# Rectification

### Related : Curve Tracing

The process of rectification involves computation of lengths of curves. The logic behind this is based on right triangle geometry and calculus.

Consider a curve in $XY$ plane. Let the curve be divided into infinitesimally smaller parts. Let $dS$ be the length of a part. As we take the limit, the arc length approximates to a straight line and we have,

${(ds)^2 = (dx)^2 + (dy)^2}$

On integrating,

${S = \int \limits_{x_1}^{x_2} \sqrt {1 + \Big (\frac {dy}{dx} \Big)^2}dx}$

This gives the length of curve between the lines $x=x_1$ and $x=x_2$.

For parametric curves $x=f(t),y=g(t)$,

${S = \int \limits_{t_1}^{t_2} \sqrt { \Big ( \frac {dx}{dt} \Big )^2 + \Big ( \frac {dy}{dt} \Big )^2}dt}$

For polar curves $r= f(\theta)$,

${S = \int \limits_{\theta_1}^{\theta_2} \sqrt {r^2 + \Big (\frac {dr}{d \theta} \Big )^2}d \theta}$

For polar curves $\theta = f(r)$,

${S = \int \limits_{r_1}^{r_2} \sqrt {1 + r^2 \Big (\frac {d \theta}{dr} \Big )^2}dr}$