Arc Length of a Curve

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Description

Calculates the arc length of a parametrized curve.

Usage

1
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

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##  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)

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