Parametric Arc Length

Parametric Arc Length

A parametric curve is defined by two functions of a parameter $t$:

$x = x(t), quad y = y(t), quad t in [t_0, t_1]$

The arc length is:

$L = int_{t_0}^{t_1} sqrt{[x'(t)]^2 + [y'(t)]^2} , dt$

Why It Works

At parameter $t$, the curve moves $x'dt$ in $x$ and $y'dt$ in $y$. The Pythagorean theorem gives length $sqrt{x'^2+y'^2},dt$.

Examples

Diagonal line $x=t, y=t$ on $[0,1]$: $x'=y'=1$, so $L = sqrt{2} approx 1.4142$

Line with slope $x=t, y=2t$ on $[0,1]$: $x'=1$, $y'=2$, so $L = sqrt{5} approx 2.2361$

Circle $x=cos(t), y=sin(t)$ on $[0,2pi]$:

$x'=-sin(t), y'=cos(t) implies sqrt{sin^2 t + cos^2 t} = 1$ $L = int_0^{2pi} 1 , dt = 2pi approx 6.2832$

Numerical Approach

for each midpoint t:
    double xp = (x_fn(t+h) - x_fn(t-h)) / (2*h);
    double yp = (y_fn(t+h) - y_fn(t-h)) / (2*h);
    sum += my_sqrt(xp*xp + yp*yp);

Your Task

Implement double param_arc_length(double (*x_fn)(double), double (*y_fn)(double), double t0, double t1, int n, double h).