R: Arc Length of a Curve

arclength {pracma} | R Documentation

## Arc Length of a Curve

### Description

Calculates the arc length of a parametrized curve.

### Usage

    arclength(f, a, b, nmax = 20, tol = 1e-05, ...)

### Arguments

`f` | parametrization of a curve in n-dim. space.
`a,b` | begin and end of the parameter interval.
`nmax` | maximal number of iterations.
`tol` | relative tolerance requested.
`...` | additional arguments to be passed to the function.

### Details

Calculates the arc length of a parametrized curve in `R^n`. It applies Richardson's extrapolation by refining polygon approximations to the curve.

The parametrization of the curve must be vectorized: if `t-->F(t)` is the parametrization, `F(c(t1,t1,...))` must return `c(F(t1),F(t2),...)`.

Can be directly applied to determine the arc length of a one-dimensional function `f:R-->R` by defining `F` (if `f` is vectorized) as `F:t-->c(t,f(t))`.

### Value

Returns a list with components `length` the calculated arc length, `niter` the number of iterations, and `rel.err` the relative error generated from the extrapolation.

### Note

If by chance certain equidistant points of the curve lie on a straight line, the result may be wrong, then use `polylength` below.

### Author(s)

HwB <hwborchers@googlemail.com>

### See Also

`poly_length`

### Examples

    ##  Example: parametrized 3D-curve with t in 0..3*pi
    f <- function(t) c(sin(2*t), cos(t), t)
    arclength(f, 0, 3*pi)
    # $length:  17.22203            # true length 17.222032...

    ##  Example: length of the sine curve
    f <- function(t) c(t, sin(t))
    arclength(f, 0, pi)             # true length  3.82019...

    ## Example: Length of an ellipse with axes a = 1 and b = 0.5
    # parametrization x = a*cos(t), y = b*sin(t)
    a <- 1.0; b <- 0.5
    f <- function(t) c(a*cos(t), b*sin(t))
    L <- arclength(f, 0, 2*pi, tol = 1e-10)     #=> 4.84422411027
    # compare with elliptic integral of the second kind
    e <- sqrt(1 - b^2/a^2)                      # ellipticity
    L <- 4 * a * ellipke(e^2)$e                 #=> 4.84422411027

    ## Not run:
    ##  Example: oscillating 1-dimensional function (from 0 to 5)
    f <- function(x) x * cos(0.1*exp(x)) * sin(0.1*pi*exp(x))
    F <- function(t) c(t, f(t))
    L <- arclength(F, 0, 5, tol = 1e-12, nmax = 25)
    print(L$length, digits = 16)
    # [1] 82.81020372882217         # true length 82.810203728822172...

    # Split this computation in 10 steps (run time drops from 2 to 0.2 secs)
    L <- 0
    for (i in 1:10)
    	L <- L + arclength(F, (i-1)*0.5, i*0.5, tol = 1e-10)$length
    print(L, digits = 16)
    # [1] 82.81020372882216

    # Alternative calculation of arc length
    f1 <- function(x) sqrt(1 + complexstep(f, x)^2)
    L1 <- quadgk(f1, 0, 5, tol = 1e-14)
    print(L1, digits = 16)
    # [1] 82.81020372882216

    ## End(Not run)

    ## Not run:
    #-- --------------------------------------------------------------------
    #   Arc-length parametrization of Fermat's spiral
    #-- --------------------------------------------------------------------
    # Fermat's spiral: r = a * sqrt(t)
    f <- function(t) 0.25 * sqrt(t) * c(cos(t), sin(t))

    t1 <- 0; t2 <- 6*pi
    a  <- 0; b  <- arclength(f, t1, t2)$length
    fParam <- function(w) {
        fct <- function(u) arclength(f, a, u)$length - w
        urt <- uniroot(fct, c(a, 6*pi))
        urt$root
    }

    ts <- linspace(0, 6*pi, 250)
    plot(matrix(f(ts), ncol=2), type='l', col="blue",
         asp=1, xlab="", ylab = "",
         main = "Fermat's Spiral", sub="20 subparts of equal length")

    for (i in seq(0.05, 0.95, by=0.05)) {
        v <- fParam(i*b); fv <- f(v)
        points(fv[1], f(v)[2], col="darkred", pch=20)
    }
    ## End(Not run)

[Package pracma version 2.0.4 Index]