vectorialcalculus (version 1.0.5)

curvature_torsion3d: Curvature and torsion of a 3D parametric curve

Description

Computes numerical curvature and torsion of a three-dimensional parametric curve at a specific value of the parameter. The curve is described by three functions that give the coordinate components. All derivatives are approximated using centered finite differences of first, second and third order.

Usage

curvature_torsion3d(
  X,
  Y,
  Z,
  t0,
  h = 1e-04,
  plot = FALSE,
  window = 1,
  n_samples = 200,
  line = list(color = "red", width = 2, dash = "solid"),
  point = list(color = "black", size = 5, symbol = "circle"),
  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"),
  tol = 1e-10
)

Value

A list with:

• kappa: numerical curvature at the evaluation point.

• tau: numerical torsion at the evaluation point, or NA if the computation is unstable.

• t0: the parameter value where the evaluation was made.

• point: a numeric vector containing the coordinates of the curve at t0.

• r1, r2, r3: numeric approximations to the first, second and third derivative vectors at t0.

Arguments

X, Y, Z

Functions of t returning the three coordinate components of the parametric curve.

t0

Value of the parameter at which curvature and torsion are evaluated.

h

Step size for the finite-difference approximations. Smaller values give more accuracy but may amplify numerical noise.

plot

Logical; if TRUE, displays a 3D plot of a short segment of the curve around the evaluation point.

window

Length of the parameter interval shown when plot = TRUE. The interval is centered at t0.

n_samples

Number of points used to draw the curve segment in the 3D plot.

line

A list defining the visual style of the curve in the 3D plot.

point

A list defining the visual style of the marker placed at the evaluation point.

scene

A list with 3D axis settings for plotly.

bg

Background colors for the plotly figure.

tol

Numeric tolerance used to detect degenerate situations in which curvature or torsion cannot be reliably computed.

Details

The curvature at the evaluation point measures how sharply the curve bends at that location. It is computed from the first and second derivative vectors. The torsion measures how the curve deviates from being planar and is computed from the first, second and third derivative vectors. If the first derivative vector is nearly zero, or if the first and second derivative vectors are nearly parallel, the torsion becomes undefined; in such cases the function returns NA and provides a diagnostic message.

Optionally, the function can display a small segment of the curve around the evaluation point using plotly. The point where the curvature and torsion are computed is highlighted in the 3D plot.

Examples

# Example curve
X <- function(t) t^2 * cos(t)
Y <- function(t) t^3 * sin(3 * t)
Z <- function(t) t
res <- curvature_torsion3d(X, Y, Z, t0 = pi)
res$kappa
res$tau

# \donttest{ if (requireNamespace("plotly", quietly = TRUE)) {
#   curvature_torsion3d(
#     X, Y, Z, t0 = pi, plot = TRUE,
#     window = 1.0, n_samples = 200,
#     line  = list(color = "red",  width = 2),
#     point = list(color = "black", size = 5, symbol = "circle")
#   )
# } }