# arclength: Arc Length of a Curve In pracma: Practical Numerical Math Functions

 arclength 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)

`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)
``````

pracma documentation built on Nov. 10, 2023, 1:14 a.m.