating_ribbon3d(
  X,
  Y,
  Z,
  a,
  b,
  h = 1e-04,
  plot = FALSE,
  n_t = 400,
  n_u = 25,
  u_max = 1,
  colorscale = "Blues",
  opacity = 0.35,
  show_curve = TRUE,
  show_centers = TRUE,
  curve_line = list(color = "black", width = 2),
  centers_line = list(color = "red", width = 2),
  show_surface_grid = TRUE,
  surface_grid_color = "rgba(60,80,200,0.25)",
  surface_grid_width = 1,
  show_axis_grid = FALSE,
  scene = list(aspectmode = "data", xaxis = list(title = "x(t)"), yaxis = list(title =
    "y(t)"), zaxis = list(title = "z(t)")),
  bg = list(paper = "white", plot = "white"),
  lighting = list(ambient = 1, diffuse = 0.15, specular = 0, roughness = 1, fresnel = 0),
  tol = 1e-10
)

Value

A list with two components:

data

A tibble containing the sampled parameter values, the coordinates of the curve, and the corresponding tangent, normal and binormal directions.

plot

A plotly object if plot = TRUE, otherwise NULL.

Arguments

X, Y, Z

Functions returning the coordinate components of the curve as functions of the parameter t.

a, b

Numeric values giving the endpoints of the parameter interval.

h

Step size used in the finite-difference approximations.

plot

Logical; if TRUE, produces a 3D visualization of the ribbon using plotly.

n_t

Number of sample points along the curve.

n_u

Number of subdivisions across the width of the ribbon.

u_max

Half-width of the ribbon, measured in units of the normal direction.

colorscale

Character string giving the plotly colorscale used for the ribbon surface.

opacity

Numeric value between 0 and 1 controlling the transparency of the ribbon.

show_curve

Logical; if TRUE, draws the base curve.

show_centers

Logical; if TRUE, draws the centerline joining the midpoints of the ribbon cross-sections.

curve_line

List with plotly style options for the base curve.

centers_line

List with plotly style options for the centerline.

show_surface_grid

Logical; if TRUE, draws a grid on the surface of the ribbon.

surface_grid_color

Character string giving the color of the grid lines on the ribbon.

surface_grid_width

Numeric value giving the width of the surface grid lines.

show_axis_grid

Logical; if TRUE, displays gridlines on the coordinate axes in the plotly scene.

scene

List with 3D scene settings for the plotly figure.

bg

List defining the background colors of the figure, typically with entries paper and plot.

lighting

List with lighting settings for the surface in plotly.

tol

Numeric tolerance used to detect numerical instabilities when computing the derivative-based frame vectors.

Details

The function samples the curve at n_t points and computes the numerical tangent, normal and binormal directions using finite-difference approximations of the derivatives. At each sampled point, a short segment is taken in the normal direction to define the width of the ribbon. These segments are interpolated across the curve and subdivided according to n_u to produce a mesh that represents the ribbon surface.

Optionally, the function can display the ribbon in an interactive 3D plot using plotly. The base curve, the centerline, and optional grid lines on the ribbon surface can be shown or hidden independently.

Examples

X <- function(t) cos(t)
Y <- function(t) sin(t)
Z <- function(t) 0.2 * t
osculating_ribbon3d(X, Y, Z, a = 0, b = 4*pi, plot = FALSE)