Arc length of vector-valued functional data

For a vector-valued curve f: [a, b] -> R^d, the arc length is \(\int_a^b \lVert f'(t) \rVert\, dt\) – the length traced out by f in R^d.

Usage

tf_arclength(f, ...)

# Default S3 method
tf_arclength(f, ...)

# S3 method for class 'tf_mv'
tf_arclength(
  f,
  arg = NULL,
  lower = tf_domain(f)[1],
  upper = tf_domain(f)[2],
  definite = TRUE,
  method = c("polyline", "derive"),
  ...
)

Arguments

f

a tf_mv object.

...

forwarded to tf_integrate() when method = "derive".

arg, lower, upper

optional evaluation/integration grid and limits.

definite

TRUE (default) returns a numeric vector of total arc lengths per curve; FALSE returns the cumulative arc length \(s(t) = \int_a^t \lVert f'(u) \rVert\, du\) as a univariate tfd.

method

"polyline" (default) or "derive".

Value

a numeric vector (definite) or a univariate tfd (indefinite).

Details

Two methods are supported:

• "polyline" (default): sum of the Euclidean lengths of the line segments between consecutive sample points (in R^d). Each curve is evaluated on the union of its components' argument grids (or a supplied arg) and the segment-sum is computed in closed form. For raw tfd_mv data this is more accurate than "derive" because it avoids the compounding error of numerical differentiation followed by quadrature.

• "derive": composes the existing verbs – per-component differentiation (tf_derive()), pointwise speed tf_speed(), then tf_integrate(). Best for tfb_mv (analytical derivatives) or when a custom tf_integrate(...) argument is needed.

See also

Other tf_mv-class: plot.tf_mv(), tf_geom, tf_mv_methods, tfb_mfpc(), tfb_mv(), tfd_mv()

Examples

# unit circle parameterised on [0, 1] -- arc length is 2*pi
t <- seq(0, 1, length.out = 401)
circ <- tfd_mv(list(
  x = tfd(matrix(cos(2 * pi * t), nrow = 1), arg = t),
  y = tfd(matrix(sin(2 * pi * t), nrow = 1), arg = t)
))
tf_arclength(circ)
#> [1] 6.283121
tf_arclength(circ, lower = 0, upper = 0.25) # quarter -> pi/2
#> [1] 1.57078
tf_arclength(circ, definite = FALSE)        # cumulative s(t)
#> tfd[1]: [0,1] -> [0,6.283121] based on 401 evaluations each
#> interpolation by tf_approx_linear 
#> [1]: ▁▁▁▂▂▂▂▃▃▃▄▄▄▅▅▅▆▆▆▇▇▇▇███