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}

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